A Macroeconomic Model with Financial Panics

K.7 A Macroeconomic Model with Financial Panics Gertler, Mark, Nobuhiro Kiyotaki, and Andrea Prestipino Please cite paper as: Gertler, Mark, Nobuhior Kiyotaki, and Andrea Prestipino (2017). A Macroeconomic Model with Financial Panics. International Finance Discussion Papers 1219. https://doi.org/10.17016/IFDP.2017.1219 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1219 December 2017

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1219 December 2017 A Macroeconomic Model with Financial Panics Mark Gertler, Nobuhiro Kiyotaki and Andrea Prestipino NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

A Macroeconomic Model with Financial Panics (cid:3) Mark Gertler, Nobuhiro Kiyotaki and Andrea Prestipino NYU, Princeton and Federal Reserve Board December, 2017 Abstract This paper incorporates banks and banking panics within a conventional macroeconomic framework to analyze the dynamics of a financial crisis of the kind recently experienced. We are particularly interested in characterizing the sudden and discrete nature of the banking panics as well as the circumstances that makes an economy vulnerable to such panics in some instances but not in others. Having a conventional macroeconomic model allows us to study the channels by which the crisis a⁄ects real activity and the e⁄ects of policies in containing crises. Keywords: Bank runs, financial crisis, New Keynesian DSGE JEL Classification: E23, E32, E44, G01, G21, G33. Theviewsexpressedinthispaperaresolelythoseoftheauthorsanddonotnecessarily (cid:3) re(cid:135)ect those of the Board of Governors of the Federal Reserve or the Federal Reserve System. We thank for their helpful comments Frederic Boissay, Pat Kehoe, John Moore as well as participants in various seminars and conferences. Financial support from the National Science Foundation and the Macro Financial Modeling group at the University of Chicago is gratefully acknowledged. 1

1 Introduction As both Bernanke (2010) and Gorton (2010) argue, at the heart of the recent (cid:133)nancial crisis was a series of bank runs that culminated in the precipitous demise of a number of major (cid:133)nancial institutions. During the period where thepanicsweremostintenseinOctober2008, allthemajorinvestmentbanks e⁄ectively failed, the commercial paper market froze, and the Reserve Primary Fund (a major money market fund) experienced a run. The distress quickly spilled over to the real sector. Credit spreads rose to Great Depression era levels. There was an immediate sharp contraction in economic activity: From 2008:Q4 through 2009:Q1 real output dropped at an eight percent annual rate, driven mainly by a nearly forty percent drop in investment expenditure. Also relevant is that this sudden discrete contraction in (cid:133)nancial and real economic activity occurred in the absence of any apparent large exogenous disturbance to the economy. In this paper we incorporate banks and banking panics within a conventional macroeconomic framework - a New Keynesian model with capital accumulation. Our goal is to develop a model where it is possible to analyze both qualitatively and quantitatively the dynamics of a (cid:133)nancial crisis of the kind recently experienced. We are particularly interested in characterizing thesuddenanddiscretenatureofbankingpanicsaswellasthecircumstances that make the economy vulnerable to such panics in some instances but not in others. Having a conventional macroeconomic model allows us to study the channels by which the crisis a⁄ects aggregate economic activity and the e⁄ects of various policies in containing crises. Our paper (cid:133)ts into a lengthy literature aimed at adapting core macroeconomic models to account for (cid:133)nancial crises1. Much of this literature emphasizes the role of balance sheets inconstraining borrowers fromspending when (cid:133)nancialmarketsareimperfect. Becausebalancesheetstendtostrengthenin booms and weaken in recessions, (cid:133)nancial conditions work to amplify (cid:135)uctuations in real activity. Many authors have stressed that this kind of balance sheet mechanism played a central role in the crisis, particularly for banks and households, but at the height of the crisis also for non-(cid:133)nancial (cid:133)rms. Nonetheless, as Mendoza (2010), He and Krishnamurthy (2017) and Brunnermeier and Sannikov (2014) have emphasized, these models do not capture the highly nonlinear aspect of the crisis. Although the (cid:133)nancial mechanisms 1See Gertler and Kiyotaki (2011) and Brunnermeier et. al (2013) for recent surveys. 2

in these papers tend to amplify the e⁄ects of disturbances, they do not easily capture sudden discrete collapses. Nor do they tend to capture the run-like behavior associated with (cid:133)nancial panics. Conversely, beginning with Diamond and Dybvig (1983), there is a large literature on banking panics. An important common theme of this literature is how liquidity mismatch, i.e. partially illiquid long-term assets funded by short-term debt, opens up the possibility of runs. Most of the models in this literature, though, are partial equilibrium and highly stylized (e.g. three periods). They are thus limited for analyzing the interaction between (cid:133)nancial and real sectors. Our paper builds on our earlier work - Gertler and Kiyotaki (GK, 2015) and Gertler, Kiyotaki and Prestipino (GKP, 2016) - which analyzed bank runs in an in(cid:133)nite horizon endowment economy. These papers characterize runs as self-ful(cid:133)lling rollover crises, following the Calvo (1988) and Cole and Kehoe (2001) models of sovereign debt crises. Both GK and GKP emphasize the complementary nature of balance sheet conditions and bank runs. Balance sheet conditions a⁄ect not only borrower access to credit but also whether the banking system is vulnerable to a run. In this way the model is able to capture the highly nonlinear nature of a collapse: When bank balance sheets are strong, negative shocks do not push the (cid:133)nancial system to the verge of collapse. When they are weak, the same size shock leads the economy into a crisis zone in which a bank run equilibrium exists.2 While our earlier work restricted attention to a simple endowment economy, here we extend the analysis to a conventional macroeconomic model. By doing so, we can explicitly capture both qualitatively and quantitatively the e⁄ect of the (cid:133)nancial collapse on investment, output and employment. In particular, we proceed to show that a calibrated version of our model is capable of capturing the dynamics of key (cid:133)nancial and real variables over the course of the recent crisis. Also related is important recent work on occasionally binding borrowing constraints as a source of nonlinearity in (cid:133)nancial crises such as Mendoza (2010) and He and Krishnamurthy (2017). There, in good times the bor- 2Somerecentexampleswhereself-ful(cid:133)lling(cid:133)nancialcrisescanemergedependingonthe state of the economy include Benhabib and Wang (2013), Bocola and Lorenzoni (2017), Farhi and Maggiori (2017) and Perri and Quadrini (forthcoming). For further attempts to incorporate bank runs in macro models, see Angeloni and Faia (2013), Cooper and Ross(1998),Martin,SkeieandVonThadden(2014),Robatto(2014)andUhlig(2010)for example. 3

rowing constraint is not binding and the economy behaves much the way it does with frictionless (cid:133)nancial markets. However, a negative disturbance can move the economy into a region where the constraint is binding, amplifying the e⁄ect of the shock on the downturn. In a similar spirit, Brunnermeier and Sannikov (2014) generate nonlinear dynamics based on the precautionary saving behavior by intermediaries worried about survival in the face of a sequence of negative aggregate shocks. Our approach also allows for occasionally binding (cid:133)nancial constraints and precautionary saving. However, in quantitative terms, bank runs provide the major source of nonlinearity. Section 2 presents the behavior of bankers and workers, the sectors where the novel features of the model are introduced. Section 3 describes the features that are standard in the New Keynesian model: the behavior of (cid:133)rms, price setting, investment and monetary policy. Section 4 describes the calibration and presents a variety of numerical exercises designed to illustrate the main features of the model, including how the model can capture the dynamics of some of the main features of the recent (cid:133)nancial crisis. 2 Model: outline, households, and bankers The baseline framework is a standard New Keynesian model with capital accumulation. In contrast to the conventional model, each household consists of bankers and workers. Bankers specialize in making loans and thus intermediate funds between households and productive capital. Households may also make these loans directly, but they are less e¢ cient in doing so than bankers.3 On the other hand, bankers may be constrained in their ability to raise external funds and also may be subject to runs. The net e⁄ect is that the cost of capital will depend on the endogenously determined (cid:135)ow of funds between intermediated and direct (cid:133)nance. We distinguish between capital at the beginning of period t, K , and t capital at the end of the period, S : Capital at the beginning of the period is t used in conjunction with labor to produce output at t. Capital at the end of period is the sum of newly produced capital and the amount of capital left after production: 3Assection2.2. makesclear,technicallyitistheworkerswithinthehouseholdthatare lefttomanageanydirect(cid:133)nance. Butsincetheseworkerscollectivelydecideconsumption, labor and portfolio choice on of behalf the household, we simply refer to them as the (cid:145)household(cid:146)going forward. 4

I t S = (cid:0) K +(1 (cid:14))K ; (1) t t t K (cid:0) (cid:18) t(cid:19) where (cid:14) is the rate of depreciation. The quantity of newly produced capital, (cid:0)(I =K )K , depends upon investment I and the capital stock. We suppose t t t t that (cid:0)( ) is an increasing and concave function of I =K to capture convex t t (cid:1) adjustment costs. A (cid:133)rm wishing to (cid:133)nance new investment as well as old capital issues a state-contingent claim on the earnings generated by the capital. Let S t be the total number of claims (e⁄ectively equity) outstanding at the end of period t (one claim per unit of capital), Sb be the quantity intermediated by t bankers and Sh be the quantity directly held by households. Then we have: t Sb +Sh = S : (2) t t t Both the total capital stock and the composition of (cid:133)nancing are determined in equilibrium. The capital stock entering the next period K di⁄ers from S due to t+1 t a multiplicative "capital quality" shock, (cid:24) ; that randomly transforms the t+1 units of capital available at t+1: K = (cid:24) S : (3) t+1 t+1 t The shock (cid:24) provides an exogenous source of variation in the return to t+1 capital. To capture that households are less e¢ cient than bankers in handling investments, we assume that they su⁄er a management cost that depends on the share of capital they hold, Sh=S . The management cost re(cid:135)ects their t t disadvantage relative to bankers in evaluating and monitoring investment projects. The cost is in utility terms and takes the following piece-wise form: 2 (cid:31) S t h (cid:13) S ; if S t h > (cid:13) > 0 &(S t h;S t ) = 2 St (cid:0) t St (4) ( (cid:16) 0;(cid:17)otherwise with (cid:31) > 0.4 4For a deeper model of the costs that non-experts face in (cid:133)nancial markets see Kurlat (2016). Our assumption that households intermediation costs are non pecuniary is made for simplicity only. All of our results go through if we assume that households(cid:146)intermediated capital is less productive, as in e.g. Brunnermeier and Sannikov (2014), as long as productivity losses increase with the quantity of capital intermediated by households. 5

For Sh=S (cid:13) there is no e¢ ciency cost: Households are able to manage t t (cid:20) a limited fraction of capital as well as bankers. As the share of direct (cid:133)nance exceeds (cid:13), the e¢ ciency cost &( ) is increasing and convex in Sh=S : In (cid:1) t t this region, constraints on the household(cid:146)s ability to manage capital become relevant. The convex form implies that the marginal e¢ ciency losses rise with the size of the household(cid:146)s direct capital holdings, capturing limits on its capacity to handle investments. Weassumethatthee¢ ciencycostishomogenousinSh andS tosimplify t t the computation. As the marginal e¢ ciency cost is linear in the share Sh=S , t t it reduces the nonlinearity in the model. An informal motivation is that, as the capital stock S increases, the household has more options from which to t select investments that it is better able to manage, which works to dampen the marginal e¢ ciency cost. Given the e¢ ciency costs of direct household (cid:133)nance, absent (cid:133)nancial frictions banks will intermediate at least the fraction 1 (cid:13) of the capital (cid:0) stock. However,whenbanksareconstrainedintheirabilitytoobtainexternal funds,householdswilldirectlyholdmorethantheshare(cid:13) ofthecapitalstock. As the constraints tighten in a recession, as will happen in our model, the share of capital held by households will expand. The reallocation of capital holdings from banks to less e¢ cient households raises the cost of capital, reducing investment and output in equilibrium. In the extreme event of a systemic bank run, banks liquidate all their holdings, and the (cid:133)resale of assetsfrombankstohouseholdswillleadtoasharpriseinthecostofcapital, leading to a deep contraction in investment and output. In the rest of this section we characterize the behavior of households and bankers which are the non-standard parts of the model. 2.1 Households We formulate this sector in a way that allows for (cid:133)nancial intermediation yet preservesthetractabilityoftherepresentativehouseholdsetup. Inparticular, each household (family) consists of a continuum of members with measure unity. Within the household there are 1 f workers and f bankers. Workers (cid:0) supply labor and earn wages for the household. Each banker manages a bank and transfers non-negative dividend back to the household. Within the family there is perfect consumption sharing. In order to preclude a banker from retaining su¢ cient earnings to permanently relax any (cid:133)nancial constraint, we assume the following: In each 6

period, with i.i.d. probability 1 (cid:27), a banker exits. Upon exit it then gives (cid:0) all its accumulated earnings to the household. This stochastic exit in conjunction with the payment to the household upon exit is in e⁄ect a simple way to model dividend payouts.5 After exiting, a banker returns to being a worker. To keep the population of each occupation constant, each period, (1 (cid:27))f workers become bankers. (cid:0) At this time the household provides each new banker with an exogenously given initial equity stake in the form of a wealth transfer, e . The banker t receivesnofurthertransfersfromthehouseholdandinsteadoperatesatarms length. Households save in the form of deposits at banks and direct claims on capital. Bank deposits at t are one period bonds that promise to pay a noncontingent gross real rate of return R in the absence of default. In the t+1 event of default at t+1, depositors receive the fraction x of the promised t+1 return, where the recovery rate x [0;1) is the value of bank assets per t+1 2 unit of promised deposit obligations. There are two reasons the bank may default: First, a su¢ ciently negative return on its portfolio may make it insolvent. Second, even if the bank is solvent at normal market prices, the bank(cid:146)s creditors may "run" forcing the bank to liquidate assets at (cid:133)resale prices. We describe each of these possibilities in detail in the next section. Let p be the probability that the t bank defaults in period t+1. Given p and x ; we can express the gross rate t t of return on the deposit contract R as t+1 R with probability 1 p R = t+1 (cid:0) t : (5) t+1 x R with probability p t+1 t+1 t (cid:26) Similar to the Cole and Kehoe (2001) model of sovereign default, a run in our model will correspond to a panic failure of households to roll over deposits. Thiscontrastswiththe"earlywithdrawal"mechanismintheclassic Diamond and Dybvig (1983) model. For this reason we do not need to impose a "sequential service constraint" which is necessary to generate runs in Diamond and Dybvig. Instead we make the weaker assumption that all households receive the same pro rata share of output in the event of default, whether it be due to insolvency or a run. 5Assection2.2makesclear,becauseofthe(cid:133)nancialconstraint,itwillalwaysbeoptimal for a bank to retain earnings until exit. 7

Let C be consumption, L labor supply, and (cid:12) (0;1) the household(cid:146)s t t 2 subjective discount factor. As mentioned before, &(Sh;S ) is the household t t utility cost of direct capital holding Sh, where the household takes the agt gregate quantity of claims S as given. Then household utility U is given t t by U = E 1 (cid:12)(cid:28) t (C (cid:28) )1 (cid:0) (cid:13) h (L (cid:28) )1+’ &(Sh;S ) ; t t (cid:0) 1 (cid:13) (cid:0) 1+’ (cid:0) (cid:28) (cid:28) ( (cid:28)=t " (cid:0) h #) X LetQ betherelativepriceofcapital,Z therentalrateoncapital,w thereal t t t wage rate, T lump sum taxes, and (cid:5) dividend distributions net transfers to t t new bankers, all of which the household takes as given. Then the household chooses C ;L Sh and deposits D to maximize expected utility subject to t t t t the budget constraint C +D +Q Sh = w L T +(cid:5) +R D +(cid:24) [Z +(1 (cid:14))Q ]Sh : (6) t t t t t t (cid:0) t t t t (cid:0) 1 t t (cid:0) t t (cid:0) 1 The (cid:133)rst order condition for labor supply is given by: w (cid:21) = (L )’; (7) t t t where (cid:21) t (C t ) (cid:0) (cid:13) h denotes the marginal utility of consumption. (cid:17) The (cid:133)rst order condition for bank deposits takes into account the possibility of default and is given by 1 = [(1 p )E ((cid:3) no def)+p E ((cid:3) x def)] R (8) t t t+1 t t t+1 t+1 t+1 (cid:0) j j (cid:1) where E ( no def) (and E ( def)) are expected value of conditional on t t (cid:1) j (cid:1) j (cid:1) no default (and default) at date t+1. The stochastic discount factor (cid:3) t+1 satis(cid:133)es (cid:21) t+1 (cid:3) = (cid:12) : (9) t+1 (cid:21) t Observe that the promised deposit rate R that satis(cid:133)es equation (8) det+1 pends on the default probability p as well as the recovery rate x :6 t t+1 Finally, the (cid:133)rst order condition for capital holdings is given by 6Notice that we are already using the fact that in equilibrium all banks will choose the same leverage so that all deposits have the same probability of default. 8

Z +(1 (cid:14))Q t+1 t+1 E (cid:3) (cid:24) (cid:0) = 1; (10) t 2 t+1 t+1 Q + @&(S t h;St)=(cid:21) 3 t @Sh t t 4 5 where @&(Sh;S ) Sh t t =(cid:21) = Max (cid:31) t (cid:13) =(cid:21) ; 0 (11) @Sh t S (cid:0) t t (cid:20) (cid:18) t (cid:19) (cid:21) is the household(cid:146)s marginal cost of direct capital holding. The (cid:133)rst order condition given by (10) will be key in determining the market price of capital. Observe that the market price of capital will tend to be decreasing in the share of capital held by households above the threshold (cid:13) since the e¢ ciency cost &(Sh;S ) is increasing and convex. As will become t t clear, in a panic run banks will sell all their securities to households, leading to a sharp contraction in asset prices. The severity of the drop will depend on the curvature of the e¢ ciency cost function given by (4), which controls asset market liquidity in the model. 2.2 Bankers The banking sector we characterize corresponds best to the shadow banking systemwhichwasattheepicenterofthe(cid:133)nancialinstabilityduringtheGreat Recession. In particular, banks in the model are completely unregulated, hold long-term securities, issue short-term debt, and as a consequence are potentially subject to runs. 2.2.1 Bankers optimization problem Eachbankermanagesa(cid:133)nancialintermediarywiththeobjectiveofmaximizing the expected utility of the household. Bankers fund capital investments by issuing short term deposits d to households as well as by using their own t equity, or net worth, n . Due to (cid:133)nancial market frictions, described later, t bankers may be constrained in their ability to obtain deposits. So long as there is a positive probability that the banker may be (cid:133)nanciallyconstrainedatsomepointinthefuture,itwillbeoptimalforthebanker to delay dividend payments until exit (as we will verify later). At this point the dividend payout will simply be the accumulated net worth. Accordingly, we can take the banker(cid:146)s objective as to maximize the discounted expected value of net worth upon exit. Given that (cid:27) is the survival probability and 9

given that the banker uses the household(cid:146)s intertemporal marginal rate of substitution (cid:3) = (cid:12)(cid:28) t(cid:21) =(cid:21) to discount future payouts, we can express the t;(cid:28) (cid:0) (cid:28) t objective of a continuing banker at the end of period t as e 1 V = E (cid:3) (1 (cid:27))(cid:27)(cid:28) t 1n t t t;(cid:28) (cid:0) (cid:0) (cid:28) (cid:0) " # (cid:28)=t+1 X = E (cid:3) [(e1 (cid:27))n +(cid:27)V ] ; (12) t t+1 t+1 t+1 f (cid:0) g where (1 (cid:27))(cid:27)(cid:28) t 1 is probability of exiting at date (cid:28); and n is terminal (cid:0) (cid:0) (cid:28) (cid:0) net worth if the banker exits at (cid:28): Duringeachperiodt;acontinuingbank(eitherneworsurviving)(cid:133)nances asset holdings Q sb with newly issued deposits and net worth: t t Q sb = d +n : (13) t t t t We assume that banks can only accumulate net worth by retained earnings and do not issue new equity. While this assumption is a reasonable approximation of reality, we do not explicitly model the agency frictions that underpin it. The net worth of surviving bankers, accordingly, is the gross return on assets net the cost of deposits, as follows: n = RbQ sb R d ; (14) t t t (cid:0) 1 t (cid:0) 1 (cid:0) t t (cid:0) 1 where Rb is the gross rate of return on capital intermediated by banks, given t by: Z +(1 (cid:14))Q Rb = (cid:24) t (cid:0) t : (15) t t Q t 1 (cid:0) So long as n is strictly positive the bank does not default. In this instance it t pays its creditors the promised rate R : If the value of assets, RbQ sb ; is t t t 1 t 1 below the promised repayments to depositors R (cid:22) d (due to either (cid:0) a ru(cid:0)n or t t 1 (cid:0) simply a bad realization of returns), n goes to zero and the bank defaults. It t then pays creditors the product of recovery rate x and R ; where x is given t t t by: RbQ sb x t = t t (cid:0) 1 t (cid:0) 1 < 1: (16) R d t t 1 (cid:0) For each new banker at t, net worth simply equals the start-up equity e t it receives from the household: n = e : (17) t t 10

To motivate a limit on a bank(cid:146)s ability to issue deposits, we introduce the following moral hazard problem: After accepting deposits and buying assets at the beginning of t, but still during the period, the banker decides whethertooperate"honestly"ortodivertassetsforpersonaluse. Tooperate honestly means holding assets until the payo⁄s are realized in period t + 1 and then meeting deposit obligations. To divert means selling a fraction (cid:18) of assets secretly on a secondary market in order to obtain funds for personal use. We assume that the process of diverting assets takes time: The banker cannot quickly liquidate a large amount of assets without the transaction being noticed. Accordingly, the banker must decide whether to divert at t; prior to the realization of uncertainty at t+1: Further, to remain undetected, he can only sell up to a fraction (cid:18) of the assets. The cost to the banker of the diversion is that the depositors force the intermediary into bankruptcy at the beginning of the next period.7 The banker(cid:146)s decision on whether or not to divert funds at t boils down to comparing the franchise value of the bank V ; which measures the present t discounted value of future payouts from operating honestly, with the gain from diverting funds, (cid:18)Q sb. In this regard, rational depositors will not lend t t to the banker if he has an incentive to cheat. Accordingly, any (cid:133)nancial arrangement between the bank and its depositors must satisfy the incentive constraint: (cid:18)Q sb V : (18) t t (cid:20) t To characterize the banker(cid:146)s optimization problem it is useful to let (cid:30) t denote the bank(cid:146)s ratio of assets to net worth, Q sb=n , which we will call t t t the "leverage multiple." Then, combining the balance sheet constraint (13) and the (cid:135)ow of funds constraint (14) yields the expression for the evolution of net worth for a surviving bank that does not default as: n = [(Rb R )(cid:30) +R ]n : (19) t+1 t+1 (cid:0) t+1 t t+1 t (cid:22) where we used (5) to substitute the promised rate R for the deposit rate t+1 R in case of no default. t+1 7We assume households deposit funds in banks other than the ones they own. Hence, diverting involves stealing funds from families other than the one to which the banker belongs. 11

Usingtheevolutionofnetworthequation(19);wecanwritethefranchise value of the bank (12) as V = ((cid:22) (cid:30) +(cid:23) )n ; (20) t t t t t where (cid:22) = (1 p )E (cid:10) (Rb R ) no def (21) t (cid:0) t t f t+1 t+1 (cid:0) t+1 j g (cid:23) = (1 p )E (cid:10) R no def (22) t t t t+1 t+1 (cid:0) f j g with (cid:10) = (cid:3) (1 (cid:27) +(cid:27) ); and t+1 t+1 t+1 (cid:0) V t+1 : t+1 (cid:17) n t+1 The variable (cid:22) is the expected discounted excess return on banks assets t relative to deposits and (cid:23) is the expected discounted cost of a unit of det posits. Intuitively, (cid:22) (cid:30) is the excess return the bank receives from having on t t additional unity of net worth (taking into account the ability to use leverage), while (cid:23) is the cost saving from substituting equity (cid:133)nance for deposit t (cid:133)nance. Notice that the bank uses the stochastic discount factor (cid:10) to value t+1 returns in t + 1. (cid:10) is the banker(cid:146)s discounted shadow value of a unit of t+1 net worth at t+1; averaged across the likelihood of exit and the likelihood of survival. We can think of in the expression for (cid:10) as the bank(cid:146)s t+1 t+1 "Tobin(cid:146)s Q ratio", i.e., the ratio of the franchise value to the replacement cost of the bank balance sheet. With probability 1 (cid:27) the banker exits, (cid:0) implying the discounted shadow value of a unit of net worth simply equals the household discount factor (cid:3) . With probability (cid:27) the banker survives t+1 implying the discounted marginal value of n equals the discounted value t+1 of the bank(cid:146)s Tobin(cid:146)s Q ratio, (cid:3) . As will become clear, to the extent t+1 t+1 that an additional unit of net worth relaxes the (cid:133)nancial market friction, in general will exceed unity provided that the bank does not default. t+1 Thebanker(cid:146)soptimizationproblemisthentochoosetheleveragemultiple (cid:30) to solve t = max ((cid:22) (cid:30) +(cid:23) ); (23) t t t t (cid:30) t subject to the incentive constraint (obtained from equations (18) and (20)): (cid:18)(cid:30) (cid:22) (cid:30) +(cid:23) ; (24) t t t t (cid:20) 12

and the deposit rate constraint (obtained from equations (8) and (16)): R = [(1 p )E ((cid:3) no def)+p E ((cid:3) x def)] 1; (25) t+1 t t t+1 t t t+1 t+1 (cid:0) (cid:0) j j where x is the following function of (cid:30) : t+1 t (cid:30) Rb x = t t+1: t+1 (cid:30) 1R t t+1 (cid:0) and (cid:22) and (cid:23) are given by (21) and (22): t t Notice that since individual bank net worth does not appear in the bank optimization problem, the optimal choice of (cid:30) is independent of n : This imt t (cid:22) pliesthatthedefaultprobability, p ; thepromisedrateondeposits, R ; and t t+1 the bank(cid:146)s Tobin(cid:146)s Q are all independent from bank(cid:146)s speci(cid:133)c characteristics. Since the franchise value of the bank V is proportionate to n by a factor t t that only depends on the aggregate state of the economy, a bank cannot operate with zero net worth. In this instance V falls to zero, implying that t the incentive constraint (18) would always be violated if the bank tried to issue deposits. That banks require positive equity to operate is vital to the possibility of bank runs. In fact, as we show below, a necessary condition for a bank run equilibrium to exist is that banks cannot operate with zero net worth. 2.2.2 Banker(cid:146)s decision rules We derive the optimal portfolio choice of banks by restricting attention to a symmetric equilibrium in which all banks choose the same leverage.8 Let(cid:22)r betheexpecteddiscountedmarginal return to increasingthelevert 8Inthissectionwedescribetheleveragechoiceofbanksasdeterminedbythe(cid:133)rstorder conditions of the banks(cid:146)optimization problem. The Appendix discusses the assumptions underwhich(cid:133)rstorderconditionsactuallyselectaglobaloptimumforthebank(cid:146)sproblem, ensuring that a symmetric strategy equilibrium exists. 13

age multiple9 d (cid:23) dR ((cid:30) ) (cid:22)r = t = (cid:22) ((cid:30) 1) t t+1 t < (cid:22) : (26) t d(cid:30) t (cid:0) t (cid:0) R d(cid:30) t t t+1 t Thesecondtermontherightofequation(26)re(cid:135)ectsthee⁄ectoftheincrease in R that arises as the bank increases (cid:30) . An increase in (cid:30) reduces the t+1 t t recovery rate, forcing R up to compensate depositors, as equation (25) t+1 suggests. The term ((cid:30) 1)(cid:23) =R then re(cid:135)ects the reduction in the bank t t t+1 (cid:0) franchise value that results from a unit increase in R : Due to the e⁄ect on t+1 R from expanding (cid:30) ; the marginal return (cid:22)r is below the average excess t+1 t t return (cid:22) . t The solution for (cid:30) depends on whether or not the marginal return of t increasing leverage multiple (cid:22)r is positive. If it is positive, the incentive t constraint (24) binds and limits the bank from increasing leverage to acquire more assets. Then from (24) with equality, we get the following solution for (cid:30) : t (cid:23) (cid:30) = t ; if (cid:22)r > 0: (27) t (cid:18) (cid:22) t t (cid:0) Theconstraint(27)limitstheleveragemultipletothepointwherethebank(cid:146)s gainfromdivertingfunds perunit of net worth(cid:18)(cid:30) is exactlybalancedbythe t cost per unit of net worth of losing the franchise value, which is measured by = (cid:22) (cid:30) +(cid:23) : Note that (cid:22) tends to move countercyclically since the excess t t t t t return on bank capital E Rb R widens as the borrowing constraint t t+1 (cid:0) t+1 tightens in recessions. As a result, (cid:30) tends to move countercyclically. As we t show later, the countercyclical movement in (cid:30) contributes to making bank t runs more likely in bad economic times.10 9Note that, although (cid:30) a⁄ects the default probability p , the indirect e⁄ect of (cid:30) on t t t (cid:133)rmvalueV throughthechangeofp iszero. Thisisbecauseattheborderlineofdefault, t t n = 0 and thus V = 0. Thus a small shift in the probability mass from the not+1 t+1 default to the default state has no impact on V . Similarly, the indirect e⁄ect of (cid:30) on t t the promised deposit rate R through p is zero, since the recovery rate x is unity at the t t t borderlineofdefault. SeeAppendixfordetails. Importanttotheargumentistheabsence of deadweight loss associated with default. 10In the data, net worth of our model corresponds to the mark-to-market di⁄erence between assets and liabilities of the bank balance sheet. It is di⁄erent from the book value often used in the o¢ cial report, which is slow in reacting to market conditions. Also bank assets here are securities and loans to the non-(cid:133)nancial sector, which exclude those to other (cid:133)nancial intermediaries. In the data, the net mark-to-market leverage 14

Conversely, if the marginal return to increasing the leverage multiple becomeszerobeforetheincentiveconstraintbecomesbinding, thebankchooses leverage as, (cid:23) (cid:22)r = 0; if (cid:30) < t : (28) t t (cid:18) (cid:22) t (cid:0) When the constraint does not bind, even if discounted excess returns are strictly postive, E (cid:3) Rb R > 0, the bank still chooses to limit the t t+1 t+1 (cid:0) t+1 leverage multiple so long as there is a possibility that the incentive constraint (cid:0) (cid:1) could bind in the future. In this instance, as in Brunnermeier and Sannikov (2014) and He and Krishnamurthy (2015), banks have a precautionary motive for scaling back their respective leverage multiples.11 The precautionary motive is re(cid:135)ected by the presence of the discount factor (cid:10) in the measure t+1 of the discounted excess return (cid:22) . The discount factor (cid:10) , which re(cid:135)ects t t+1 the shadow value of net worth, tends to vary countercyclically given that borrowing constraints tighten in downturns. By reducing their leverage multiples, banks reduce the risk of taking losses when the shadow value of net worth is high. 2.2.3 Aggregation of the (cid:133)nancial sector absent default We now characterize the aggregate (cid:133)nancial sector during periods where banks do not default. Given that the optimal leverage multiple (cid:30) is int dependent of bank-speci(cid:133)c factors, individual bank portfolio decisions, sb t and d ; are homogenous in net worth. Accordingly, we can sum across banks t to obtain the following relation between aggregate bank asset holdings Q Sb t t and the aggregate quantity of net worth N in the banking sector: t Q Sb = (cid:30) N : (29) t t t t multiple of the (cid:133)nancial intermediation sector - the ratio of securities and loans to the non(cid:133)nancial sector to the net worth of the aggregate (cid:133)nancial intermediaries - tends to move counter-cyclically, even though the gross leverage multiple - the ratio of book value totalassets(includingsecuritiesandloanstotheotherintermediaries)tothenetworthof some individual intermediaries may move procyclically. Concerning the debate about the procyclicality and countercyclicality of the leverage rate of the intermediaries, see Adrian and Shin (2010) and He, Khang and Krishnamurthy (2010). 11One di⁄erence of our model from these papers is that, because default occurs in equilibrium, the bank(cid:146)s leverage a⁄ects the promised deposit rate and the cost of funds. This e⁄ect provides an additional motive for the bank to reduce its leverage multiple as implied by the fact that when the constraint is not binding (cid:22) >(cid:22)r =0. t t 15

TheevolutionofN dependsonboththeretainedearningsofbankersthat t survived fromthe previous period and the injection of equity to newbankers. For technical convenience again related to computational considerations, we suppose that the household transfer e to a each new banker is proportionate t to the stock of capital at the end of the previous period, S ; with e = t 1 t (cid:16) S :12 Aggregating across both surviving and entering b (cid:0) ankers yields (1 (cid:27))f t 1 th(cid:0)e follo (cid:0) wing expression for the evolution of net worth N = (cid:27)[(Rb R )(cid:30) +R ]N +(cid:16)S : (30) t t (cid:0) t t (cid:0) 1 t t (cid:0) 1 t (cid:0) 1 The (cid:133)rst term is the total net worth of bankers that operated at t 1 and (cid:0) survived until t: The second, (cid:16)S , is the total start-up equity of entering t 1 (cid:0) bankers. 2.3 Runs, insolvency and the default probability We now turn to the case of default due to either runs or insolvency. After describing bank runs and the condition for a bank run equilibrium to exist, we characterize the overall default probability. 2.3.1 Conditions for a bank run equilibrium AsinDiamondandDybvig(1983),therunsweconsiderarerunsontheentire banking system and not an individual bank. A run on an individual bank will not have aggregate e⁄ects as depositors simply shu› e their funds from one bank to another. As we noted earlier, though, we di⁄er from Diamond andDybviginthatrunsre(cid:135)ectapanicfailuretorolloverdepositsasopposed to early withdrawal. Consider the behavior of a household that acquired deposits at t 1: (cid:0) Suppose further that the banking system is solvent at the beginning of time t : assets valued at normal market prices exceed liabilities. The household must then decide whether to roll over deposits at t: A self-ful(cid:133)lling "run" equilibrium exists if and only if the household correctly believes that in the event all other depositors run, thus forcing the banking system into liquidation, the household will lose money if it rolls over its deposits individually. Note that this condition is satis(cid:133)ed if and only if the liquidation forces banks 12Here we value capital at the steady state price Q = 1: If we use the market price instead, the (cid:133)nancial accelerator would be enhanced but not signi(cid:133)cantly. 16

into default, i.e. reduces the value of bank assets below promised obligations to depositors driving aggregate bank net worth to zero. A household that deposits funds in a zero net worth bank will simply lose its money as the bank will divert the money for personal use.13 If instead bank net worth is positive even at liquidation prices, banks would be able to o⁄er a pro(cid:133)table deposit contract to an individual household deciding to roll over. The condition for a bank run equilibrium to exist at t, accordingly, is that in the event of liquidation following a run, bank net worth goes to zero. Recall that earlier we de(cid:133)ned the depositor recovery rate, x , as the ratio of t the value of bank assets to promised obligations to depositors. Therefore, a bank run equilibrium exists at t if and only if the recovery rate conditional on a run, xR, is less than unity: t (cid:24) [(1 (cid:14))Q +Z ]Sb xR = t (cid:0) (cid:3)t t(cid:3) t 1 (31) t (cid:0) R D t t 1 (cid:0) Rb (cid:30) = t(cid:3) t 1 < 1 (cid:0) R (cid:1) (cid:30) 1 t t 1 (cid:0) (cid:0) whereQ istheassetliquidationprice, Z isrentalrate, andRb isthereturn (cid:3)t t(cid:3) t(cid:3) on bank assets conditional on a run. Since the liquidation price Q is below (cid:3)t the normal market price Q ; a run may occur even if the bank is solvent at t normal market prices. Moreover, as we will show below, when deteriorating economic conditions cause bank leverage (cid:30) to increase substantially, t 1 even relatively small new disturbances which d (cid:0) ecrease R t b (cid:3)can open up the Rt possibility of a banking panic. 2.3.2 The liquidation price Keytotheconditionforabankrunequilibriumisthebehavioroftheliquidation price Q : A depositor run at t induces all the existing banks to liquidate (cid:3)t their assets by selling them to households. We suppose that new banks can only store their net worth during a run and start raising deposit one period 13As we mention below, we assume that new entrant banks during a run do not setup their banking operations until a period after the run. Thus an individual depositor who does not run would be forced to save in a bank with zero net worth instead of in a new bank. 17

after the panic.14 Accordingly in the wake of the run: Sh = S : (32) t t The banking system then rebuilds itself over time as new banks enter. The evolution of net worth following the run at t is given by N = (cid:16)S +(cid:27)(cid:16)S : (33) t+1 t t 1 (cid:0) N = (cid:27)[(Rb R )(cid:30) +R ]N +(cid:16)S ; for all (cid:28) t+2: (cid:28) (cid:28) (cid:0) (cid:28) (cid:28) (cid:0) 1 (cid:28) (cid:28) (cid:0) 1 (cid:28) (cid:0) 1 (cid:21) To obtain Q , we invert the household Euler equation: (cid:3)t 1 (cid:28) Sh Q = E (cid:3) (1 (cid:14))(cid:28) t 1 (cid:24) Z (cid:31) (cid:28) (cid:13) =(cid:21) (cid:3)t t t;(cid:28) (cid:0) (cid:0) (cid:0) j (cid:1) (cid:28) (cid:0) S (cid:0) (cid:28) ( (cid:28)=t+1 j=t+1 ! (cid:20) (cid:18) (cid:28) (cid:19) (cid:21) ) X Y (cid:31)(1 (cid:13))e=(cid:21) : (34) t (cid:0) (cid:0) where the term (cid:31)(1 (cid:13))=(cid:21) is the period t marginal e¢ ciency cost following t (cid:0) a run at t (given Sh=S = 1 in this instance).15 The liquidation price is thus t t equal to the expected discounted stream of dividends net the marginal e¢ ciencylossesfromhouseholdportfoliomanagement. Sincemarginale¢ ciency losses are at a maximum when Sh equal S , Q is at a minimum, given the t t (cid:3)t expected future path of Sh: Further, the longer it takes the banking system (cid:28) to recover (so Sh falls back to its steady state value) the lower will be Q . (cid:28) (cid:3)t Finally, note that Q will vary positively with the expected path of (cid:24) and (cid:3)t (cid:28) Z and with the stochastic discount factor (cid:3) : (cid:28) t;(cid:28) 2.3.3 The default probability: illiquideity versus insolvency In the run equilibrium, banks default even though they are solvent at normal market prices. It is the forced liquidation at (cid:133)resale prices during a run that pushes these banks into bankruptcy. Thus, in the context of our model, a bank run can be viewed as a situation of illiquidity. By contrast, default is also possible if banks enter period t insolvent at normal market prices. 14Although goods are storable one-for-one, people do not use storage in equilibrium except for a period of bank run. 15We are imposing that S t h (cid:13) 0 as is the case in all of our numerical simulations. St (cid:0) (cid:21) 18

Accordingly, the total probability of default in the subsequent period, p , t is the sum of the probability of a run pR and the probability of insolvency t pI : t p = pR +pI: (35) t t t We begin with pI. By de(cid:133)nition, banks are insolvent if the ratio of assets t valued at normal market prices is less than liabilities. In our economy, the only exogenous shock to the aggregate economy is a shock to the quality of capital (cid:24) . Let us de(cid:133)ne (cid:24)I to be the value of capital quality, (cid:24) , that t t+1 t+1 makes the depositor recovery rate at normal market prices, x((cid:24)I ) equal to t+1 unity: (cid:24)I [Z ((cid:24)I )+(1 (cid:14))Q ((cid:24)I )]Sb x((cid:24)I ) = t+1 t+1 t+1 (cid:0) t+1 t+1 t = 1: (36) t+1 R D t t For values of (cid:24) below (cid:24)I , the bank will be insolvent and must default. t+1 t+1 The probability of default due to insolvency is then given by pI = prob (cid:24) < (cid:24)I ; (37) t t t+1 t+1 where prob t ( ) is the probability of(cid:0)satisfying (cid:1)conditional on date t infor- (cid:1) (cid:1) mation. We next turn to the determination of the run probability. In general, the time t probability of a run at t + 1 is the product of the probability a run equilibrium exists at t+1 times the probability a run will occur when it is feasible. We suppose the latter depends on the realization of a sunspot. Let (cid:19) be a binary sunspot variable that takes on a value of 1 with probability t+1 { and a value of 0 with probability 1 {. In the event of (cid:19) t+1 = 1, depositors (cid:0) coordinate on a run if a bank run equilibrium exists. Note that we make the sunspot probability { constant so as not to build in exogenous cyclicality in the movement of the overall bank run probability pR: t A bank run arises at t + 1 i⁄ (i) a bank run equilibrium exists at t + 1 and (ii) (cid:19) = 1. Let ! be the probability at t that a bank run equilibrium t+1 t exists at t+1: Then the probability pR of a run at t+1 is given by t pR t = ! t (cid:1) {: (38) To (cid:133)nd the value of ! ; let us de(cid:133)ne (cid:24)R as the value of (cid:24) that makes t t+1 t+1 the recovery rate conditional on a run xR unity when evaluated at the t+1 19

(cid:133)resale liquidation price Q and rental rate during run Z : (cid:3)t+1 t(cid:3)+1 (cid:24)R [(1 (cid:14))Q ((cid:24)R )+Z ((cid:24)R )]Sb x((cid:24)R ) = t+1 (cid:0) (cid:3) t+1 (cid:3) t+1 t = 1: (39) t+1 R D t t For values of (cid:24) below (cid:24)R , xR is below unity and a bank run equilibrium t+1 t+1 t+1 is feasible. Therefore, the probability that a bank run equilibrium exists is given by the probability that (cid:24) lies in the interval below (cid:24)R but above t+1 t+1 the threshold for insolvency (cid:24)I : In particular, t+1 ! = prob (cid:24)I (cid:24) < (cid:24)R : (40) t t t+1 (cid:20) t+1 t+1 Given equation (40), we can dis(cid:0)tinguish regions o(cid:1)f (cid:24) where insolvency t+1 emerges ((cid:24) < (cid:24)I ) from regions where an illiquidity problem may emerge t+1 t+1 ((cid:24)I (cid:24) < (cid:24)R ): t+1 (cid:20) t+1 t+1 Overall, the probability of a run varies inversely with the expected recovery rate E x : The lower the forecast of the depositor recovery rate, the t t+1 higher ! and thus the higher p : In this way the model captures that an t t expected weakening of the banking system raises the likelihood of a run. Finally, comparing equations (37) and (40) makes clear that the possibility of a run equilibrium signi(cid:133)cantly expands the chances for a banking collapse, beyond the probability that would arise simply from default due to insolvency. In this way the possibility of runs makes the system more fragile. Indeed, within the numerical exercises we present the probability of a fundamental shock that induces an insolvent banking system is negligible. However, the probability of a shock that induces a bank run equilibrium is not negligible. 3 Production, market clearing and policy The rest of the model is fairlystandard. There is a production sector consisting of producers of (cid:133)nal goods, intermediate goods and capital goods. Prices are sticky in the intermediate goods sector. In addition there is a central bank that conducts monetary policy. 3.1 Final and intermediate goods (cid:133)rms There is a continuum of measure unity of (cid:133)nal goods producers and intermediate goods producers. Final goods (cid:133)rms make a homogenous good Y t 20

that may be consumed or used as input to produce new capital goods. Each intermediate goods (cid:133)rm f [0;1] makes a di⁄erentiated good Y (f) that is t 2 used in the production of (cid:133)nal goods. Final goods (cid:133)rm transforms intermediate goods into (cid:133)nal output according to the following CES production function: " Y t = 1 Y t (f) " (cid:0)" 1 df " (cid:0) 1 ; (41) (cid:20)Z0 (cid:21) where " > 1 is the elasticity of substitution between intermediate goods. Let P (f) be the nominal price of intermediate good f. Then cost mint imization of (cid:133)nal goods (cid:133)rms yields the following demand function for each intermediate good f (after integrating across the demands of by all (cid:133)nal goods (cid:133)rms): P (f) " t (cid:0) Y (f) = Y ; (42) t t P (cid:20) t (cid:21) where P is the price index as t 1 1 1 " P = P (f)1 "df (cid:0) : t t (cid:0) (cid:20)Z0 (cid:21) There is a continuum of intermediate good (cid:133)rms owned by consumers, indexed by f [0;1]. Each produces a di⁄erentiated good and is a mo- 2 nopolistic competitor. Intermediate goods (cid:133)rm f uses both labor L (f) and t capital K (f) to produce output according to: t Y (f) = A K (f)(cid:11)L (f)1 (cid:11); (43) t t t t (cid:0) where A is a technology parameter and 0 > (cid:11) > 1 is the capital share. t Both labor and capital are freely mobile across (cid:133)rms. Firms rent capital from owners of claims to capital (i.e. banks and households) in a competitive market on a period by period basis. Then from cost minimization, all (cid:133)rms choose the same capital labor ratio, as follows K (f) (cid:11) w K t t t = = : (44) L (f) 1 (cid:11)Z L t t t (cid:0) where, as noted earlier, w is the real wage rate and Z is the rental rate of t t capital. The (cid:133)rst order conditions from the cost minimization problem imply that marginal cost is given by 21

1 w 1 (cid:11) Z (cid:11) t (cid:0) t MC = : (45) t A 1 (cid:11) (cid:11) t (cid:18) (cid:0) (cid:19) (cid:18) (cid:19) Observe that marginal cost is independent of (cid:133)rm-speci(cid:133)c factors. Following Rotemberg (1982), each monopolistically competitive (cid:133)rm f faces quadratic costs of adjusting prices. Let (cid:26)r ("r" for Rotemberg) be the parameter governing price adjustment costs. Then each period, it chooses P (f) and Y (f) to maximize the expected discounted value of pro(cid:133)t: t t 1 P (cid:28) (f) (cid:26)r P (cid:28) (f) 2 E (cid:3) MC Y (f) Y 1 ; (46) t t;(cid:28) (cid:28) (cid:28) (cid:28) P (cid:0) (cid:0) 2 P (f) (cid:0) ( X (cid:28)=t " (cid:18) (cid:28) (cid:19) (cid:18) (cid:28) (cid:0) 1 (cid:19) #) e subject to the demand curve (42). Here we assume that the adjustment cost is proportional to the aggregate demand Y . t Taking the (cid:133)rm(cid:146)s (cid:133)rst order condition for price adjustment and imposing symmetry implies the following forward looking Phillip(cid:146)s curve: " " 1 Y t+1 ((cid:25) 1)(cid:25) = MC (cid:0) +E (cid:3) ((cid:25) 1)(cid:25) ; (47) t (cid:0) t (cid:26)r t (cid:0) " t t+1 Y t+1 (cid:0) t+1 (cid:18) (cid:19) (cid:20) t (cid:21) where (cid:25) = Pt is the realized gross in(cid:135)ation rate at date t. t Pt 1 (cid:0) 3.2 Capital goods producers There is a continuum of measure unity of competitive capital goods (cid:133)rms. Each produces new investment goods that it sells at the competitive market price Q : By investing I (j) units of (cid:133)nal goods output, (cid:133)rm j can produce t t (cid:0)(I (j)=K ) K new capital goods, with (cid:0) > 0; (cid:0) < 0; and where K is the t t t 0 00 t (cid:1) aggregate capital stock.16 The decision problem for capital producer j is accordingly I (j) t maxQ (cid:0) K I (j): (48) t t t It(j) (cid:18) K t (cid:19) (cid:0) 16Forsimplicityweareassumingthattheaggregatecapitalstockentersintoproduction function of investment goods as an externality. Alternatively, we could make an assumpiton similar to Cao, Lorenzoni and Walentin (2016): Each capital goods producer buys capital after being used to produce intermediated goods and combines the capital with (cid:133)nal output goods to produce the total capital stock. One can then obtain a (cid:133)rst order condition like (49). 22

Given symmetry for capital producers (I (j) = I ); we can express the (cid:133)rst t t order condition as the following "Q" relation for investment: I 1 t (cid:0) Q = (cid:0) (49) t 0 K (cid:20) (cid:18) t(cid:19)(cid:21) which yields a positive relation between Q and investment. t 3.3 Monetary Policy Let (cid:2) be a measure of cyclical resource utilization, i.e., resource utilization t relative to the (cid:135)exible price equilibrium. Next let R = (cid:12) 1 denote the real (cid:0) interestrateinthedeterministicsteadystatewithzeroin(cid:135)ation. Wesuppose that the central bank sets the nominal rate on the riskless bondRn according t to the following Taylor rule: 1 Rn = ((cid:25) )(cid:20)(cid:25) ((cid:2) )(cid:20)y (50) t (cid:12) t t with (cid:20) > 1. Note that, if the net nominal rate cannot go below zero, the (cid:25) policy rule would become Rn = max 1 ((cid:25) )(cid:20)(cid:25) ((cid:2) )(cid:20)y ;1 . t (cid:12) t t A standard way to measure (cid:2) t ins to use the ratiooof actual output to a hypothetical (cid:135)exible price equilibrium value of output. Computational considerations lead us to use a measure which similarly captures the cyclical e¢ ciency of resource utilization but is much easier to handle numerically. Speci(cid:133)cally, we take as our measure of cyclical resource utilization the ratio of the desired markup, 1+(cid:22) = "=(" 1) to the current markup 1+(cid:22) :17 t (cid:0) 1+(cid:22) (cid:2) = (51) t 1+(cid:22) t with (1 (cid:11))(Y =L ) 1+(cid:22) = MC 1 = (cid:0) t t : (52) t t(cid:0) L’C (cid:13) h t t The markup corresponds to the ratio of the marginal product of labor to the marginal rate of substitution between consumption and leisure, which corresponds to the labor market wedge. The inverse markup ratio (cid:2) thus isolates t 17In the case of consumption goods only, our markup measure of e¢ ciency corresponds exactly to the output gap. 23

the cyclical movement in the e¢ ciency of the labor market, speci(cid:133)cally the component that is due to nominal rigidities. Finally, one period bonds which have a riskless nominal return have zero net supply. (Bank deposits have default risk). Nonetheless we can use the following household Euler equation to price the nominal interest rate of these bonds Rn as t Rn E (cid:3) t = 1: (53) t t+1 (cid:25) (cid:18) t+1(cid:19) 3.4 Resource constraints and equilibrium Total output is divided between consumption, investment, the adjustment cost of nominal prices and a (cid:133)xed value of government consumption G: (cid:26)r Y = C +I + ((cid:25) 1)2Y +G: (54) t t t t t 2 (cid:0) Given a symmetric equilibrium, we can express total output as the following function of aggregate capital and labor: Y = A K(cid:11)L1 (cid:11): (55) t t t t(cid:0) Although we consider a limiting case in which supply of government bond and money is zero, government adjusts lump-sum tax to satisfy the budget constraint. Finally, labor market must clear, which implies that aggregate labor demand of producers equals aggregate labor supply of households. This completes the description of the model. See Appendix for details. 4 Numerical exercises 4.1 Calibration Table 1 lists the choice of parameter values for our model. Overall there are twenty one parameters. Thirteen are conventional as they appear in standard New Keynesian DSGE models. The other eight parameters govern the behavior of the (cid:133)nancial sector, and hence are speci(cid:133)c to our model. We begin with the conventional parameters. For the discount rate (cid:12); the risk aversion parameter (cid:13) ; the inverse Frisch elasticity ’, the elasticity of h substitution between goods ", the depreciation rate (cid:14) and the capital share (cid:11) 24

weusestandardvaluesintheliterature. Threeadditionalparameters((cid:17);a;b) involve the investment technology, which we express as follows: I I 1 (cid:17) t t (cid:0) (cid:0) = a +b: K K (cid:18) t(cid:19) (cid:18) t(cid:19) We set (cid:17), which corresponds to the elasticity of the price of capital with respect to investment rate, equal to 0:25, a value in line with panel data estimates. Wethenchooseaandbtohittwotargets: (cid:133)rst,aratioofquarterly investment to the capital stock of 2:5% and, second, a value of the price of capital Q equals unity in the risk-adjusted steady state. We set the value of (cid:133)xed government expenditure G to 20% of steady state output. Next we choose the cost of price adjustment parameter (cid:26)jr to generate an elasticity of in(cid:135)ation with respect to marginal cost equal to 1 percent, which is roughly in line with the estimates.18 Finally, we set the feedback parameters in the Taylorrule, (cid:20) and(cid:20) totheirconventional values of 1:5 and0:5 respectively. (cid:25) y We nowturn to the (cid:133)nancial sector parameters. There are six parameters that directly a⁄ect the evolution of bank net worth and credit spreads: the banker(cid:146)ssurvivalprobability(cid:27);theinitialequityinjectiontoenteringbankers as a share of capital (cid:16); the asset diversion parameter (cid:18); the threshold share for costless direct household (cid:133)nancing of capital, (cid:13); the parameter governing the convexity of the e¢ ciency cost of direct (cid:133)nancing (cid:31); and the probability of observing a sunspot {. We choose the values of these parameter to hit the following six targets: (i) the average arrival rate of a systemic bank run equals 4 percent annually, corresponding to a frequency of banking panics of once every 25 years, which is in line with the evidence for advanced economies19; (ii) the average bank leverage multiple equals 10;20 (iii) the average excess rate of return on bank assets over deposits equals 2%; based on Philippon (2015); (iv) the average share of bank intermediated assets equals 0:5; which is a reasonable estimate of the share of intermediation performed by investment banks and large commercial banks; (v) and(vi) the increase inexcess returns (measuredbycredit 18See, for example, Del Negro, Giannoni and Shorfheide (2015) 19See, for example, Bordo et al (2001), Reinhart and Rogo⁄(2009) and Schularick and Taylor (2012). 20We think of the banking sector in our model as including both investment banks andsomelargecommercialbanksthatoperatedo⁄balancesheetvehicleswithoutexplicit guarantees. Tenisonthehighsideforcommercialbanksandonthelowsideforinvestment banks. See Gertler, Kiyotaki and Prestipino (2016). 25

spreads) and the dropin investment following a bankrun match the evidence from the recent crisis. The remaining two parameters determine the serial correlation of the capital quality (cid:26) and and the standard deviation of the innovations (cid:27) : (cid:24) (cid:24) That is we assume that the capital quality shock obeys the following (cid:133)rst order process : (cid:24) = 1 (cid:26) +(cid:26) (cid:24) +(cid:15) t+1 (cid:24) (cid:24) t t+1 (cid:0) with 0 < (cid:26) (cid:24) < 1 and where (cid:15)(cid:0)t+1 a ((cid:1)truncated) normally distributed i.i.d. random variable with mean zero and standard deviation (cid:27) .21 We choose (cid:24) (cid:26) and (cid:27) so that the unconditional standard deviations of investment and (cid:24) (cid:24) output match the ones observed over the 1983Q1-2008Q3 period. Given that our policy functions are non linear we obtain model implied momentsbysimulatingoureconomyfor100thousandperiods. Table2shows unconditional standard deviations for some key macroeconomic variables in the model and in the data. The volatilities of output, investment and labor are reasonably in line with the data. Consumption is too volatile, but the variability of the sum of consumption and investment matches the evidence. 4.2 Experiments In this section we perform several experiments that are meant to illustrate how our model economy behaves and compares with the data. We (cid:133)rst show the response of the economy to a capital quality shock with and without runs to illustrate how the model generates a (cid:133)nancial panic. We then compare how runs versus occasionally binding constraints can generate nonlinear dynamics. Finally, we turn to an experiment that shows how the model can replicate salient features of the recent (cid:133)nancial crisis. 4.2.1 Response to a capital quality shock: no bank run case We suppose the economy is initially in a risk-adjusted steady state. Figure 1 shows the response of the economy to a negative one standard deviation 21In practice we assume that (cid:15) is a truncated normal with support ( 10(cid:27) ;10(cid:27) ). t+1 (cid:24) (cid:24) (cid:0) Given our calibration for (cid:27) and (cid:26) the probability that (cid:24) goes below zero is computa- (cid:16) (cid:24) t tionally zero. 26

(.75%) shock to the quality of capital.22 The solid line is our baseline model andthedottedlineisthecasewherethereareno(cid:133)nancial frictions. Forboth cases the shock reduces the expected return to capital, reducing investment and in turn aggregate demand. In addition, for the baseline economy with (cid:133)nancialfrictions,theweakeningofbankbalancesheetsampli(cid:133)esthecontractionindemandthroughthe(cid:133)nancialacceleratororcreditcyclemechanismof Bernanke Gertler and Gilchrist (1999) and Kiyotaki and Moore(1997). Poor asset returns following the shock cause bank net worth to decrease by about 15%. As bank net worth declines, incentive constraints tighten and banks decrease their demand for assets causing the price of capital to drop. The drop in asset prices feeds back into lower bank net worth, an e⁄ect that is magni(cid:133)ed by the extent of bank leverage. As (cid:133)nancial constraints tighten and asset prices decline, excess returns rise by 75 basis points which allows bankstoincreasetheirleveragebyabout10%: Overall, a0:75 percentdecline in the quality of capital results in a drop in investment by 5 percent and a drop in output by slightly more than 1 percent. The drop in investment is roughly double the amount in the case absent (cid:133)nancial frictions, while the drop in output is about thirty percent greater. In the experiment of Figure 1, the economy is always ex post in a "safe zone", where a bank run equilibrium does not exist. Under our parametrization, a bank run cannot happen in the risk-adjusted steady state: bank leverage is too low. The dashed line in the (cid:133)rst panel of Figure 1 shows the size of the shock in the subsequent period needed to push the economy into the run region: In our example, a two standard deviation shock is needed to open up the possibility of runs starting from the risk adjusted steady state, which is double the size of the shock considered in Figure 1. Even though in this case the economy is always in a safe region ex post, it is possible ex ante that a run equilibrium could occur in the subsequent period. In particular, the increase in leverage following the shock raises the probability that a su¢ ciently bad shock in the subsequent period pushes the economy into the run region. As the top middle panel of Figure 1 shows, the overall probability of a run increases following the shock. 22In all of the experiments we trace the response of the economy to the shocks considered assuming that after these shocks capital quality is exactly equal to its conditional expectations, i.e. setting future " to 0: t 27

4.2.2 Bank runs In the previous experiment the economy was well within a safe zone. A one standarddeviationshockdidnotandcouldnotproducea(cid:133)nancialpanic. We nowconsideracasewheretheeconomystartsinthesafezonebutisgradually pushed to the edge of the crisis zone, where a run equilibrium exists. We then show how an arrival of sunspot induces a panic with damaging e⁄ects on the real economy. To implement this experiment, we assume that the economy is hit by a sequence of three equally sized negative shocks that push the economy to the run threshold. That is, we (cid:133)nd a shock (cid:15) that satis(cid:133)es: (cid:3) (cid:24)R = 1+(cid:15) 1+(cid:26) +(cid:26)2 3 (cid:3) (cid:24) (cid:24) where (cid:24)R is the threshold level for th(cid:0)e capital qu(cid:1)ality below which a run is 3 possible in period 3; given that the economy is in steady state in period 0 and is hit by two equally sized shocks in periods 1 and 2, i.e. (cid:15) = (cid:15) = (cid:15) : 1 2 (cid:3) The (cid:133)rst two shocks push the economy to the edge of the crisis zone. The third pushes it just in. ThesolidlineinFigure2showstheresponseoftheeconomystartingfrom period two onwards under the assumption that the economy experiences a run with arrival of a sunspot in period 3. For comparison, the dashed line shows the response of the economy to the same exact capital quality shocks but assuming that no sunspot is observed and so no run happens. As shown in panel 1 the size of the threshold innovation of capital quality shock turns out to be roughly equal to one standard deviation, i.e. (cid:15) = :83%:, which is the size of the shock in Figure 1. After the (cid:133)rst (cid:3) (cid:0) two innovations, the capital quality is 1:4% below average and the run probability is about 2% quarterly. The last innovation pushes the economy into the run region. When the sunspot is observed and the run occurs, bank net worth is wiped out which forces banks to liquidate assets. In turn, households absorb the entire capital stock. Households however are only willing to increase their portfolio holdings of capital at a discount, which leads excess returnstospikeandinvestmenttocollapse. Whentherunoccurs, investment dropsanadditional25% resultinginanoveralldropof35%. Comparingwith the case of no run clari(cid:133)es that almost none of this additional drop is due to the capital quality shock itself: The additional drop in investment absent a run is only 2.5%. The collapse in investment demand causes in(cid:135)ation to decrease and induces monetary policy to ease by reducing the policy rate to 28

slightly below zero. However, reducing the nominal interest rate to roughly zero is not su¢ cient to insulate output which drops by 7%. As new bankers enter the economy, bank net worth is slowly rebuilt and the economy returns to the steady state. This recovery is slowed down by a persistent increase in the run probability following the banking panic. The increase in the run probability reduces the amount of leverage that banks are willing to take on. To get a sense of the role that nominal rigidities are playing, Figure 3 describes the e⁄ect of bank runs in the economy with (cid:135)exible prices. For comparison, with the analogous experiment in our baseline (in Figure 2) we hit the (cid:135)ex price economy with the same sequence of shocks that would take the baseline economy to the run threshold.23 There are two main takeaways from Figure 3. First, the output drop in the (cid:135)exible price case is only about half that in our benchmark sticky price case. The New Keynesian features thus magnify the e⁄ects of the banking crisis. The reason is that the banking crisis generates a steep decline in the natural rate of interest by inducing a collapse in investment demand. As a result, in the (cid:135)exible price case the real interest rate, which is equals the natural rate, drops roughly eight hundred basis points below zero leading to a temporary expansion in consumption demand and hence dampening the output contraction. Clearly, such a dramatic drop in real rates would not be feasible with nominal rigidities and a zero lower bound. Second, even in the (cid:135)exible price case, a bank run will amplify the contraction in output by inducing a large drop in investment demand. In our example, relative to the no run case, the run increases the drop in output from about one percent to three and a half percent. 4.2.3 Nonlinearities: occasionally binding constraints vs runs We now turn to nonlinearities within our baseline model. We will start by considering the e⁄ects of occasionally binding constraints. Figure 4 shows the behavior of the economy when it transits from slack to binding (cid:133)nancing (incentive) constraints. In our calibration, the risk adjusted steady state lies at the borderline for the (cid:133)nancing constraint to be binding. If the shock to capital quality is positive the constraint is slack, while it becomes bind- 23However, since in the (cid:135)ex price economy there is much less ampli(cid:133)cation, the expost run that we consider is actually not an equilbrium. As the (cid:133)rst panel in the (cid:133)gure shows, evenafterthe(cid:133)rsttwoshockstheshockthatisneededtopushtheeconomytothe threshold is still very large in the (cid:135)ex price economy, i.e. around -4%. 29

ing with negative capital quality shocks. Overall, nonlinearities are present, thoughtheydonotturnouttobeaslargeasinthecaseofbankruns. Anegative capital quality shock a⁄ects investment, asset prices and credit spreads only a little more, in absolute value, than does a similar magnitude increase. The asymmetries arising in our framework are somewhat dampened for two reasons: First, in many frameworks the maximum leverage multiple is (cid:133)xed (e.g. Mendoza, 2010). However, in our model, as the economy moves into the constrained region the maximum leverage multiple increases (see section 2.2.2). This relaxing of the leverage constraint mitigates the decline in real activity and asset prices and the rise in credit spreads. Second, it is often assumedthattherealinterestrateis(cid:133)xed. Inourmodel, however, therealrate declines as the economy weakens, which also works to dampen the decline in the constrained region. Next we consider bank runs. Figure 5 shows the response of the economy to a capital quality shock starting from the same initial state considered in Figure2.Thedashedlinedepictstheresponseinthecaseinwhichnosunspot occurs(sothatabankruncannothappen)andthesolidoneshowsthecasein which a sunspot appears (so that a run will occur if a run equilibriumexists). As long as capital quality shocks are above the run threshold the responses areidentical inthebothcasessinceinthisregionarunisnotpossible. When the shock lies below the run threshold, however, a run equilibrium exists. In this region, when agents observe a sunspot they run on (cid:133)nancial institutions pushingtheeconomytoanequilibriuminwhichbanksareforcedtoliquidate assets at (cid:133)re sale prices. The discrete and highly nonlinear behavior during (cid:133)nancial crisis (which we described in Introduction) then emerges: excess returns spike and investment and asset prices collapse. 4.2.4 Crisis experiment: model versus data Figure 6 illustrates how the model can replicate some salient features of the recent (cid:133)nancial crisis. We hit the economy with a series of capital quality shocks over the period 2007Q4 until 2008Q3. The starting point is the beginning of the recession, which roughly coincides with the time credit markets (cid:133)rst came under stress following Bear Stearns(cid:146)losses on its MBS portfolios. We pick the size of the capital quality shocks to match the observed decline in investment during this period, in panel 1. We then assume that a run happens in 2008Q4, the quarter in which Lehman failed and the shadow banking system collapsed. The solid line shows the observed response of some key 30

macroeconomic variables.24 The dashed line shows the response of the economy when a run occurs in 2008Q4 and the dotted line shows the response under the assumption that a run does not happen. As indicated in panel 2, the sequence of negative surprises in the quality of capital needed to match the observed contraction in investment leads to a gradual decline in banks net worth that matches closely the observed decline in (cid:133)nancial sector equity as measured by the XLF index, which is an index of S&P 500 (cid:133)nancial stocks. Given that banks net worth is already depleted by poor asset returns, a very modest innovation in 2008Q4 pushes the economy into the run region. When the run occurs, the model economy generates a suddenspikeinexcessreturnsandadropininvestment,output,consumption and employment of similar magnitudes as those observed during the crisis in panels 3 - 6. The dotted line shows how, absent a run, the same shocks would generate a much less severe downturn. The model economy also predicts a rather slow recovery following the (cid:133)nancial crisis, although faster than what we observed in the data. It is important however to note that in the experiment we are abstracting from any disturbances after 2008Q4. This implies a rather swift recovery of (cid:133)nancial equity and excess returns to their long run value. On the other hand, the observed recovery of net worth and credit spreads was much slower with bothvariables still farfromtheirpre-crisis values as of today. Various factors that are not captured in our model economy, such as a drastic change in the regulatory framework of (cid:133)nancial institutions, increased uncertainty following the crisis and slow adjustment of household balance sheets, have likely contributed to the very slow recovery of these (cid:133)nancial variables. Incorporating these factors could help the model account for the very slow recovery of investment and employment. However we leave this extension for future research. 5 Conclusion We have developed a macroeconomic model with a banking sector where costly (cid:133)nancial panics can arise. A panic or run in our model is a selfful(cid:133)lling failure of creditors to roll over their short-term credits to banks. 24For output, investment and consumption we show deviation from a trend computed by using CBO estimates of potential output and similarly for hours worked we let the CBO estimate of potentail labor represent the trend. 31

When the economy is close to the steady state a self-ful(cid:133)lling rollover crises cannot happen because banks have su¢ ciently strong balance sheets. In this situation, "normal size" business cycle shocks do not lead to (cid:133)nancial crises. However, in a recession, banks may have su¢ ciently weak balance sheet so as to open up the possibility of a run. Depending on the circumstances either a small shock or no further shock can generate a run that has devastating consequences for the real economy. We show that our model generates the highly nonlinear contraction in economic activity associated with (cid:133)nancial crises. It also captures how crises may occur even in the absence of large exogenous shocks to the economy. We then illustrate that the model is broadly consistent with the recent (cid:133)nancial crisis. One issue we save for further work is the role of macroprudential policy. Aswithothermodelsofmacroprudentialpolicy,externalitiesarepresentthat lead banks to take more risk than is socially e¢ cient. Much of the literature is based on the pecuniary externality analyzed by Lorenzoni (2008), where individual banks do not properly internalize the exposure of the system to asset price (cid:135)uctuations that generate ine¢ cient volatility, but not runs. A distinctive feature of our model is that the key externality works through the e⁄ect of leverage on the bank run probability: Because the run probability depends on the leverage of the banking system as a whole, individual banks do not fully take into account the impact of their own leverage decisions on the exposure of the entire system. In this environment, the key concern of the macroprudential policy becomes reducing the possibility of a (cid:133)nancial collapse in the most e¢ cient way. Our model will permit us to explore the optimal design of policies qualitatively and quantitatively. 32

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Table 1: Calibration Parameter Description Value Target Standard Parameters β Impatience .99 Risk Free Rate γ Risk Aversion 2 Literature h ϕ Frish Elasticity 2 Literature (cid:15) Elasticity of subst across varieties 11 Markup 10% α Capital Share .33 Capital Share δ Depreciation .025 I =.025 K η Elasticity of q to i .25 Literature a Investment Technology Parameter .53 Q=1 b Investment Technology Parameter -.83% I =.025 K G Government Expenditure .45 G =.2 Y ρjr Price adj costs 1000 Slope of Phillips curve .01 κ Policy Response to Inflation 1.5 Literature π κ Policy Response to Output .5 Literature y Financial Intermediation Parameters σ Banker Survival rate .93 Leverage QSb =10 N New Bankers Endowments ζ .1% % ∆ I in crisis ≈35% as a share of Capital θ Share of assets divertible .23 Spread Increase in Crisis = 1.5% Threshold for γ .432 Sb =.5 HH Intermediation Costs S χ HH Intermediation Costs .065 ERb−R=2% Annual κ Sunspot Probability .15 Run Probability 4% Annual σ((cid:15)ξ) std of innovation to capital quality .75% std Output ρξ serial correlation of capital quality .7 std Investment 1

Table 2: Standard Deviations Data vs. Model Data Model 1 9_83_-_2_0_07_q_3__ N o _R_u_n_s _H_a_p_pen Y 1.9 2.4 C+I 2.7 3.0 I 7.2 6.9 C 1.3 3.1 L 3.1 3.4 All values in percentages. NOTE: For output, investment, consumption, and government spending we compute real per capita terms by dividing the nominal variables by the population and adjusting by the GDP deflator. For labor we compute per capita hours worked by dividing total labor hours by the population. We then show the standard deviations of the logged variables in deviations from a linear trend starting in 1983q1 and ending in 2007q3. SOURCE: Output, investment (gross private domestic investment plus durable good consumption), consumption (personal consumption expenditure less durable good consumption), government spending, and the GDP deflator are from the Bureau of Economic Analysis. Total labor hours (aggregate hours, nonfarm payrolls) and population (civilian noninstitutional, 16 years and over) are from the Bureau of Labor Statistics.

0 -0.5 -1 -1.5 -2 0 20 40 60 SS morf " % Capital Quality 0.01 0.008 0.006 Capital Quality Run Threshold 0.004 0.002 0 0 20 40 60 leveL Run Probability 5 0 -5 -10 -15 -20 0 20 40 60 SS morf " % Bank Net Worth 12 10 8 6 4 2 0 0 20 40 60 SS morf " % Leverage Multiple: ? 1 0 -1 -2 -3 -4 -5 0 20 40 60 SS morf " % Investment 0.5 0 -0.5 -1 -1.5 0 20 40 60 SS morf " % Output 250 200 150 100 50 0 0 20 40 60 Quarters stnioP sisaB launnA leveL Excess Return: ERb-Rfree 450 400 350 300 0 20 40 60 Quarters stnioP sisaB launnA leveL Policy Rate 15 10 5 0 -5 -10 0 20 40 60 Quarters stnioP sisaB launnA leveL Fig. 1. Response to a Capital Quality Shock (1 std): No Run Case Baseline No Financial Fricitons Inflation

1 0.5 0 -0.5 -1 -1.5 -2 2 20 40 60 SS morf " % Capital Quality 0.12 0.1 0.08 Capital Quality 0.06 Run Threshold Initial Threshold 0.04 0.02 0 2 20 40 60 leveL Run Probability 0 -20 -40 -60 -80 -100 2 20 40 60 SS morf " % Bank Net Worth 1500 1000 500 0 -500 2 20 40 60 SS morf " % Leverage Multiple: ? 10 0 -10 -20 -30 -40 2 20 40 60 SS morf " % Investment 0 -2 -4 -6 -8 2 20 40 60 SS morf " % Output 2000 1500 1000 500 0 2 20 40 60 Quarters stnioP sisaB launnA leveL Excess Return: ERb-Rfree 500 400 300 200 100 0 -100 2 20 40 60 Quarters stnioP sisaB launnA leveL Policy Rate 50 0 -50 -100 2 20 40 60 Quarters stnioP sisaB launnA leveL Fig. 2. Response to a Sequence of Shocks: Run VS No Run RUN (Run Threshold Shock and Sunspot) NO RUN (Run Threshold Shock and No Sunspot) Inflation

2 0 -2 -4 -6 2 20 40 60 SS morf " % Capital Quality 0.1 0.08 0.06 Capital Quality Run Threshold 0.04 Initial Threshold 0.02 0 2 20 40 60 leveL Run Probability 0 -20 -40 -60 -80 -100 2 20 40 60 SS morf " % Bank Net Worth 1500 1000 500 0 -500 2 20 40 60 SS morf " % Leverage: ? 5 0 -5 -10 -15 -20 -25 2 20 40 60 SS morf " % Investment 0 -1 -2 -3 -4 2 20 40 60 SS morf " % Output 2000 1500 1000 500 0 2 20 40 60 Quarters stnioP sisaB launnA leveL Excess Return: ERB-Rfree 500 0 -500 -1000 2 20 40 60 Quarters stnioP sisaB launnA leveL Natural Rate 1 0 -1 -2 -3 2 20 40 60 Quarters SS morf " % Fig. 3. Response to the Same Sequence of Shocks in Flex Price Economy: Run VS No Run RUN (Off-Equilibrium) NO RUN Consumption

30 20 10 0 -10 -20 -30 -40 - 1% 0 + 1% Capital Quality Shock "% Bank Net Worth 10 9 8 7 6 - 1% 0 + 1% Capital Quality Shock leveL Leverage multiple: ? 2.5 2 1.5 1 0.5 0 - 1% 0 + 1% Capital Quality Shock )% launnA( leveL 10 5 0 -5 -10 -15 - 1% 0 + 1% Capital Quality Shock b Excess Returns: ER -R "% Investment 2 1 0 -1 -2 -3 - 1% 0 + 1% Capital Quality Shock "% Price of Capital 7 6 5 4 3 2 - 1% 0 + 1% Capital Quality Shock )% launnA( leveL teN Fig. 4. Non-Linearities due to Occasionally Binding Constraints Constraint Binds Constraint Slack free Real Interest Rate: R

40 20 0 -20 -40 -60 -80 -100 r 0 1% 0 Capital Quality Shock "% Net Worth 25 20 15 10 5 0 r 0 1% 0 Capital Quality Shock leveL Leverage 25 20 15 10 5 0 r 0 1% 0 Capital Quality Shock )% launnA( leveL 4 2 0 -2 -4 -6 -8 -10 r 0 1% 0 Capital Quality Shock b Excess Returns: ER -R "% Price of Capital 6 4 2 0 -2 -4 -6 r 0 1% 0 Capital Quality Shock )% launnA( leveL teN free Real Interest Rate: R 10 0 -10 -20 -30 r 0 1% 0 Capital Quality Shock "% Fig. 5. Non-linearities from Runs No Sunspot Sunspot Run Threshold: 0 r = - 0.9% Investment

Fig. 6. Financial Crisis: Model vs. Data Shocks* : -0.3% -0.6% -0.5% -0.8 % -0.7 % 2007q4 2008q1 2008q2 2008q3 2008q4 1.Investment 2.XLF Index and Net Worth 10 25 Bear Stearns Lehman Brothers 0 0 -10 -25 -20 -50 -30 Data -75 Model -40 Model No Run -50 -100 2004 2007q3 2008q4 2016 2004 2007q3 2008q4 2016 3.Spreads (AAA-Risk Free) 4.GDP 4 5 3 0 2 -5 1 0 -10 2004 2007q3 2008q4 2016 2004 2007q3 2008q4 2016 5.Labor (hours) 6.Consumption 5 5 0 0 -5 -10 -5 2004 2007q3 2008q4 2016 2004 2007q3 2008q4 2016 NOTE: The data for GDP, Investment, and Consumption are computed as logged deviations from trend where the trend is the CBO potential GDP. Labor data is computed as logged deviations from trend where the trend is the CBO potential hours worked. The XLF Index data is computed as the percent deviation from its 2007q3 level.

6 Appendix This Appendix describes the details of the equilibrium. The aggregate state of the economy is summarized by the vector of state variables ~ = (S ;Sb ;R D ;(cid:24) ); with sunspot realization (cid:19) at time t, where S M = t capi t t (cid:0) a 1 l st t o(cid:0)c 1 k at t th t (cid:0) e 1 en t d of t 1; Sb = bank capit t al holdings in t 1; t (cid:0) R 1 D = bank deposit obligatio (cid:0) n at th t (cid:0)e 1 beginning of t; and (cid:24) = t t 1 t (cid:0) (cid:0) capital quality shock realized in t: 6.1 Producers As described in the text, the capital stock for production in t is given by K = (cid:24) S ; (56) t t t 1 (cid:0) The capital quality shock is serially correlated as follows (cid:24) F (cid:24) (cid:24) = F (cid:24) t+1 t+1 t t t+1 (cid:24) j with a continuous density: (cid:0) (cid:1) (cid:0) (cid:1) F (cid:24) = f (cid:24) ; for (cid:24) (0; ): t0 t+1 t t+1 t+1 2 1 Capital at the end of(cid:0)perio(cid:1)d is (cid:0) (cid:1) I t S = (cid:0) K +(1 (cid:14))K : (57) t t t K (cid:0) (cid:18) t(cid:19) As we described in the text, capital goods producer(cid:146)s (cid:133)rst order condition for investment is I t Q (cid:0) = 1: (58) t 0 K (cid:18) t(cid:19) A (cid:133)nal goods (cid:133)rms chooses intermediate goods Y (f) to minimize the t f g cost 1 P (f)Y (f)df t t Z0 subject to the production function: " Y t = 1 Y t (f) " (cid:0)" 1 df " (cid:0) 1 : (59) (cid:20)Z0 (cid:21) 36

The cost minimization then yields a demand function for each intermediate good f : P (f) " t (cid:0) Y (f) = Y ; (60) t t P (cid:20) t (cid:21) where P is the price index, given by t 1 1 1 " P = P (f)1 "df (cid:0) : t t (cid:0) (cid:20)Z0 (cid:21) Conversely, an intermediate goods producer f chooses input to minimize the production cost w L (f)+Z K (f) t t t t subject to A [K (f)](cid:11)[L (f)]1 (cid:11) = Y (f): t t t (cid:0) t The (cid:133)rst order conditions yield K (f) (cid:11) w K t t t = = ; (61) L (f) 1 (cid:11)Z L t t t (cid:0) and the following relation for marginal cost: 1 Z (cid:11) w 1 (cid:11) t t (cid:0) MC = : (62) t A (cid:11) 1 (cid:11) t (cid:18) (cid:19) (cid:18) (cid:0) (cid:19) Each period, the intermediate goods producer chooses P (f) and Y (f) to t t maximize the expected discounted value of pro(cid:133)ts: 1 P (cid:28) (f) (cid:26)r P (cid:28) (f) 2 E (cid:3) MC Y (f) Y 1 ; t t;(cid:28) (cid:28) (cid:28) (cid:28) P (cid:0) (cid:0) 2 P (f) (cid:0) ( X (cid:28)=t " (cid:18) (cid:28) (cid:19) (cid:18) (cid:28) (cid:0) 1 (cid:19) #) subject to th e e demand curve (60), where (cid:3) t;(cid:28) = (cid:12)(cid:28) (cid:0) t(C (cid:28) =C t ) (cid:0) (cid:13) h is the discount factor of the representative household. Taking the (cid:133)rm(cid:146)s (cid:133)rst order condition for price adjustment and imposineg symmetry implies the following forward looking Phillip(cid:146)s curve: " " 1 Y t+1 ((cid:25) 1)(cid:25) = MC (cid:0) +E (cid:3) ((cid:25) 1)(cid:25) ; (63) t (cid:0) t (cid:26)r t (cid:0) " t t;t+1 Y t+1 (cid:0) t+1 (cid:18) (cid:19) (cid:20) t (cid:21) where (cid:25) = Pt is the realized gross in(cid:135)eation rate at date t. The cost t Pt 1 minimizationco(cid:0)nditionswithsymmetryalsoimplythataggregateproduction is simply Y = A K (cid:11)L 1 (cid:11): (64) t t t t (cid:0) 37

6.2 Households We modify the household(cid:146)s maximization problem in the text by allowing for a riskless nominal bond which will be in zero supply. We do so to be able the pin down the riskless nominal rate Rn: Let B be real value of this t t riskless bond. The household then chooses C ;L ;B ;D and Sh to maximize t t t t t expected discounted utility U : t U = E 1 (cid:12)(cid:28) t (C (cid:28) )1 (cid:0) (cid:13) h (L (cid:28) )1+’ &(Sh;S ) ; t t (cid:0) 1 (cid:13) (cid:0) 1+’ (cid:0) (cid:28) (cid:28) ( (cid:28)=t (cid:20) (cid:0) h (cid:21) ) X subject to the budget constraint Rn C +D +Q Sh+B = w L T +(cid:5) +R D + t 1B +(cid:24) [Z +(1 (cid:14))Q ]Sh : t t t t t t t (cid:0) t t t t (cid:0) 1 (cid:25) (cid:0) t t (cid:0) 1 t t (cid:0) t t (cid:0) 1 As explained in the text, the rate of return on deposits is given by (cid:24) [Z +(1 (cid:14))Q ]Sb R = Max R ; t t (cid:0) t t 1 t t (cid:0) D (cid:26) t 1 (cid:27) (cid:0) (cid:24) [Z +(1 (cid:14))Q ] Q Sb = Max R t ; t t Q (cid:0) t Q S t (cid:0) b 1 t (cid:0) 1 N (cid:26) t (cid:0) 1 t (cid:0) 1 t (cid:0) 1 (cid:0) t (cid:0) 1(cid:27) (cid:30) = Max R ;Rb t 1 ; t t(cid:30) (cid:0) 1 (cid:18) t (cid:0) 1 (cid:0) (cid:19) where R t b = (cid:24) t [Zt+ Q ( t 1 (cid:0) 1 (cid:14))Qt] and where (cid:30) t = Q t S t b=N t is the bank leverage multiple. (cid:0) We obtain the (cid:133)rst order conditions for labor, riskless bonds, deposits and direct capital holding, as follows: w = (C )(cid:13) h(L )’ (65) t t t Rn E (cid:3) t = 1 (66) t t+1 (cid:25) (cid:18) t+1(cid:19) (cid:30) E (cid:3) Max R ;Rb t = 1 (67) t t+1 t+1 t+1(cid:30) 1 (cid:20) (cid:18) t (cid:0) (cid:19)(cid:21) Z +(1 (cid:14))Q t+1 t+1 E (cid:3) (cid:24) (cid:0) = 1; (68) t t+1 t+1Q + @ &(Sh;S ) C (cid:13) ( t @S t h t t (cid:1) t h) 38

where C t+1 (cid:0) (cid:13) h (cid:3) = (cid:3) = (cid:12) ; and t+1 t;t+1 C (cid:18) t (cid:19) @ Sh &(Sh;S ) = M e ax (cid:31) t (cid:13) ; 0 : @Sh t t S (cid:0) t (cid:20) (cid:18) t (cid:19) (cid:21) 6.3 Bankers For ease of exposition, the description of the banker(cid:146)s problem in the text does not specify how the individual choice of bank(cid:146)s leverage a⁄ects its own probabilityofdefault. Thiswaspossiblebecause, asarguedinfootnote9, the indirect marginal e⁄ect of leverage on the objective of the (cid:133)rm, V ; through t the change in p is zero. Therefore the (cid:133)rst order conditions for the bank(cid:146)s t problem, equations (27) and (28); can be derived irrespectively of how the individual choice of bank(cid:146)s leverage a⁄ects its own probability of default. Wenowformalizetheargumentinfootnote9anddescribehowthedefault thresholds for individual banks vary with individual bank leverage. As will become clear in section 6.5 below, this analysis is key in order to study global optimality of the leverage choice selected by using the (cid:133)rst order conditions in the text, equations (27) and (28). As in the text, (cid:19) is a sunspot which takes on values of either unity or zero. We can then express the rate of return on bank capital Rb t+1 Z +(1 (cid:14))Q Rb = (cid:24) t+1 (cid:0) t+1 = Rb ((cid:24) ;(cid:19) ); t+1 t+1 Q t+1 t+1 t+1 t The individual bank defaults at date t+1 if and only if (cid:24) [Z +(1 (cid:14))Q ]sb Rb ((cid:24) ;(cid:19) ) Q sb 1 > t+1 t+1 (cid:0) t+1 t = t+1 t+1 t+1 t t ; R d R Q sb n t+1 t t+1 t t t (cid:0) or (cid:30) Rb ((cid:24) ;(cid:19) ) < R t : t+1 t+1 t+1 t+1 (cid:30) 1 t (cid:0) Let (cid:4)D ((cid:30)) be the set of capital quality shocks and sunspot realizations t+1 which make the individual bank with a leverage multiple of (cid:30) default and conversely let (cid:4)N ((cid:30)) be the set that leads to non-default at date t+1: t+1 (cid:30) 1 (cid:4)D ((cid:30)) = ((cid:24) ;(cid:19) ) Rb ((cid:24) ;;(cid:19) ) < (cid:0) R ((cid:30)) ; t+1 t+1 t+1 j t+1 t+1 t+1 (cid:30) t+1 (cid:26) (cid:27) 39

(cid:30) 1 (cid:4)N ((cid:30)) = ((cid:24) ;(cid:19) ) Rb ((cid:24) ;(cid:19) ) (cid:0) R ((cid:30)) : t+1 t+1 t+1 j t+1 t+1 t+1 (cid:21) (cid:30) t+1 (cid:26) (cid:27) where R ((cid:30))isthepromiseddepositinterestratewhentheindividualbank t+1 chooses (cid:30) which satis(cid:133)es the condition for the household to hold deposits: (cid:30) 1 = R ((cid:30)) (cid:3) dF + (cid:3) Rb ((cid:24) ;(cid:19) )dF : (69) t+1 t+1 t (cid:30) 1 t+1 t+1 t+1 t+1 t Z(cid:4)N t+1 ((cid:30)) (cid:0) Z(cid:4)D t+1 ((cid:30)) e e Here F ((cid:24) ;;(cid:19) ) denotes the distribution function of ((cid:24) ;(cid:19) ) condit t+1 t+1 t+1 t+1 tional on date t information: e Assume that the aggregate leverage multiple is given by (cid:30) . When the t individual banker chooses the leverage multiple (cid:30); which can be di⁄erent from (cid:30) ; the individual bank defaults at date t+1 if and only if t (cid:24) < (cid:24)I ((cid:30)) and (cid:19) = 0 t+1 t+1 t+1 where (70) (cid:24)I ((cid:30))Rb ((cid:24)I ((cid:30));0) = (cid:30) 1R ((cid:30)) t+1 t+1 t+1 (cid:0) (cid:30) t+1 or (cid:24) < (cid:24)R ((cid:30)) and (cid:19) = 1 t+1 t+1 t+1 where (71) (cid:24)R ((cid:30)) = sup (cid:24) s.t. (cid:24) Rb (cid:24) ;1 < (cid:30) 1R ((cid:30)) : t+1 t+1 t+1 t+1 t+1 (cid:0) (cid:30) t+1 n o Thusthesetofcapitalqualityshocksand(cid:0)sunspo(cid:1)tswhichmaketheindividual bank default (cid:4)D ((cid:30)) is t+1 (cid:24) < (cid:24)I ((cid:30)) and (cid:19) = 0 t+1 t+1 t+1 (cid:4)D ((cid:30)) = ((cid:24) ;(cid:19) ) or : (72) t+1 8 t+1 t+1 (cid:12) 9 (cid:12) (cid:24) < (cid:24)R ((cid:30)) and (cid:19) = 1 < (cid:12) t+1 t+1 t+1 = (cid:12) : (cid:12) (cid:12) (cid:24) t+1 (cid:21) (cid:24)I t+1 ((cid:30)) and (cid:19) t+1 = 0 ; (cid:4)N ((cid:30)) = ((cid:24) ;(cid:19) ) or : (73) t+1 8 t+1 t+1 (cid:12) 9 (cid:12) (cid:24) (cid:24)R ((cid:30)) and (cid:19) = 1 < (cid:12) t+1 (cid:21) t+1 t+1 = (cid:12) The behaviorof (cid:24)I ((cid:30)) is stra(cid:12)ightforwardandcanbeeasilycharacterized :t+1 (cid:12) ; from (70) under the natural assumption that Rb is increasing in the quality t+1 of capital at t+1: This gives: d(cid:24)I ((cid:30)) t+1 > 0; for (cid:30) (1; ) d(cid:30) 2 1 (74) lim (cid:24)I ((cid:30)) = 0: t+1 (cid:30) 1 # 40

The behavior of (cid:24)R ((cid:30)) is more complicated because, when a sunspot is t+1 observed, thefunctionRb (cid:24) ;1 thatdeterminesreturnsonbank(cid:146)sassets t+1 t+1 as a function of the capital quality is discontinuous around the aggregate run threshold (cid:24)R = (cid:24)R ((cid:30) (cid:22) ) (cid:0) : at th (cid:1) e threshold (cid:24)R asset prices jump from t+1 t+1 t t+1 liquidation prices up to their normal value (See Figure 5): lim Rb (cid:24) ;1 = Rb (cid:24)R ;0 > lim Rb (cid:24) ;1 : (75) t+1 t+1 t+1 t+1 t+1 t+1 (cid:24) (cid:24)R (cid:24) (cid:24)R t+1# t+1 t+1" t+1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) Thisimpliesthat,ifthecapitalqualityshockisattheaggregaterunthreshold (cid:24)R ;anincreaseinleveragefromthevaluethatmakestherecoveryrateequal t+1 to unity at liquidation prices, does not induce default as long as it is not so large that the bank becomes insolvent even at normal prices. By de(cid:133)nition of the run threshold (cid:24)R ; the value of leverage that makes t+1 the recovery rate at liquidation prices equal to unity is exactly the aggregate (cid:22) leverage (cid:30) ; that is t (cid:22) (cid:30) 1 t (cid:0) R (cid:30) (cid:22) = lim Rb (cid:24) ;1 : (cid:30) (cid:22) t t+1 t (cid:24) t+1" (cid:24)R t+1 t+1 t+1 (cid:0) (cid:1) (cid:0) (cid:1) ^ On the other hand, we let (cid:30) denote the value above which the bank defaults t attheaggregaterunthreshold(cid:24)R evenatnormalprices. Thisvaluesatis(cid:133)es t+1 ^ (cid:30) 1 t (cid:0) R (cid:30) ^ = Rb (cid:24)R ;0 ^ t+1 t t+1 t+1 (cid:30) t (cid:16) (cid:17) (cid:0) (cid:1) ^ (cid:22) and (75) implies that (cid:30) > (cid:30) . t t (cid:22) ^ For any value of leverage above the aggregate level (cid:30) but below (cid:30) , when t t a sunspot is observed, the bank defaults if and only if a system wide run happens. That is (cid:24)R ((cid:30)) is insensitive to variation in individual bank(cid:146)s t+1 leverage in this region: (cid:24)R ((cid:30)) = (cid:24)R ((cid:30) ) for (cid:30) [(cid:30) ;(cid:30) ]: t+1 t+1 t 2 t t For values of leverage above (cid:30) the bank is alwaybs insolvent even at non t liquidationprices wheneverdefaults, i:e: (cid:24)R ((cid:30)) = (cid:24)I ((cid:30)) for(cid:30) > (cid:30) ^ : When t+1 t+1 t (cid:30) is smaller than aggregate (cid:30) ; thbe bank is less vulnerable to the run so that t (cid:24)R ((cid:30)) < (cid:24)R . In the extreme when the leverage multiple equals unity, the t+1 t+1 individual bank is not vulnerable to run so that (cid:24)R (1) = 0. t+1 41

To summarize, the behavior of (cid:24)R ((cid:30)) can be characterized as follows: t+1 lim (cid:24)R ((cid:30)) = 0 t+1 (cid:30) 1 d(cid:24)R ((cid:30)) # t+1 > 0; for (cid:30) 1;(cid:30) d(cid:30) 2 t (76) (cid:24)R ((cid:30)) = (cid:24)R ; for (cid:30) [(cid:30) ;(cid:30) ] where (cid:24)I ((cid:30) ) = (cid:24)R t+1 t+1 2 t t (cid:0) (cid:1)t+1 t t+1 (cid:24)R ((cid:30)) = (cid:24)I ((cid:30)); for (cid:30) [(cid:30) ; ): t+1 t+1 2 t 1 b b See Figure A-1. b We can now rewrite the problem of the bank as in the text, but incorporating explicitly the dependence of the default and non default sets on the individual choice of leverage; as captured by (cid:4)D ((cid:30)) and (cid:4)N ((cid:30)) : t+1 t+1 max ((cid:22) (cid:30)+(cid:23) ); (77) t t (cid:30) subject to the incentive constraint: (cid:18)(cid:30) (cid:22) (cid:30)+(cid:23) ; (78) t t (cid:20) the deposit rate constraint obtained from (69) : 1 (cid:30) (cid:3) Rb ((cid:24) ;(cid:19) )dF R ((cid:30)) = (cid:0) (cid:30) (cid:0) 1 (cid:4)D t+1 ((cid:30)) t+1 t+1 t+1 t+1 t : (79) t+1 h i R (cid:3) dF (cid:4)N ((cid:30)) t+1 t e t+1 R (cid:22) and (cid:23) given by e t t (cid:22) = (cid:10) [Rb R ((cid:30))]dF (80) t t+1 t+1 (cid:0) t+1 t Z(cid:4)N t+1 ((cid:30)) e (cid:23) = (cid:10) R ((cid:30))dF : (81) t t+1 t+1 t Z(cid:4)N t+1 ((cid:30)) e and where (cid:4)D ((cid:30)) and (cid:4)N ((cid:30)) are given by (72) (73), (cid:24)I ((cid:30)) and (cid:24)R ((cid:30)) t+1 t+1 (cid:0) t+1 t+1 satisfy (70) (71). (cid:0) Using (80) (81) in the objective we can write the objective function as (cid:0) (cid:9) ((cid:30)) = (cid:10) [Rb ((cid:24) ;(cid:19) ) R ((cid:30))](cid:30)+R ((cid:30)) dF : (82) t t+1 f t+1 t+1 t+1 (cid:0) t+1 t+1 g t Z(cid:4)N t+1 ((cid:30)) e 42

Before proceeding with di⁄erentiation of the objective above, we introduce some notation that will be helpful in what follows. For any function (cid:22) ^ G (cid:30);(cid:24) ;(cid:19) and for any (cid:30) di⁄erent from (cid:30) or (cid:30) we let t+1 t+1 t t (cid:0) (cid:1) d @G (G) G((cid:30);(cid:24);(cid:19))dF ((cid:24);(cid:19)) ((cid:30);(cid:24);(cid:19))dF ((cid:24);(cid:19)) (cid:3) (cid:30)t (cid:17) d(cid:30) t (cid:0) @(cid:30) t " Z(cid:4)D t+1 ((cid:30)) # Z(cid:4)D t+1 ((cid:30)) e e d(cid:24)I ((cid:30)) d(cid:24)R ((cid:30)) = (1 (cid:0) {)G (cid:30);(cid:24)I t+1 ((cid:30));0 f t (cid:24)I t+1 ((cid:30)) t d + (cid:30) 1 +{G (cid:30);(cid:24)R t+1 ((cid:30));1 f t (cid:24)R t+1 ((cid:30)) t d + (cid:30) 1 : (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (83)(cid:1) denote the marginal e⁄ect of (cid:30) on G only through its e⁄ect on the default probability. Then we know that as long as G( ) is continuous at (cid:24)I ((cid:30)) and (cid:1) t+1 (cid:24)R ((cid:30)) we have t+1 d @G G((cid:30);(cid:24);(cid:19))dF ((cid:24);(cid:19)) ((cid:30);(cid:24);(cid:19))dF ((cid:24);(cid:19)) = (G) : d(cid:30) t (cid:0) @(cid:30) t (cid:0) (cid:3) (cid:30)t " Z(cid:4)N t+1 ((cid:30)) # Z(cid:4)N t+1 ((cid:30)) e e (cid:22) ^ Notice that we have not de(cid:133)ned (G) for (cid:30) = (cid:30) or (cid:30) = (cid:30) because (cid:3) (cid:30)t t t t t d(cid:24)R ((cid:30)) t+1 does not exist at that point. d(cid:30) (cid:22) ^ Di⁄erentiation of (82) at any value di⁄erent from (cid:30) and (cid:30) yields t t (cid:23) dR ((cid:30)) (cid:9) 0t ((cid:30)) = (cid:22) t (cid:0) ((cid:30) (cid:0) 1) R t t d + (cid:30) 1 (cid:0) (cid:10) t+1 [R t b +1 ((cid:24) t+1 ;(cid:19) t+1 ) (cid:0) R t+1 ((cid:30))](cid:30)+R t+1 ((cid:30)) (cid:3) (cid:30)t t+1 (cid:0) (cid:8) (cid:9)(cid:1) (cid:22) ^ Now notice that for (cid:30) [1;(cid:30) ) and (cid:30) > (cid:30) we have that the bank networth t t 2 is zero at both thresholds, that is [Rb ((cid:24)I ((cid:30));0) R ((cid:30))](cid:30)+R ((cid:30)) = 0 t+1 t+1 (cid:0) t+1 t+1 [Rb ((cid:24)R ((cid:30));1) R ((cid:30))](cid:30)+R ((cid:30)) = 0 t+1 t+1 (cid:0) t+1 t+1 implying (cid:10) [Rb ((cid:24) ;(cid:19) ) R ((cid:30))](cid:30)+R ((cid:30)) (cid:3) = 0: t+1 t+1 t+1 t+1 (cid:0) t+1 t+1 (cid:30)t (cid:22) ^ For (cid:30) (cid:0) (cid:30) ;(cid:8)(cid:30) we have that at the insolvency thresh(cid:9)o(cid:1)ld net worth is still t t 2 zero (cid:16) (cid:17) [Rb ((cid:24)I ((cid:30));0) R ((cid:30))](cid:30)+R ((cid:30)) = 0 t+1 t+1 (cid:0) t+1 t+1 while the run threshold is (cid:133)xed at the aggregate level d(cid:24)R ((cid:30)) t+1 = 0 d(cid:30) 43

so that again (cid:10) [Rb ((cid:24) ;(cid:19) ) R ((cid:30))](cid:30)+R ((cid:30)) (cid:3) = 0: t+1 t+1 t+1 t+1 (cid:0) t+1 t+1 (cid:30)t (cid:22) ^ Therefore we have that for all (cid:30) di⁄erent from (cid:30) and (cid:30) (cid:0) (cid:8) t t(cid:9)(cid:1) (cid:23) dR ((cid:30)) t t+1 (cid:9) ((cid:30)) = (cid:22) ((cid:30) 1) 0t t (cid:0) (cid:0) R d(cid:30) t+1 and by continuity of (cid:9) ((cid:30)) and (cid:22) ((cid:30) 1) (cid:23)t dRt+1((cid:30)) it can be extended t t (cid:0) (cid:0) Rt+1 d(cid:30) (cid:22) ^ to (cid:30) and (cid:30) as well. t t Then, as reported in the text, the (cid:133)rst order condition is (cid:30) = (cid:23)t ; if (cid:22)r > 0; and t (cid:22)r = (cid:18) (cid:0) (cid:22) 0 t ; if (cid:30) t < (cid:23)t ; (84) t t (cid:18) (cid:22) (cid:0) t (cid:23) dR ((cid:30) ) (cid:22)r = (cid:22) ((cid:30) 1) t t+1 t : (85) t t (cid:0) t (cid:0) R d(cid:30) t+1 t (Here we assume (cid:22) < (cid:18) which we will verify later). t As explained below in section 6:5; we make assumptions such that conditions (84) (85) characterize the unique global optimum for the bank(cid:146)s (cid:0) choice of leverage. Since these conditions don(cid:146)t depend on the individual net worth of a banker, every banker chooses the same leverage multiple and has the same Tobin(cid:146)s Q = (cid:22) (cid:30) +(cid:23) : (86) t t t t Thus from the discussion in the text, it follows that there is a system wide default if and only if Z +(1 (cid:14))Q (cid:30) 1 Rb ((cid:24) ;0) = (cid:24) t+1 (cid:0) t+1 < t (cid:0) R ((cid:30) ); or t+1 t+1 t+1 Q (cid:30) t+1 t t t Z +(1 (cid:14))Q (cid:30) 1 Rb ((cid:24) ;1) = (cid:24) t(cid:3)+1 (cid:0) (cid:3)t+1 < t (cid:0) R ((cid:30) ); t+1 t+1 t+1 Q (cid:30) t+1 t t t where R ((cid:30) ) is the aggregate promised deposit interest rate. t+1 t A systemic default occurs if and only if Z +(1 (cid:14))Q (cid:30) 1 (cid:24) < (cid:24)I , where (cid:24)I t+1 (cid:0) t+1 = t (cid:0) R ((cid:30) ); (87) t+1 t+1 t+1 Q (cid:30) t+1 t t t 44

or Z +(1 (cid:14))Q (cid:30) 1 (cid:24) < (cid:24)R and (cid:19) = 1, where (cid:24)R t(cid:3)+1 (cid:0) (cid:3)t+1 = t (cid:0) R ((cid:30) ): t+1 t+1 t+1 t+1 Q (cid:30) t+1 t t t (88) It follows that the probability of default at date t+1 conditional on date t information in the symmetric equilibrium is given by p t = F t ((cid:24)I t+1 )+{ F t ((cid:24)R t+1 ) (cid:0) F t ((cid:24)I t+1 ) : (89) (cid:2) (cid:3) The aggregate capital holding of the banking sector is proportional to the aggregate net worth as Q Sb = (cid:30) N : (90) t t t t The aggregate net worth of banks evolves as (cid:27)max (cid:24) (Z +(1 (cid:14))Q )Sb R D ; 0 +(cid:16)S if no default at t N = t t (cid:0) t t (cid:0) 1 (cid:0) t t (cid:0) 1 t (cid:0) 1 : t 8 (cid:8) (cid:9) 0 otherwise < (91) Banks:(cid:133)nance capital holdings by net worth and deposit, which implies D = ((cid:30) 1)N : (92) t t t (cid:0) 6.4 Market Clearing The market for capital holding implies S = Sb +Sh: (93) t t t The (cid:133)nal goods market clearing condition implies (cid:26)r Y = C +I + (cid:25) 2Y +G: (94) t t t t t 2 As is explained in the text, the monetary policy rule is given by 1 MC ’ y Rn = ((cid:25) )’ (cid:25) t : (95) t (cid:12) t " 1 (cid:18) (cid:0)" (cid:19) 45

The recursive equilibrium is given by a set of ten quantity variables (K ;S ;I ;L ;Y ;C ;Sh;Sb;D ;N ),sevenpricevariables(w ;Z ;MC ;(cid:25) ;R , t t t t t t t t t t t t t t t+1 Q ;Rn) and eight bank coe¢ cients ;(cid:22) ;(cid:23) ;(cid:22)r;(cid:30) ;p ;(cid:24)R ;(cid:24)I as a funct t t t t t t t t+1 t+1 tion of the four state variables M = (S ;Sb ;R D ;(cid:24) ) and a sunspot t (cid:0) t 1 t 1 t t 1 t (cid:1) variable(cid:19) ;whichsatis(cid:133)estwenty(cid:133)veequa (cid:0) tions,(cid:0)givenby (cid:0) : (56,57,58,61,62,63, t 64,65,66,67,68,70,71,80,81,84,f85,86,89,90,91,92,93,94,95). Here, the capital quality shocks follow a Markov process (cid:24) F (cid:24) (cid:24) and the t+1 t+1 t (cid:24) j sunspot is iid. with (cid:19) t = 1 with probability {: (cid:0) (cid:1) 6.5 On the Global Optimum for Individual Bank(cid:146)s Choice To study global optimality of the individual leverage choice selected by the (cid:133)rstorderconditionsin(84)weneedtoanalyzethecurvatureoftheobjective function (cid:9) ((cid:30)) in (82): t To do so we use (79) to derive an expression for dR(cid:22) and substitute it into d(cid:30) (85) to obtain (cid:10) dF (cid:4)N ((cid:30)) t+1 t (cid:9) ((cid:30)) = (cid:10) Rb dF 1 (cid:3) Rb dF t+1 ; 0t Z(cid:4)N t+1 ((cid:30)) t+1 t+1 t (cid:0) " (cid:0) Z(cid:4)D t+1 ((cid:30)) t+1 t+1 t # R (cid:4)N ((cid:30)) (cid:3) t+1 dF et t+1 e e (96) R Proceedingasinsection6.3todi⁄erentiate(96)foranyvalueof(cid:30)di⁄ereent (cid:22) ^ from (cid:30) and (cid:30) ; we get t t (cid:0) (cid:10) t+1 R t b +1 (cid:3) (cid:30)t + (cid:3) t+1 R t b +1 (cid:3) (cid:30)t (cid:1) R (cid:4) (cid:4) N t N t + + 1 1 ( ( (cid:30) (cid:30) ) ) (cid:10) (cid:3)t t + + 1 1 d d F F et t (cid:9) "((cid:30)) = (cid:0) (cid:1) (cid:0) + (cid:1) R t e 1 (cid:3) Rb dF (cid:4)N t+1((cid:30)) (cid:10)t+1dFt ((cid:10)t+1)(cid:3)(cid:30)t ((cid:3)t+1)(cid:3)(cid:30)t : (cid:0) (cid:4)D t+1 ((cid:30)) t+1 t+1 t R (cid:4)N t+1((cid:30)) (cid:3)t+1dF et (cid:20) (cid:4)N t+1((cid:30)) (cid:10)t+1dFt (cid:0) (cid:4)N t+1((cid:30)) (cid:3)t+1dFt (cid:21) h i R R R R (97) e e e e (cid:22) Note that for (cid:30) [1;(cid:30) ) t 2 (cid:30) 1 Rb (cid:24)I ((cid:30));(cid:19) = Rb (cid:24)R ((cid:30));1 = (cid:0) R ((cid:30)): t+1 t+1 t+1 t+1 (cid:30) t+1 (cid:0) (cid:1) (cid:0) (cid:1) (cid:22) ^ d(cid:24)R ((cid:30)) For (cid:30) (cid:30) ;(cid:30) we have t+1 = 0 which implies that for any function 2 t t d(cid:30) G (cid:24) ;(cid:19) (cid:16) (cid:17) t+1 t+1 (cid:0) (cid:1) d(cid:24)I ((cid:30)) (G) (cid:3) (cid:30)t = (1 (cid:0) {)G((cid:30);(cid:24)I t+1 ((cid:30));0)f t (cid:30);(cid:24)I t+1 ((cid:30)) t d + (cid:30) 1 for (cid:30) 2 (cid:30) (cid:22) t ;(cid:30) ^ t (98) (cid:16) (cid:17) (cid:0) (cid:1) 46

and also (cid:30) 1 Rb (cid:24)I ((cid:30));0 = (cid:0) R ((cid:30)) t+1 t+1 (cid:30) t+1 (cid:0) (cid:1) Then, we learn (cid:30) 1 (cid:10) t+1 R t b +1 (cid:3) (cid:30)t = ((cid:10) t+1 ) (cid:3) (cid:30)t (cid:1) (cid:0) (cid:30) R t+1 ((cid:30)) (cid:0) (cid:1) (cid:30) 1 (cid:3) t+1 R t b +1 (cid:3) (cid:30)t = ((cid:3) t+1 ) (cid:3) (cid:30)t (cid:1) (cid:0) (cid:30) R t+1 ((cid:30)): (cid:0) (cid:1) Substituting this back into (97) and using (79) to substitute for R ((cid:30)) t+1 we get (cid:9) "((cid:30)) = 1 (cid:4)N t+1 ((cid:30)) (cid:10) t+1 dF t ((cid:10) t+1 ) (cid:3) (cid:30)t ((cid:3) t+1 ) (cid:3) (cid:30)t (99) t (cid:30)R (cid:3) dF 2 (cid:10) dF (cid:0) (cid:3) dF 3 (cid:4)N ((cid:30)) t+1 et (cid:4)N ((cid:30)) t+1 t (cid:4)N ((cid:30)) t+1 t t+1 t+1 t+1 4 5 R R R for any (cid:30) di⁄erent from (cid:30) (cid:22) aend (cid:30) ^ :25 e e t t We assume that a bank that individually survives a systemic bank run by choosing its own leverage below the aggregate level (cid:30) behaves just like new t entrants during the panic: it stores its net worth and starts operating the period right after the crisis. Given that both leverage and spreads increase dramatically aftera crisis, newbanker(cid:146)s Tobin(cid:146)s Qis very high during a crisis so that (cid:10) (cid:24) ;1 (cid:3) (cid:24) ;1 t+1 t+1 > t+1 t+1 for (cid:24) = (cid:24)R ((cid:30)) < (cid:24)R : t+1 t+1 t+1 (cid:10) dF (cid:3) dF (cid:4)N ((cid:30) (cid:0) ) t+1 (cid:1)t (cid:4)N ((cid:30) (cid:0) ) t+1 (cid:1)t t+1 t+1 25Notic R e that ((cid:10) t+1 )(cid:3)(cid:30)t eand ((cid:3) R t+1 )(cid:3)(cid:30)t are not ceontinuous at (cid:30)(cid:22) t since, for instance d(cid:24)I ((cid:30)) d(cid:24)R ((cid:30)) (cid:30) li " m (cid:30)(cid:22) t ((cid:10) t+1 )(cid:3)(cid:30)t = (1 (cid:0) {)(cid:10) t+1 ((cid:24)I t+1 ;0)f t (cid:16) (cid:24)I t+1 (cid:17) t+ d(cid:30) 1 +{(cid:10) t+1 ((cid:24)R t+1 ;1)f t (cid:16) (cid:24)R t+1 (cid:17) " t+ d(cid:30) 1 # (cid:0) d(cid:24)I ((cid:30)) > (1 (cid:0) {)(cid:10) t+1 ((cid:24)I t+1 ;0)f t (cid:24)I t+1 t+ d(cid:30) 1 = (cid:30) lim (cid:30)(cid:22) ((cid:10) t+1 )(cid:3)(cid:30)t (cid:16) (cid:17) # t where d(cid:24)R t+1 ((cid:30)) is the left derivative of (cid:24)R ((cid:30)) at (cid:30)(cid:22) : This implies that (cid:9)"((cid:30)) does not d(cid:30) t+1 t exist ath(cid:30)(cid:22) . i(cid:0) t 47

By the same argument, we also have that (cid:10) (cid:24) ;0 (cid:3) (cid:24) ;0 t+1 t+1 > t+1 t+1 for (cid:24) = (cid:24)I ((cid:30)) < (cid:24)I t+1 t+1 t+1 (cid:10) dF (cid:3) dF (cid:4)N ((cid:30) (cid:0) ) t+1 (cid:1)t (cid:4)N ((cid:30) (cid:0) ) t+1 (cid:1)t t+1 t+1 R R Given this, equation (e99) and (83) implyethat the objective function of the banker is strictly convex in the region where leverage is below the aggregate (cid:22) (cid:22) level (cid:30) ; that is (cid:9) "((cid:30)) > 0 for (cid:30) [1;(cid:30) ); as long as the probability of a run t t t 2 is still positive, i:e: f (cid:24)R ((cid:30)) > 0. If, on the other hand, leverage is so low t t+1 that default is not possible, i.e. f (cid:24)R ((cid:30)) = f (cid:24)I ((cid:30)) = 0; the second (cid:0) (cid:1) t t+1 t t+1 derivative is zero. (cid:22) ^ (cid:0) (cid:1) (cid:0) (cid:1) For (cid:30) (cid:30) ;(cid:30) equations (98) and (99) imply that (cid:9) "((cid:30)) depends on t t t 2 the relative(cid:16)increas(cid:17)e in the marginal value of wealth of the banker and of the households only at the insolvency threshold, See Figure A1. Therefore in this case we have that the objective is convex, (cid:9) "((cid:30)) > 0; as long as t f (cid:24)I ((cid:30)) > 0: t t+1 Summing up we have: (cid:0) (cid:1) = 0 if (cid:30) [1;(cid:30) (cid:22) ) and f (cid:24)R ((cid:30)) = 0 = f (cid:24)I ((cid:30)) > 0 2 if (cid:30) t [1;(cid:30) (cid:22) t ) an t+ d 1 f (cid:24)R ((cid:30)) > t 0 t+1 8 2 t (cid:0) t (cid:1)t+1 (cid:0) (cid:1) (cid:9) t "((cid:30)) > = 0 if (cid:30) (cid:30) (cid:22) ;(cid:30) ^ and f (cid:24)I ((cid:30)) = 0 (100) > > > 2 t t t(cid:0) t+1 (cid:1) < > 0 if (cid:30) (cid:16)(cid:30) (cid:22) ;(cid:30) ^ (cid:17) and f (cid:0)(cid:24)I ((cid:30))(cid:1) > 0 2 t t t t+1 > > > > (cid:16) (cid:17) (cid:0) (cid:1) Equati:on (100) implies that the objective of the bank is weakly convex. Thus, to study global optimality it is su¢ cient to compare the equilibrium (cid:22) choice of leverage, (cid:30) ; to deviations to corner solutions. t When the incentive constraint is binding, i.e. (cid:9) (cid:30) = (cid:22)r > 0 at (cid:30) = 0t t t t (cid:23)t ; a bank cannot increase its own leverage above (cid:30) (cid:22) so that the only d (cid:18) (cid:0)ev (cid:22) t iation that we need to check is (cid:30) = 1: Therefore, t (cid:0) he (cid:1) c t ondition for global optimality in this case is: (cid:23) t (cid:9) (1) < (cid:9) : (101) t t (cid:18) (cid:22) (cid:18) (cid:0) t(cid:19) When the constraint is not binding, i.e. (cid:9) (cid:30) = (cid:22)r = 0 and (cid:30) < (cid:23)t ; 0t t t t (cid:18) (cid:22) an individual bank could deviate to either (cid:30) = 1 or (cid:30) = (cid:30)IC; where (cid:0) (cid:30)I t C (cid:0) (cid:1) t t 48

is the maximum level of leverage compatible with incentive constraints, i.e. (cid:9) ((cid:30)IC) = (cid:18)(cid:30)IC:Inthiscasegivenweakconvexityoftheobjective, theglobal t optimality condition is satis(cid:133)ed if and only if (cid:9) (1) = (cid:9) (cid:30) (cid:22) = (cid:9) (cid:30)IC : (102) t t t t t Notice that equation (100) implies t(cid:0)ha(cid:1)t the ab(cid:0)ove(cid:1)equality is satis(cid:133)ed if and only if the probability of default is zero for any feasible choice of leverage (cid:30) 1;(cid:30)IC which would result in a (cid:135)at objective function: 2 t Weverifynumericallythatcondition(101)issatis(cid:133)edintheneighborhood (cid:2) (cid:3) of the risk adjusted steady state, where the constraint is binding. Moreover, in our calibration, whenever the incentive constraint is not binding in equilibriumthe probability of insolvency is zero for any feasible choice of leverage above the equilibrium level, i.e. f (cid:24)I ((cid:30)) = 0 for (cid:30) ((cid:30) (cid:22) ;(cid:30)IC]; so that a t t+1 2 t t deviation by an individual bank to a higher level of leverage is never strictly (cid:0) (cid:1) preferred: Howevertheeconomydoesoccasionallytransittoextremestatesinwhich the constraint is binding but the probability of the run is high enough that equation (101) is violated and to states in which the constraint is slack and the probability of the run is positive thus violating (102). In such states a bank would gain by a deviation to (cid:30) = 1; See Figure A2. The only equilibrium in these cases would then be one in which a fraction of banks decrease their leverage in anticipation of a run while all of the others are against the constraint, i.e. there is no symmetric equilibrium. In order to focus on the symmetric equilibrium, we introduce a small cost to a bank to deviating to a position of taking no leverage (cid:30) = 1. This cost could re(cid:135)ect expenses involved in a major restructuring of the bank(cid:146)s portfolio. It could also re(cid:135)ect reputation costs associated with the bank(cid:146)s refusal to accept deposits in a givenperiodinordertosurvivearuninthesubsequentperiod. Inparticular, we posit that the objective of the bank is given by (cid:22) (cid:22) V (n ) = (cid:9) ((cid:30))n 1 (cid:28)(cid:30) for (cid:30) [1;(cid:30) ): t t t t t t (cid:0) 2 That is, a deviation of a bank th(cid:0)at reduc(cid:1)es leverage below the aggregate (cid:22) (cid:22) value (cid:30) entails a (cid:133)xed cost (cid:28)(cid:30) per unit of net worth. We check computat t (cid:22) tionally that the deviation is never pro(cid:133)table, i.e. (cid:9) (cid:30) > (cid:9) (1); in all of t t t our experiments for values of (cid:28) which are greater than or equal to 0:77%:26 (cid:0) (cid:1) 26The value of deviating can increase in very extreme cases but in a simulation of 100 thoushands periods it is still below 1.7% for 99 percent of the times. 49

Examining asymmetric equilibrium without such reputation cost is a topic of future research. 6.6 Computation It is convenient for computations to let the aggregate state of the economy be given by =(S ;N ;(cid:24) ;(cid:19) ): t t 1 t t t M (cid:0) Noticethatbanknetworthreplaces thespeci(cid:133)casset andliabilitypositionof banksinthenaturalstatethatwehaveusedsofar ~ =(S ;Sb ;D R (cid:22) ;(cid:24) ): To see that this state is su¢ cient to compute the M equ t ilibr t i (cid:0) u 1 m w t (cid:0)e 1 rew t (cid:0) ri 1 te t the t evolution of net worth, equation (91), forward. Using the de(cid:133)nition of the leverage multiple and the budget constraint of the banker we get that whenever there is no run at time t, so that N > 0; the evolution of net worth is t given by if there is no default : (cid:27)N (cid:30) (cid:24) Zt+1+(1 (cid:14))Qt+1 R +R +(cid:16)S t t t+1 Q (cid:0) t (cid:0) t+1 t t (cid:24) (cid:4)N ((cid:30) ) 8 t+1 2 t+1 t n (cid:16) (cid:17) o > > > > if there is a run : N = > > 0 : t+1 > > (cid:24) < (cid:24)R and (cid:19) = 1 > > t+1 t+1 t+1 < if banks are insolvent: > > (cid:16)S > > t (cid:24) < (cid:24)I and (cid:19) = 0 > > t+1 t+1 t+1 > > (103) > > : Otherwise, if a run has happened at time t so that N = 0; the evolution of t net worth is given by equation (33); which we report for convenience: S t 1 N t+1 = (cid:16)S t 1+(cid:27) (cid:0) : (104) S (cid:18) t (cid:19) We can then look for a recursive equilibrium in which each equilibrium variable is a function of and the evolution of net worth is given by a function t M N ; (cid:24) ;(cid:19) thatdependsontherealizationoftheexogenousshocks t+1 t t+1 t+1 M (cid:24) ;(cid:19) and satis(cid:133)es equations (103) and (104) above. t+1(cid:0) t+1 (cid:1) We use time iteration in order to approximate the functions (cid:0) (cid:1) # = Q( );C( ); ( );(cid:24)R ( );(cid:24)I ( );T( ;(cid:24) ;(cid:19)) M M M t+1 M t+1 M M 0 0 (cid:8) (cid:9) 50

where T( ;(cid:24) ;(cid:19)) is the transition law determining the stochastic evolution t 0 0 M of the state. The computational algorithm proceeds as follows: 1. Determine a functional space to use for approximating equilibrium functions. (We use piecewise linear). 2. FixagridofvaluesforthestateG Sm;SM 0;NM 1 4(cid:27)(cid:24);1+4(cid:27)(cid:24) (cid:26) (cid:2) (cid:2) (cid:0) 0;1 (cid:2) f g (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) 3. Set j = 0 and guess initial values for the equilibrium objects of interest on the grid # = Q ( );C ( ); ( );(cid:24)R ( );(cid:24)I ( );T/ ( ;(cid:24) ;(cid:19)) j j M j M j M t+1;j M t+1;j M j M 0 0 G n oM2 4. Assume that # has been found for i < M where M is set to 10000: Use i # to (cid:133)nd associated functions # in the approximating space, e.g. Q i i i is the price function that satis(cid:133)es Q ( ) = Q ( ) for each G. i i M M M 2 5. Compute all time t+1 variables in the system of equilibrium equations by using the functions # from the previous step; e.g. for each G i M 2 let Q = Q T/ ( ;(cid:24) ;(cid:19)) ; and then solve the system to get the t+1 i j M 0 0 implied # i+1 (cid:16) (cid:17) 6. Repeat 4 and 5 until convergence of # i 51

Fig. A1: Run and Insolvency thresholds Threshold capital quality 𝑅𝑅 ξ 𝑡𝑡+1 𝑅𝑅 ξ 𝑡𝑡+1 ϕ 𝐼𝐼 ξ 𝑡𝑡+1 ϕ Leverage: ф 0 1 ϕ � ϕ �

Fig. A2: Global conditions not satisfied without cost of deviation CASE A: Constraint binds with strong CASE B: Constraint slack with positive precautionary motive run probability Ψ (ф) Ψ (ф) t t Ψ (1) Ψ (1) t t Ψ (ф ) Ψ (ф ) t t t t (1-τ ф )Ψ (1) (1-τ ф )Ψ (1) t t t t IC IC 1 ф ф 1 ф = ф t t t t