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ArticleReview of Economic Studies (2020) 87, 240–288
doi:10.1093/restud/rdz032© The Author(s) 2019. Published by Oxford University Press on behalf of The Review of Economic Studies Limited.
Advance access publication 29 May 2019

A Macroeconomic Model withFinancial Panics
MARK GERTLER
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New York University

NOBUHIRO KIYOTAKI
Princeton University

andANDREA PRESTIPINO
Board of Governors of the Federal Reserve
First version received December 2017; Editorial decision January 2019; Accepted May 2019 (Eds.)
This article incorporates banks and banking panics within a conventional macroeconomic framework
to analyse the dynamics of a financial crisis of the kind recently experienced. We are particularly interested
in characterizing the sudden and discrete nature of banking panics as well as the circumstances thatmake 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 affects real activity both
qualitatively and quantitatively. In addition to modelling the financial collapse, we also introduce a belief
driven credit boom that increases the susceptibility of the economy to a disruptive banking panic.
Key words: Financial panic, Great recession, Credit boom.
JEL Codes: E23, E32, E44, G01, G21, G33

1. INTRODUCTION
As both Bernanke (2010, 2018) and Gorton (2010) argue, at the heart of the recent financial crisis
was a series of bank runs that culminated in the precipitous demise of a number of major financial
institutions. During the period where the panics were most intense in October 2008, all the major
investment banks effectively failed, the commercial paper market froze, and the Reserve PrimaryFund (a major money market fund) experienced a run. The distress quickly spilled over to thereal 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 8%
annual rate, driven mainly by a nearly 40% drop in investment expenditures. Also relevant is that
this sudden discrete contraction in financial and real economic activity occurred in the absenceof any apparent large exogenous disturbance to the economy.
In this article, we incorporate banks and banking panics within a conventional macroeconomic
framework—a New Keynesian model with investment. Our goal is to develop a model where it
The editor in charge of this paper was Veronica Guerrieri.
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Downloaded from https://academic.oup.com/restud/article-abstract/87/1/240/5505450 by guest on 14 May 2020is possible to analyse both qualitatively and quantitatively the dynamics of a financial crisis of
the kind recently experienced. We are particularly interested in characterizing the sudden anddiscrete nature of banking panics as well as the circumstances that make the economy vulnerable
to such panics in some instances but not in others. Having a conventional macroeconomic modelallows us to study the channels by which the crisis affects aggregate economic activity and theeffects of various policies in containing crises.
While our baseline framework focuses on capturing a financial collapse, we then extend
the analysis to capture a credit boom that leads to a crisis. As Schularick and Taylor (2012)and many others have emphasized, financial crises are typically preceded by credit booms
that lay the seeds for the subsequent collapse. In the spirit of Geanakoplos (2010) andBordalo Gennaioli and Schleifer (2017), we model the boom by introducing optimistic beliefs.
We then characterize how the boom increases the susceptibility of the economy to a disruptivebanking panic.
Our article fits into a lengthy literature aimed at adapting core macroeconomic models toaccount for financial crises.1 Much of this literature emphasizes the role of balance sheets inconstraining borrowers from spending when financial markets are imperfect. Because balance
sheets tend to strengthen in booms and weaken in recessions, financial conditions work to amplify
fluctuations 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 ofthe crisis also for non-financial firms. Nonetheless, as Mendoza (2010), He and Krishnamurthy(2017) and Brunnermeier and Sannikov (2014) have emphasized, these models do not capture
the highly non-linear aspect of the crisis. Although the financial mechanisms in these papers tend
to amplify the effects of disturbances, they do not easily capture sudden discrete collapses. Nor
can they capture the run-like behaviour associated with financial 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. partiallyilliquid long-term assets funded by short-term debt, opens up the possibility of runs. Most of themodels in this literature, though, are partial equilibrium and highly stylized (e.g. three periods).
They are thus limited for analysing the interaction between financial and real sectors.
Our article builds on our earlier work—Gertler and Kiyotaki (GK, 2015) and
Gertler Kiyotaki and Prestipino (GKP, 2016)—which analysed bank runs in an infinite
horizon endowment economy. These papers characterize runs as self-fulfilling rollover crises,
following the Calvo (1988) and Cole and Kehoe (2000) models of sovereign debt crises. Both
GK and GKP emphasize the complementary nature of balance sheet conditions and bank runs.
Balance sheet conditions affect 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 non-linear natureof a collapse: when bank balance sheets are strong, negative shocks do not push the financial
system to the verge of collapse. When they are weak, the same size shock leads the economy intoa 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 effect of
the financial collapse on investment, output, and employment. In particular, we proceed to showthat a calibrated version of our model is capable of capturing the dynamics of key financial and1. See Gertler and Kiyotaki (2011) and Brunnermeier et al. (2013) for recent surveys.
2. Some recent examples where self-fulfilling financial crises can emerge depending on the state of the economy
include Benhabib and Wang (2013), Bocola and Lorenzoni (2017), Farhi and Maggiori (2018), and Perri and Quadrini(2018). For further attempts to incorporate bank runs in macro models, see Angeloni and Faia (2013), Cooper and Ross
(1998), Martin Skeie and Von Thadden (2014), Robatto (2019), and Uhlig (2010) for example.

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real variables over the course of the recent crisis. Another important difference from our earlier
work is that we also model the credit boom preceding the crisis.
There are several other strands of literature relevant to our article. The first is the recent workon occasionally binding borrowing constraints as a source of non-linearity in financial crises suchas Mendoza (2010), He and Krishnamurthy (2017), and Brunnermeier and Sannikov (2014). Our
approach also allows for occasionally binding financial constraints and precautionary saving.
However, in quantitative terms, bank runs provide the major source of non-linearity. Second,
there is a literature on credit booms and busts, including Boissay Collard and Smets (2016) andBordalo Gennaioli and Schleifer (2017). We differ mainly by explicitly modelling banking panics
and by integrating the analysis within a reasonably conventional quantitative macroeconomic
framework.
Section 2 presents the behaviour of bankers and workers, the sectors where the novel features ofthe model are introduced. Section 3 describes the features that are standard in the New Keynesianmodel: the behaviour of firms, price setting, investment, and monetary policy. Section 4 describes
the calibration and presents a variety of numerical exercises designed to illustrate the main featuresof the model, including how the model can capture the dynamics of the recent financial crisis.
We also show how it is possible to generate a credit boom that increases the financial fragility ofthe economy.
2. MODEL: OUTLINE, HOUSEHOLDS, AND BANKERS
The baseline framework is a standard New Keynesian model with investment.3 In contrast to
the conventional model, each household consists of bankers and workers. Bankers specialize inmaking loans and thus intermediate funds between households and productive capital. Households
may also make these loans directly, but they are less efficient in doing so than bankers.4 On theother hand, bankers may be constrained in their ability to raise external funds and also may
be subject to runs. The net effect is that the cost of capital will depend on the endogenously
determined flow of funds between intermediated and direct finance.
We distinguish between capital at the beginning of period t, Kt , and capital at the end of theperiod, St . Capital at the beginning of the period is used in conjunction with labour 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:
St = (It )+(1−δ)Kt ,(2.1)
where δ is the rate of depreciation. The quantity of newly produced capital, (It ), is an increasing
and concave function of investment expenditure, It , to capture convex adjustment costs.
A firm wishing to finance new investment as well as old capital, issues a state-contingent claimon the earnings generated by the capital. Let St be the total number of claims (effectively equity)
outstanding at the end of period t (one claim per unit of capital), Stb be the quantity intermediatedby bankers, and Sth be the quantity directly held by households. Then we have:
Stb +Sth = St .

(2.2)

Both the total capital stock and the composition of financing are determined in equilibrium.
3. For simplicity, we do not distinguish between non-residential and residential investment, though the latterclearly played an important role in the crisis. It should become obvious how to extend the model to make this distinction.
4. As Section 2.2 makes clear, technically it is the workers within the household that are left to manage any direct
finance. But since these workers collectively decide consumption, labour, and portfolio choice on of behalf the household,we simply refer to them as the “household” going forward.

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The capital stock entering the next period, Kt+1 , differs from St due to a multiplicative “capitalquality” shock, ξt+1 , that randomly transforms the units of capital available at t +1:
Kt+1 = ξt+1 St .

(2.3)

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The shock ξt+1 provides an exogenous source of variation in the return to capital.5
To capture that households are less efficient than bankers in handling investments, we assume
that they suffer a management cost that depends on the share of capital they hold, Sth /St . The
management cost reflects their disadvantage relative to bankers in evaluating and monitoring
capital projects. The cost is in utility terms and takes the following piece-wise form:  hς(Sth ,St ) =

χ
2

2
St
Sh
−γ
St , if Stt > γ > 0
St

(2.4)

0, otherwise

with χ > 0.6
For Sth /St ≤ γ there is no efficiency cost: households are able to manage a limited fraction ofcapital as well as bankers. As the share of direct finance exceeds γ , the efficiency cost ς (·) is
increasing and convex in Sth /St . In this region, limitations on the household’s ability to manage
capital become relevant. The convex form implies that the marginal efficiency losses rise withthe size of the household’s direct capital holdings, capturing limits on its capacity to handle
investments.
We assume that the efficiency cost is homogenous in Sth and St to simplify the computation.
Making the marginal efficiency cost linear in the share Sth /St reduces the non-linearity in themodel. An informal motivation is that, as the capital stock St increases, the household has more
options from which to select investments that it is better able to manage, which works to dampen
the marginal efficiency cost.
Given the efficiency costs of direct household finance, absent financial frictions banks willintermediate at least the fraction 1−γ of the capital stock. However, when banks are constrainedin their ability to obtain external funds, households will directly hold more than the share γ ofthe capital stock. As the constraints tighten in a recession, as will happen in our model, the shareof capital held by households will expand. The reallocation of capital holdings from banks to lessefficient 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 firesale ofassets from banks to households will lead to a sharp rise in the cost of capital, leading to a deep
contraction in investment and output.
In the rest of this section, we characterize the behaviour of households and bankers which are
the non-standard parts of the model.

5. The advantage of the capital quality shock is that it can introduce an exogenous source of variation in the market
value of bank assets, which will play a role in making banks vulnerable to panics. The disadvantage is that the shock has
a direct supply effect on output by affecting the quantity of capital. However, we show that this direct effect is negligible
for the size of shocks we consider. In addition, we also consider how beliefs about capital quality affect market values,
which eliminates the direct supply effect altogether.
6. For a deeper model of the costs that non-experts face in financial markets see Kurlat (2016). Our assumptionthat households intermediation costs are non-pecuniary is made for simplicity only. All of our results go through if we
assume that households’ intermediated capital is less productive, as in e.g. Brunnermeier and Sannikov (2014), as longas productivity losses increase with the quantity of capital intermediated by households.

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We formulate this sector in a way that allows for financial intermediation yet preserves thetractability of the representative household setup. In particular, each household is a family thatconsists of a continuum of members with measure unity. Within the household there are 1−fworkers and f bankers. Workers supply labour and earn wages for the household. Each banker
manages a bank and transfers (non-negative) dividends back to the household. Within the family
there is perfect consumption sharing.
In order to preclude a banker from retaining sufficient earnings to permanently relax any
financial constraint, we assume the following: in each period, with i.i.d. probability 1−σ , abanker exits. Upon exit, it gives all its accumulated earnings to the household. This stochastic
exit in conjunction with the payment to the household upon exit is in effect a simple way to model
dividend payouts.7
After exiting, a banker returns to being a worker. To keep the population of each occupation
constant, each period, (1−σ )f workers become bankers. At this time, the household provides
each new banker with an exogenously given initial equity stake in the form of a wealth transfer, et .
The banker receives no further transfers from the household and instead operates at arms 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 non-contingent gross real rate of return Rt+1
in the absence of default. In the event of default at t +1, depositors receive the fraction xt+1 of
the promised return, where the recovery rate xt+1 ∈ [0,1) is the value of bank assets per unit ofpromised deposit obligations.
There are two reasons the bank may default: first, a sufficiently negative return on its portfolio
may make it insolvent. Second, even if the bank is solvent at normal market prices, the bank’s
creditors may “run” forcing the bank to liquidate assets at firesale prices. We describe each ofthese possibilities in detail in the next section. Let pt be the probability that the bank defaults inperiod t +1. Given pt and xt+1 , we can express the gross rate of return on the deposit contract
Rt+1 as
Rt+1 with probability 1−pt
Rt+1 =.
(2.5)
xt+1 Rt+1 with probability pt
Similar to the Cole and Kehoe (2000) model of sovereign default, a run in our model willcorrespond to a panic failure of households to roll over deposits. This contrasts with the “early
withdrawal” mechanism in the classic 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 thesame pro rata share of output in the event of default, whether it be due to insolvency or a run.
Let Ct be consumption, Lt labour supply, and β ∈ (0,1) the household’s subjective discountfactor. As mentioned before, ς (Sth ,St ) is the household utility cost of direct capital holding Sth ,where the household takes the aggregate quantity of claims St as given. Then household utility
Ut is given by
∞


1−γh

(Lτ )1+ϕτ −t (Cτ )hβ−−ς (Sτ ,Sτ ) .
Ut = Et
1−γh
1+ϕ
τ =t
Let Qt be the relative price of capital, Zt the rental rate on capital, wt the real wage rate, Tt
lump-sum taxes, and t dividend distributions net transfers to new bankers, all of which the

7. As Section 2.2 makes clear, because of the financial constraint, it will always be optimal for a bank to retain
earnings until exit.

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household takes as given. Then the household chooses Ct ,Lt ,Sth , and deposits Dt to maximize
expected utility subject to the budget constrainth.
Ct +Dt +Qt Sth = wt Lt −Tt + t +Rt Dt−1 +ξt [Zt +(1−δ)Qt ]St−1(2.6)

The first order condition for labour supply is given by:
wt λt = (Lt )ϕ ,(2.7)

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where λt ≡ (Ct )−γh denotes the marginal utility of consumption.
The first order condition for bank deposits takes into account the possibility of default and isgiven by
1 = [(1−pt )Et t+1 |no def +pt Et t+1 xt+1 |def ]·Rt+1 ,(2.8)
where Et (· | no def ) (and Et (· | def )) are expected value of · conditional on no default (and default)
at date t +1. The stochastic discount factor t+1 satisfies
t+1 = β

λt+1
.
λt

(2.9)

Observe that the promised deposit rate Rt+1 that satisfies equation (2.8) depends on the default
probability pt as well as the recovery rate xt+1 .8
Finally, the first-order condition for capital holdings is given by⎡⎤

Zt+1 +(1−δ)Qt+1 ⎥⎢Et ⎣t+1 ξt+1
⎦ = 1,∂ς (Sth ,St )
Qt +
/λ
t
h

(2.10)∂St

where∂ς (Sth ,St )
∂Sth

 



Sth
/λt = Max χ−γ /λt , 0
St(2.11)

is the household’s marginal cost of direct capital holding.
The first-order condition given by (2.10) 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 heldby households above the threshold γ since the efficiency cost ς (Sth ,St ) is increasing and convex.
As will become clear, in a panic run banks will sell all their securities to households, leading toa sharp contraction in asset prices. The severity of the drop will depend on the curvature of theefficiency cost function given by (2.4), which controls asset market liquidity in the model.
2.2.

Bankers

The banking sector we characterize corresponds best to the shadow banking system which wasat the epicenter of the financial instability during the Great Recession. In particular, banks inthe model are completely unregulated, hold long-term securities, issue short-term debt, and as aconsequence are potentially subject to runs.
8. Notice 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.

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ArticleREVIEW OF ECONOMIC STUDIESτ =t+1= Et t+1 [(1−σ )nt+1 +σ Vt+1 ] ,Downloaded from https://academic.oup.com/restud/article-abstract/87/1/240/5505450 by guest on 14 May 2020

2.2.1. Bankers optimization problem. Each banker manages a financial intermediarywith the objective of maximizing the expected utility of the household. Bankers fund capital
investments by issuing short-term deposits dt to households as well as by using their own equity,or net worth, nt . Due to financial market frictions described later, bankers may be constrained intheir ability to obtain deposits.
So long as there is a positive probability that the banker may be financially constrained atsome point in the future, it will be optimal for the banker 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’s objective as to maximize the discounted expectedvalue of net worth upon exit. Given that σ is the survival probability and given that the banker
t,τ = β τ −t λτ /λt to discountuses the household’s intertemporal marginal rate of substitution 
future payouts, we can express the objective of a continuing banker at the end of period t as⎡⎤∞t,τ (1−σ )σ τ −t−1 nτ ⎦V t = Et ⎣
(2.12)

where (1−σ )σ τ −t−1 is probability of exiting at date τ, and nτ is terminal net worth if the banker
exits at τ.
During each period t, a continuing bank (either new or surviving) finances asset holdings Qt stb
with newly issued deposits and net worth:Qt stb = dt +nt .

(2.13)

We assume that banks can only accumulate net worth by retained earnings and do not issue newequity. 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:bnt = Rtb Qt−1 st−1
−Rt dt−1 ,(2.14)
where Rtb is the gross rate of return on capital intermediated by banks, given by:
Rtb = ξt

Zt +(1−δ)Qt.
Qt−1(2.15)

So long as nt is strictly positive the bank does not default. In this instance, it pays its creditorsb , is below the promised repayments tothe promised rate Rt . If the value of assets, Rtb Qt−1 st−1
depositors R̄t dt−1 (due to either a run or simply a bad realization of returns), nt goes to zero andthe bank defaults. It then pays creditors the product of recovery rate xt and Rt , where xt is givenby:b
Rtb Qt−1 st−1
xt =
< 1.
(2.16)Rt dt−1
For each new banker at t, net worth simply equals the start-up equity et it receives from thehousehold:nt = et .
(2.17)
To motivate a limit on a bank’s ability to issue deposits, we introduce the following moral
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θ Qt stb ≤ Vt .

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the period, the banker decides whether to operate “honestly” or to divert assets for personal use.
To operate honestly means holding assets until the payoffs are realized in period t +1 and thenmeeting deposit obligations. To divert means selling a fraction θ 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 realizationof uncertainty at t +1. Further, to remain undetected, he can only sell up to a fraction θ of theassets. The cost to the banker of the diversion is that the depositors force the intermediary into
bankruptcy at the beginning of the next period.9
The banker’s decision on whether or not to divert funds at t boils down to comparing the
franchise value of the bank Vt , which measures the present discounted value of future payouts fromoperating honestly, with the gain from diverting funds, θ Qt stb . In this regard, rational depositors
will not lend to the banker if he has an incentive to cheat. Accordingly, any financial arrangement
between the bank and its depositors must satisfy the incentive constraint:(2.18)To characterize the banker’s optimization problem it is useful to let φt denote the bank’s ratioof assets to net worth, Qt stb /nt , which we will call the “leverage multiple”. Then, combining thebalance sheet constraint (2.13) and the flow of funds constraint (2.14) yields the expression for
the evolution of net worth for a surviving bank that does not default as:b
nt+1 = [(Rt+1−Rt+1 )φt +Rt+1 ]nt ,(2.19)

where we used (2.5) to substitute the promised rate R̄t+1 for the deposit rate Rt+1 in the case of
no default.
Using the evolution of net worth equation (2.19), we can write the franchise value of the bank
(2.12) as(2.20)
Vt = (μt φt +νt )nt ,whereb
μt = (1−pt )Et {t+1 (Rt+1−Rt+1 ) | no def }(2.21)

νt = (1−pt )Et {t+1 Rt+1 | no def }(2.22)

witht+1 = t+1 (1−σ +σ ψt+1 ), andψt+1 ≡

Vt+1
≥ 1.
nt+1

The variable μt is the expected discounted excess return on banks assets relative to deposits,and νt is the expected discounted cost of a unit of deposits. Intuitively, μt φt is the excess returnthe bank receives from having an additional unit of net worth (taking into account the bank’s
ability to use leverage), while νt is the cost saving from substituting equity finance for deposit
finance.
9. We 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.

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ArticleREVIEW OF ECONOMIC STUDIESψt = max (μt φt +νt ),φt

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Notice that the bank uses the stochastic discount factor t+1 to value returns in t +1. t+1 isthe bank’s discounted shadow value of a unit of net worth at t +1, averaged across the likelihood ofexit and the likelihood of survival. If the bank exits, which occurs with probability 1−σ, it returns
its net worth to the household. Accordingly, in this case the value of net worth is simply givenby the household’s marginal utility of consumption. If instead the bank survives, which occurs
with probability σ , the unit value of bank net worth is larger than the household’s marginal utilityof consumption, i.e. ψt+1 exceeds unity. This is because a surviving bank can use the additional
unit of net worth to expand its capital holdings rather than to pay dividends to the households. As
will become clear, the bank will always prefer to do so since financial market frictions prevent
perfect arbitrage by banks and thus capital investment is always strictly profitable at the margin.
Therefore, we can think of ψt+1 in the expression for t+1 as the bank’s “Tobin’s Q ratio”, i.e.
the shadow value of a unit of net worth within the bank.
The banker’s optimization problem is then to choose the leverage multiple φt to solve(2.23)

subject to the incentive constraint (obtained from equations (2.18) and (2.20)):
θφt ≤ μt φt +νt ,and the deposit rate constraint (obtained from equations (2.8) and (2.16)):
−1
Rt+1 = (1−pt )Et (t+1 | no def )+pt Et (t+1 xt+1 | def ),(2.24)(2.25)

where xt+1 is the following function of φt :b

xt+1 =

φt Rt+1,
φt −1 Rt+1

and μt and νt are given by (2.21) and (2.22).
Note that in addition to the direct effect of the choice of leverage φt on the objective and theconstraints, there is an indirect effect that could arise via the impact of leverage on the default
probability pt . However, this indirect effect on the banks franchise value is zero on net and hence
does not show up in the bank’s first order conditions.10 Therefore, for ease of exposition, we
relegate to the Appendix the explicit description of how an individual banker’s choice of leverage
influences its own run and insolvency thresholds.
Notice also that since individual bank net worth does not appear in the bank optimization
problem, the optimal choice of φt is independent of nt . This implies that the default probability,
pt , the promised rate on deposits, R̄t+1 , and the bank’s Tobin’s Q ratio are all independent frombank’s specific characteristics.
Since the franchise value of the bank Vt is proportionate to nt by a factor that only depends
on the aggregate state of the economy (see equation (2.20)), a bank cannot operate with zero net
worth. In this instance Vt falls to zero, implying that the incentive constraint (2.18) would always
be violated if the bank tried to issue deposits. That banks require positive equity to operate is vitalto 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.
10. This is because at the default threshold, the bank’s assets are exactly equal to its liabilities, so that nt+1 = 0
and Vt+1 = 0. Thus, a small shift in the probability mass from the no-default to the default state has no impact on Vt .
Similarly, the indirect effect of φt on the promised deposit rate Rt through pt is zero, since the recovery rate xt is unity atthe borderline of default. Important to the argument is the absence of deadweight losses associated with default.

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249

2.2.2. Banker’s decision rules. We derive the optimal portfolio choice of banks byrestricting attention to a symmetric equilibrium in which all banks choose the same leverage.11
Let μrt be the expected discounted marginal return to increasing the leverage multiple φt
μrt =

dψt
νt dRt+1 (φt )= μt −(φt −1)< μt .
dφt
dφt
Rt+1

(2.26)

μrt = 0, if φt <

νt
.
θ −μt

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The second term on the right of equation (2.26) reflects the effect of the increase in Rt+1 thatarises as the bank increases φt . An increase in φt reduces the recovery rate, forcing Rt+1 up tocompensate depositors, as equation (2.25) suggests. The term (φt −1)νt /Rt+1 then reflects the
reduction in the bank franchise value that results from a unit increase in Rt+1 . Due to the effecton Rt+1 from expanding φt , the marginal return μrt is below the average excess return μt .
The solution for φt depends on whether the incentive constraint binds. For the incentive
constraint to bind μrt must be positive at the optimum. With μrt positive, the bank will want toborrow to acquire more assets but the incentive constraint will keep it from doing so by placing
a limit on its leverage. Thus, if μrt > 0 at the optimum, (2.24) holds with equality, implying thefollowing solution for φt :νt
φt =, if μrt > 0.
(2.27)
θ −μt
The constraint (2.27) limits the leverage multiple to the point where the bank’s gain from diverting
funds per unit of net worth θ φt is exactly balanced by the cost per unit of net worth of losing the
franchise value, which is measured by ψt = μt φt +νt . Note that μt (as given by equation (2.21))b −R
tends to move countercyclically since the excess return on bank capital Et Rt+1
t+1 widens as
the borrowing constraint tightens in recessions. As a result, φt tends to move countercyclically.
As we show later, the countercyclical movement in φt contributes to making bank runs more
likely in bad economic times.12
Conversely, if the marginal return to increasing the leverage multiple becomes zero before the
incentive constraint binds, the bank chooses leverage as follows:(2.28)

Even though the constraint does not bind, in this case, the bank will still choose to limit theleverage so long as there is a possibility that the incentive constraint could bind in the future.
In this instance, as in Brunnermeier and Sannikov (2014) and He and Krishnamurthy (2017),
banks have a precautionary motive for scaling back their respective leverage multiples.13 The
11. In this section, we describe the leverage choice of banks as determined by the first-order conditions of thebanks’ optimization problem. The Appendix discusses the assumptions under which first order conditions actually selecta global optimum for the bank’s problem, ensuring that a symmetric strategy equilibrium exists.
12. In the data, net worth of our model corresponds to the mark-to-market difference between assets and liabilitiesof the bank balance sheet. It is different from the book value often used in the official report, which is slow in reacting tomarket conditions. Also bank assets here are securities and loans to the non-financial sector, which exclude those to otherfinancial intermediaries. In the data, the net mark-to-market leverage multiple of the financial intermediation sector—theratio of securities and loans to the non-financial sector to the net worth of the aggregate financial intermediaries—tends tomove countercyclically, even though the gross leverage multiple—the ratio of book value total assets (including securities
and loans to the other intermediaries) to the net worth of some individual intermediaries may move procyclically. While
Adrian and Shin (2010) show book leverage is procyclical for investment bankers, He Khang and Krishnamurthy (2010)
and He Kelly and Manela (2017) show market leverage is countercyclical.
13. One difference of our model from these papers is that, because default occurs in equilibrium, the bank’s leverage
affects the promised deposit rate and the cost of funds. This effect provides an additional motive for the bank to reduce
its leverage multiple as implied by the fact that when the constraint is not binding μt > μrt = 0.

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precautionary motive is reflected by the presence of the discount factor t+1 in the measure of
the discounted excess return μt . The discount factor t+1 , which reflects 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 valueof net worth is high.

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2.2.3. Aggregation of the financial sector absent default. We now characterize theaggregate financial sector during periods where banks do not default. Given that the optimal
leverage multiple φt is independent of bank-specific factors, individual bank portfolio decisions,stb and dt , are homogenous in net worth. Accordingly, we can sum across banks to obtain thefollowing relation between aggregate bank asset holdings Qt Stb and the aggregate quantity of net
worth Nt in the banking sector:
Qt Stb = φt Nt .
(2.29)The evolution of Nt depends on both the retained earnings of bankers that survived fromthe previous period and the injection of equity to new bankers. For technical convenience again
related to computational considerations, we suppose that the household transfer et to a eachnew banker is proportionate to the stock of capital at the end of the previous period, St−1 , withζ14et = (1−σ)f St−1 . Aggregating across both surviving and entering bankers yields the following
expression for the evolution of net worth
Nt = σ [(Rtb −Rt )φt−1 +Rt ]Nt−1 +ζ St−1 .

(2.30)

The first term is the total net worth of bankers that operated at t −1 and survived until t. The
second, ζ St−1 , is the total start-up equity of entering 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 andthe condition for a bank run equilibrium to exist, we characterize the overall default probability.
2.3.1. Conditions for a bank run equilibrium. As in Diamond and Dybvig (1983), theruns we consider are runs on the entire banking system and not an individual bank. A run on
an individual bank will not have aggregate effects as depositors simply shuffle their funds fromone bank to the others in the system.15 As we noted earlier, though, we differ from Diamond andDybvig in that runs reflect a panic failure to roll over deposits as opposed to early withdrawal.
Consider the behaviour of a household that acquired deposits at t −1. Suppose further that thebanking 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-fulfilling“run” equilibrium exists if and only if the household correctly believes that in the event all otherdepositors 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 satisfied if and only if theliquidation reduces the value of bank assets below promised obligations to depositors, driving
14. Here, we value capital at the steady state price Q = 1. If we use the market price instead, the financial accelerator
would be enhanced but not significantly.
15. Key to this argument is that banks are atomistic so that a run on a single bank has no effect on the price ofcapital.

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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.16 If instead bank net
worth is positive even at liquidation prices, banks would be able to offer a profitable deposit
contract to an individual household deciding to roll over. In this instance, a panic equilibrium
would not exist.
The condition for a bank run equilibrium to exist at t, accordingly, is that in the event ofliquidation following a run, bank net worth goes to zero. Recall that earlier we defined the depositor
recovery rate, xt , as the ratio of 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,xt∗ , is less than unity:=bξt [(1−δ)Qt∗ +Zt∗ ]St−1(2.31)Rt Dt−1
Rtb∗
Rt·

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xt∗ =

φt−1
< 1,
φt−1 −1

where Qt∗ is the asset liquidation price, Zt∗ is rental rate, and Rtb∗ is the return on bank assets, allconditional on a run. Since the liquidation price Qt∗ is below the normal market price Qt , a run
may occur even if the bank is solvent at normal market prices. Moreover, as we will show below,
when either deteriorating economic conditions or bankers optimistic beliefs cause bank leverageRb∗φt−1 to increase substantially, even relatively small new disturbances which decrease t can
Rtopen up the possibility of a banking panic.
2.3.2. The liquidation price. Key to the condition for a bank run equilibrium is the
behaviour of the liquidation price Qt∗ . A depositor panic at t induces all the existing banks toliquidate their assets by selling them to households. Accordingly in the wake of the run:
Sth = St .

(2.32)

To obtain Qt∗ , we iterate the household Euler equation:

Qt∗ = Et

⎧
∞
⎨ 
⎩

⎛
t,τ (1−δ)τ −t−1 ⎝τ =t+1

τ


⎞ 



ξj ⎠ · Zτ −(1−δ)χ

j=t+1

Sτh
Sτ

−χ (1−γ )/λt ,
−γ /λτ

⎫⎬⎭(2.33)

where the term χ (1−γ )/λt is the period t marginal efficiency cost following a run at t (given
Sth /St = 1 in this instance).17 The liquidation price is thus equal to the expected discounted streamof dividends net the marginal efficiency losses from household portfolio management. Since
marginal efficiency losses are at a maximum when Sth equal St , Qt∗ is at a minimum, given the
16. As 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 zeronet worth instead of in a new bank.
Sh

17. We are imposing that Stt −γ ≥ 0 as is the case in all of our numerical simulations.

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expected future path of Sτh . Further, the longer it takes the banking system to recover (so Sτh fallsback to its steady state value) the lower will be Qt∗ . Finally, note that Qt∗ will vary positively with
t,τ .
the expected paths of ξτ , Zτ and the stochastic discount factor 
The banking system then recapitalizes as follows: we suppose that during a run at t new banks
cannot set up business. However, they can store their initial equity and enter in the following
period along with the new banks scheduled to begin at t +1
Nt+1 = ζ St +σ ζ St−1 .

(2.34)

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Given that banks can operate in t +1 (the period after the run), it follows that starting from t +2
the evolution of bank equity follows the standard process: for all τ ≥ t +2
σ [(Rτb −Rτ )φτ −1 +Rτ ]Nτ −1 +ζ Sτ −1 if there is no run at τ.
0
if there is a run at τ

2.3.3. The default probability: illiquidity versus insolvency. In the run equilibrium,banks default even though they are solvent at normal market prices. It is the forced liquidation at
firesale prices during a run that pushes these banks into bankruptcy. Thus, in the context of ourmodel, a bank run can be viewed as a situation of illiquidity. In contrast, default is also possible
if banks enter period t insolvent at normal market prices.
Accordingly, the total probability of default in the subsequent period, pt , is the sum of theprobability of a run pRt and the probability of insolvency pIt :pt = pRt +pIt .

(2.35)We begin with pIt . By definition, banks are insolvent if the ratio of assets valued at normal
market prices is less than liabilities. In our economy, the only exogenous shock to the aggregate
I to be the value of capital quality,
economy is a shock to the quality of capital ξt . Let us define ξt+1
I ), equal to unity:
ξt+1 , that makes the depositor recovery rate at normal market prices, x(ξt+1bI )+(1−δ)QIξ I [Zt+1 (ξt+1
t+1 (ξt+1 )]StI
) = t+1= 1.
x(ξt+1R t Dt(2.36)

I , the bank will be insolvent and must default. The probability of
For values of ξt+1 below ξt+1
default due to insolvency is then given byI
pIt = probt ξt+1 < ξt+1,

(2.37)

where probt (·) is the probability of satisfying · conditional on date t information.
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 ιt+1 be a binary sunspot variable that takes on a value of 1 with probability  and a valueof 0 with probability 1−. In the event of ιt+1 = 1, depositors coordinate on a run if a bank run
equilibrium exists. Note that we make the sunspot probability  constant so as not to build inexogenous cyclicality in the movement of the overall bank run probability pRt .

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A bank run arises at t +1 iff (1) a bank run equilibrium exists at t +1 and (2) ιt+1 = 1. Let ωtbe the probability at t that a bank run equilibrium exists at t +1. Then the probability pRt of a run
at t +1 is given by
pRt = ωt ·.

(2.38)

R as the value of ξ
To find the value of ωt , let us define ξt+1
t+1 that makes the recovery rate∗
∗ and rental
conditional on a run xt+1 unity when evaluated at the firesale liquidation price Qt+1∗
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R )+Z ∗ (ξ R )]S b
ξ R [(1−δ)Q∗ (ξt+1t
t+1R) = t+1= 1.
x ∗ (ξt+1
R t Dt(2.39)R , x∗For values of ξt+1 below ξt+1
t+1 is below unity and a bank run equilibrium is feasible.
Therefore, the probability that a bank run equilibrium exists is given by the probability that
R but above the threshold for insolvency ξ I . In particular,
ξt+1 lies in the interval below ξt+1
t+1



I
R
ωt = probt ξt+1
≤ ξt+1 < ξt+1.

(2.40)I )Given equation (2.40), we can distinguish regions of ξt+1 where insolvency emerges (ξt+1 < ξt+1
I ≤ξRfrom regions where an illiquidity problem may emerge (ξt+1
t+1 < ξt+1 ).
Overall, the probability of a run varies inversely with the expected recovery rate Et xt+1 . The
lower the forecast of the depositor recovery rate, the higher ωt and thus the higher pt . In this way,
the model captures that an expected weakening of the bank balance sheets raises the likelihoodof a run. Further, because the leverage multiple is countercyclical (see Section 2.2.2), Et xt+1
is procyclical, implying that the run probability is endogenously countercyclical: runs are more
likely in bad times than in good.
Finally, comparing equations (2.37) and (2.40) makes clear that the possibility of a run
equilibrium significantly 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 probabilityof a fundamental shock that induces an insolvent banking system is negligible. However, theprobability of a shock that induces a bank run equilibrium is not negligible.

3. PRODUCTION, MARKET CLEARING, AND POLICY
The rest of the model is fairly standard. There is a production sector consisting of producers offinal 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 firms
There is a continuum of measure unity of final goods producers and intermediate goods producers.
Final goods firms make a homogenous good Yt that may be consumed or used as input to produce
new capital goods. Each intermediate goods firm f ∈ [0,1] makes a differentiated good Yt (f ) thatis used in the production of final goods.

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Final goods firms transform intermediate goods into final output according to the following
CES production function:
$
Yt =

 ε

1

Yt (f )

0

ε−1
ε

ε−1

df

,

(3.41)

where ε > 1 is the elasticity of substitution between intermediate goods.
Let Pt (f ) be the nominal price of intermediate good f . Then cost minimization of final goods
firms yields the following demand function for each intermediate good f (after integrating acrossthe demands of by all final goods firms):
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Pt (f ) −εYt (f ) =Yt ,Pt

(3.42)

where Pt is the price index as
$
Pt =

 1

1

1−ε

Pt (f )

1−ε

df

0

.

There is a continuum of intermediate good firms owned by consumers, indexed by f ∈ [0,1].
Each produces a differentiated good and is a monopolistic competitor. Intermediate goods firm f
uses both labour Lt (f ) and capital Kt (f ) to produce output according to:Yt (f ) = At Kt (f )α Lt (f )1−α ,(3.43)

where At is a technology parameter and 0 > α > 1 is the capital share.
Both labour and capital are freely mobile across firms. Firms rent capital from owners ofclaims to capital (i.e. banks and households) in a competitive market on a period by period basis.
Then from cost minimization, all firms choose the same capital labour ratio, as follows
Kt (f )
α w t Kt
= ,=Lt (f ) 1−α Zt Lt(3.44)

where, as noted earlier, wt is the real wage rate and Zt is the rental rate of capital. The first-orderconditions from the cost minimization problem imply that marginal cost is given by
1
MCt =
At

'

' (
(
wt 1−α Zt α.
1−αα(3.45)

Observe that marginal cost is independent of firm-specific factors.
Following Rotemberg (1982), each monopolistically competitive firm f faces quadratic costsof adjusting prices. Let ρ r (“r” for Rotemberg) be the parameter governing price adjustment costs.
Then each period, it chooses Pt (f ) and Yt (f ) to maximize the expected discounted value of profit:
∞
'
(
'
(2 
r

P
P
ρ
(f
)
(f
)
τ
τ
t,τ
,
(3.46)
−MCτ Yτ (f )− Yτ
Et
−1
Pτ
2
Pτ −1 (f )
τ =tsubject to the demand curve (3.42). Here, we assume that the adjustment cost is proportional tothe aggregate demand Yt .

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Taking the firm’s first-order condition for price adjustment and imposing symmetry implies
the following forward looking Phillip’s curve:
'
%
&
(
Yt+1
ε
ε −1
(3.47)
πt+1 −1 πt+1 ,+Et t+1(πt −1)πt = r MCt −ρεYttis the realized gross inflation rate at date t.
where πt = PPt−1

3.2.

Capital goods producers

max Qt (It (j))−It (j).
It (j)

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There is a continuum of measure unity of competitive capital goods firms, owned by households.
Each produces new investment goods that it sells at the competitive market price Qt . By investing
It (j) units of final goods output, firm j can produce (It (j)) new capital goods, with   > 0 and  < 0.
The decision problem for capital producer j is accordingly(3.48)

Given symmetry for capital producers (It (j) = It ), we can express the first-order condition as thefollowing “Q” relation for investment:−1
Qt =   (It )(3.49)

which yields a positive relation between Qt and investment.
3.3.

Monetary policy

Let t be a measure of cyclical resource utilization, i.e. resource utilization relative to the flexibleprice equilibrium. Next, let R = β −1 denotes the real interest rate in the deterministic steady statewith zero inflation. We suppose that the central bank sets the nominal rate on the riskless bond
Rtn according to the following Taylor rule:Rtn =1(πt )κπ (t )κyβ(3.50)

with κπ > 1. Unfortunately, due to the high degree of non-linearity, we are unable to impose thezero lower bound on the net nominal rate. We instead start with a high equilibrium real rate of
4% so that under a standard Taylor rule the nominal interest does not drop significantly below
the ZLB.
A standard way to measure t is to use the ratio of actual output to a hypothetical flexibleprice equilibrium value of output. Computational considerations lead us to use a measure whichsimilarly captures the cyclical efficiency of resource utilization but is much easier to handle
numerically. Specifically, we take as our measure of cyclical resource utilization the ratio of thedesired markup, 1+μ = ε/(ε −1) to the current markup 1+μt .18
t =

1+μ
1+μt

(3.51)

18. In the case of consumption goods only, our markup measure of efficiency corresponds exactly to the output
gap.

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with
1+μt = MCt−1 =(1−α)(Yt /Lt ).
ϕ γ
L t Ct h(3.52)

3.4.

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The markup corresponds to the ratio of the marginal product of labour to the marginal rate of
substitution between consumption and leisure, which corresponds to the labour market wedge.
The inverse markup ratio t thus isolates the cyclical movement in the efficiency of the labour
market, specifically the component that is due to nominal rigidities.
Finally, one period bonds which have a riskless nominal return have zero net supply. (Bankdeposits have default risk.) Nonetheless, we can use the following household Euler equation to
price the nominal interest rate of these bonds Rtn as'(
Rn
Et t+1 t= 1.
(3.53)
πt+1
Resource constraints and equilibrium

Total output is divided between consumption, investment, the adjustment cost of nominal prices,
and a fixed value of government consumption G:
Yt = Ct +It +ρr(πt −1)2 Yt +G.
2(3.54)

Given a symmetric equilibrium, we can express total output as the following function of aggregate
capital and labour:(3.55)
Yt = At Ktα Lt1−α .
Although we consider a limiting case in which supply of government bonds and money is zero,
the government adjusts lump sum taxes to satisfy the budget constraint. Finally, the labour market
must clear, which implies that aggregate labour demand of producers equals aggregate labour
supply of households.
This completes the description of the model. See the Appendix for details.19
4. NUMERICAL EXERCISES
4.1.

Calibration

Table 1 lists the choice of parameter values for our model. Overall there are 21 parameters.
Thirteen are conventional as they appear in standard New Keynesian DSGE models. The othereight parameters govern the behaviour of the financial sector, and hence are specific to our model.
We begin with the conventional parameters. For the discount rate β, the risk aversion parameter
γh , the inverse Frisch elasticity ϕ, the elasticity of substitution between goods ε, the depreciation
rate δ, and the capital share α we use standard values in the literature. Three additional parameters(η,a,b) involve the investment technology, which we express as follows:(It ) = a(It )1−η +b.
We set η, which corresponds to the elasticity of the price of capital with respect to investment
rate, equal to 0.25, a value within the range of estimates from panel data as in, for instance,
19. We focus throughout on the two equilibria described in the text. There is also a third equilibrium that we ruleout because it is not tatonnement stable.

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257

TABLE 1
Parameter

Description

β
γh
ϕ
α
δ
η
a
b
G
ρ jr
κπ
κy

Impatience
Risk Aversion
Inverse Frisch Elasticity
Elasticity of Subst. Across Varieties
Capital Share
Capital Depreciation
Elasticity of Q w.r.t. I
Inv. Technology
Inv. Technology
Government Expenditure
Price Adj. Costs
Policy Response to Inflation
Policy Response to Output

σ
ζ
θ
γ
χ

σ( ξ)
ρξ

Banker Survival Rate
New Banker Endowment
Share of Divertible Assets
Threshold for HH Intermediation Costs
Marginal HH Intermediation Costs
Sunspot Probability
Standard Deviation of Innovation to Capital Quality
Serial Correlation of Capital Quality

Value

Target

0.99
2
0.5
11
0.33
0.33
0.25
0.53
−0.83%
0.45
1000
1.5
0.125

Risk Free Rate
Literature
Literature
Markup 10%
Capital ShareI
K = 0.025
Literature
Q=1
(I) = I
G
Y = 0.2
Slope of Phillips curve = 0.01
Literature
LiteratureStandard parameters

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Financial intermediation parameters
0.93
0.1%
0.22
0.61
0.105
0.15
0.5%
0.7

Leverage = 10
Investment Drop in Crisis = 35%
Spread Increase in Crisis = 1.5%
Sb
S = 0.33b
ER −R = 2% Annual
Run Probability= 4% Annual
Standard Deviation of Output (C+I)
Standard Deviation of Investment

Gilchrist and Himmelberg (1995), Eberly (1997), and Hall (2004). We then choose a and b tohit two targets: first, a ratio of quarterly investment to the capital stock of 2.5% and, second, avalue of the price of capital Q equal to unity in the risk-adjusted steady state. We set the valueof government expenditures G to 20% of steady state output. Next, we choose the cost of priceadjustment parameter ρ jr to generate an elasticity of inflation with respect to marginal cost equal
to 1%, which is roughly in line with the estimates.20 Finally, we set the feedback parameters in
the Taylor rule, κπ and κy , to their conventional values of 1.5 and 0.5, respectively. As we noted
earlier, due to the high degree on non-linearity in the model, we are unable to impose the zero
lower bound. We verify, however, that the violation of the ZLB that occurs during a crisis is notlarge.
We now turn to the financial sector parameters. There are six parameters that directly affectthe evolution of bank net worth and credit spreads: the banker’s survival probability σ ; the initial
equity injection to entering bankers as a share of capital ζ ; the asset diversion parameter θ ; thethreshold share for costless direct household financing of capital, γ ; the parameter governing
the convexity of the efficiency cost of direct financing χ ; and the probability of observing asunspot .
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% annually, corresponding to a frequency of banking panics
of once every 25 years, which is in line with the evidence for advanced economies;21 (ii) the
average bank leverage multiple equals 10;22 (iii) the average excess rate of return on bank assets
20. See, for example, Del Negro et al. (2015).
21. See, for example, Bordo et al. (2001), Reinhart and Rogoff (2009), and Schularick and Taylor (2012).
22. We think of the banking sector in our model as including both investment banks and some large commercial
banks that operated off balance sheet vehicles without explicit guarantees. Ten is on the high side for commercial banks
and on the low side for investment banks. See Gertler Kiyotaki and Prestipino (2016).

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ArticleREVIEW OF ECONOMIC STUDIESTABLE 2

Y
C+I
I
C
L

Data
1983–2007 Q3

Model
No runs happen

1.9
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7.2
1.3
3.1

1.4
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Notes: For output, investment, consumption, and government expenditure, we compute real per capital values by dividing
by population and adjusting by the GDP deflator. For labour, we compute per capita hours worked by dividing total hours
by population. We then report standard deviations of the logged variables in deviations from a linear trend starting in
1983 Q1 and ending in 2007 Q3.
Source: Output, investment (gross domestic investment plus durable good consumption), consumption (personal
consumption expenditure less durables), government expenditure, and GDP deflator are from the Bureau of Economic
Analysis. Total labour hours (aggregate hours, non-farm payrolls) and population (civilian non-institutional, 16 years andover) are from the Bureau of Labor Statistics.

over deposits equals 2%, based on Philippon (2015); (iv) the average share of bank intermediatedassets equals 0.33, which is a reasonable estimate of the share of intermediation performed byshadow banks;23 (v) and (vi) the increase in excess returns (measured by credit spreads) and thedrop in investment following a bank run to match the evidence from the recent crisis.
The remaining two parameters determine the serial correlation of the capital quality ρξ andthe standard deviation of the innovations σξ . That is we assume that the capital quality shockobeys the following first-order process:ξt+1 = 1−ρξ +ρξ ξt + t+1
with 0 < ρξ < 1 and where t+1 is a (truncated) normally distributed i.i.d. random variablewith mean zero and standard deviation σξ .24 We choose ρξ and σξ so that the unconditional
standard deviations of investment and aggregate domestic demand are consistent with the evidence
observed over 1983Q1-2008Q3.
Given that our policy functions are non-linear, we obtain model implied moments bysimulating our economy for one hundred thousand periods.25 Table 2 shows unconditional
standard deviations for some key macroeconomic variables in the model and in the data. The
volatilities of output, investment, and labour are reasonably in line with the data. Consumption is
too volatile, but the variability of the sum of consumption and investment matches the evidence.26
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 show the response of the economy to a capital quality

23. The bulk of shadow bank intermediation was performed by investment banks and special purpose vehicles oflarge commercial banks. Begenau and Landvoight (2017) use a share of shadow bank intermediation of 33% based on
work by Gallin (2015). Pozsar et al. (2010) also have similar estimates for the share of shadow bank intermediation.
24. In practice, we assume that t+1 is a truncated normal with support −10σξ ,10σξ . Given our calibration forσζ and ρξ , the probability that ξt goes below zero is computationally zero.
25. As we are excluding the financial crisis from the sample, the sunspot variable is set to zero in these simulations.
26. One way to reduce consumption volatility would be to introduce habit formation though this would come at thecost of an additional state variable. Also some of the excess volatility in consumption arises because we are keeping theoutput share of government spending fixed.

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ArticleA MACRO MODEL WITH FINANCIAL PANICS

259

shock first without and then with runs to illustrate how the model generates a financial panic
with significant real effects on the economy. We also illustrate the role that the New Keynesianfeatures play. We then show how the model can replicate salient features of the recent financial
crisis. Finally, we model a credit boom by introducing optimistic beliefs and show how the boom
can help precipitate the subsequent financial collapse.

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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 (0.50%) shock to the quality of capital without runs.27 The solid line isour baseline model and the dotted line is the case where there are no financial frictions. For bothcases the shock reduces the expected return to capital, reducing investment and in turn aggregate
demand. In addition, for the baseline economy with financial frictions, the weakening of bankbalance sheets amplifies the contraction in demand through the financial accelerator/credit cycle
mechanism of Bernanke Gertler and Gilchrist (1999) and Kiyotaki and Moore (1997). Poor asset
returns following the shock cause bank net worth to decrease by about 13%. As bank net worth
declines, incentive constraints tighten and banks decrease their demand for assets causing the priceof capital to drop. The drop in asset prices feeds back into lower bank net worth, an effect that ismagnified by the extent of bank leverage. As financial constraints tighten and asset prices decline,
excess returns rise by 50 basis points which allows banks to increase their leverage multiple byabout 9%. Overall, a 0.5% decline in the quality of capital results in a drop in investment by
4% and a drop in output by slightly less than 1%. The drop in investment is roughly double theamount in the case absent financial frictions, while the drop in output is about 30% greater.
In the experiment of Figure1, the economy is always ex post in a “safe zone”, where a bankrun equilibrium does not exist. Under our parametrization, a bank run cannot happen in the riskadjusted steady state because bank leverage is too low. The dashed line in the first panel of Figure 1
shows the size of the shock needed in the subsequent period to push the economy into the run
region. In our example, a two standard deviation shock is needed to open up the possibility ofruns starting from the risk-adjusted steady state, which is double the size of the shock consideredin Figure 1.
Even though in this case the economy is always in a safe region ex post, it is possible ex antethat a run equilibrium could occur in the subsequent period. In particular, the increase in leveragefollowing the shock raises the probability that a sufficiently 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.
4.2.2. Bank runs. In the previous experiment, the economy was well within a safe zone.
A one standard deviation shock did not and could not produce a financial panic. We now considera case where the economy starts in the safe zone but is gradually pushed to the edge of the crisis
zone, where a run equilibrium exists. We then show how an arrival of a sunspot induces a panic
with damaging effects on the real economy.
To implement this experiment, we assume that the economy is hit by a sequence of threemodest and equally sized negative shocks that push the economy to the run threshold. That is, wefind a shock ∗ that satisfies:
ξ3R = 1+ ∗ 1+ρξ +ρξ2 ,27. In 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 εt to 0. See Appendix Section
6.7 for a formal definition of the risk-adjusted steady state and our impulse response functions.

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REVIEW OF ECONOMIC STUDIES
Baseline
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Figure 1
Response to a capital quality shock (1 std): no run casewhere ξ3R is the threshold level for the capital quality below which a run is possible in period 3,
given that the economy is in steady state in period 0 and is hit by two equally sized shocks inperiods 1 and 2, i.e. 1 = 2 = ∗ . The first two shocks push the economy to the edge of the crisiszone. The third pushes it just in.
The solid line in Figure 2 shows the response of the economy starting from 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 capitalquality shocks but assuming that no sunspot is observed and so no run happens.
As shown in panel 1, the size of the threshold shock to the quality of capital needed to push
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A MACRO MODEL WITH FINANCIAL PANICS

RUN (Run Threshold Shock and Sunspot)
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-1.5
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Figure 2
Response to a sequence of shocks: run versus no run

shock used in Figure 1, i.e. the threshold is ∗ = −0.63% while σ ξ = 0.5. After the first two
innovations, the capital quality shock is 1.1% 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 willingto increase their portfolio holdings of capital at a discount, which leads excess returns to spikeand investment to collapse. When the run occurs, investment drops an additional 25% resulting
in an overall drop of 35%. Comparing with the case of no run clarifies that almost none of thisadditional drop is due to the capital quality shock itself: the additional drop in investment absent
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ArticleREVIEW OF ECONOMIC STUDIES

monetary policy to ease by reducing the policy rate to slightly below zero.28 However, reducing
the nominal interest rate to roughly zero is not sufficient 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 induces banks to reduce the rate
at which they increase borrowing to expand asset holdings during the recovery, which in turnsuppresses the capital price, investment, and output.

⎛
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4.2.3. Banking panics: the role of nominal rigidities. To understand the role that the
New Keynesian features of the model play in the banking crisis, Figure 3 describes the effect ofbank runs in the economy with flexible prices. For comparison with the analogous experimentin our baseline (in Figure 2), we hit the flex-price economy with the same sequence of shocks
that would take the baseline economy to the run threshold.29 The run still produces a significant
disruption of the economy but the output drop in the flexible-price case is only about half thatin our benchmark sticky price case. Because the zero lower bound on nominal rates is not aconsideration in the flexible-price economy, the real interest rate drops roughly 800 basis points
below zero in this setting, leading to a temporary expansion in consumption demand. Whilebanking crises induce a large drop in investment also in the flexible-price model, the overall dropof about 22% is much smaller than the 35% drop in our benchmark model.
To dig deeper into how the New Keynesian features affect the banking crisis dynamics, it isuseful to consider the behaviour of the market value of capital Qt in the wake of a run: given the
Q relation in equation (3.49), the market value of capital governs investment dynamics.
Let Hτ denote the rental rate at time τ for a unit of capital in place at time t +1, adjusted for
depreciation and the capital quality shock as follows,⎞ξj ⎠ Zτ ,j=t+1

where Zτ is the rental rate per unit of capital at time τ . Then, we can express the market value ofcapital (equation (2.33)) as follows:
Q t = Et∞

Xt,τ ·Hτ ,

(4.56)

τ =t+1

Xt,τ is the rate at which the future rental stream is discounted and is given by the household’s
where stochastic discount factor, Lt,τ , adjusted for the required excess return on capital due to financial
market frictions, Xs :Xt,τ = ) t,τ.
(4.57)τ −1
s=t (1+Xs )28. Had we been able to impose the ZLB, the drop in output would have been somewhat larger.
29. However, since in the flex-price economy there is much less amplification, the ex post run that we consider isactually not an equilibrium. As the first panel in the figure shows, even after the first two shocks the shock that is needed
to push the economy to the threshold is still very large in the flex price economy, i.e. around −4%.

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A MACRO MODEL WITH FINANCIAL PANICS
RUN (Off-Equilibrium)

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NO RUNRun Probability

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Figure 3
Response to the same sequence of shocks in flex-price economy: run versus no run

The excess return Xt is the discounted spread between the rate of return on capital to banks,Rtnb , and the risk free rate, Rf
Rt+1
t+1 ≡ πt+1 . Given equations (2.10) and (3.53), we can express Xt as
follows:
 hSχ Stt −γ /λtfb−Rt+1 )] =.
(4.58)Xt ≡ Et [t+1 (Rt+1
Qt
Equations (4.56)–(4.58) show that Qt depends on the future paths of three main factors: (1)dividend payouts, {Zτ }τ ≥t ; (2) the risk free rate, which in turn is a function of the householdt,τ{Xτ }τ ≥t .
discount factor τ ≥t ; and (3) the excess returns stemming from market frictions
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ArticleREVIEW OF ECONOMIC STUDIES

three factors in the wake of a run. In particular, we can decompose the present value expression for
Qt , equation (4.56), into three components with each representing one of the factors, as follows:
Qt = !t +Rt +ϒt(4.59)with∞!t = Etτ =t+1β τ −t ·Hτ ; Rt = Et∞t,τ −β τ −t ·Hτ ; ϒt = Et

τ =t+1∞ Xt,τ − t,τ ·Hττ =t+1

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!t captures the effect on Qt of the future rental stream. It is the present discounted value of future
dividends at a fixed interest rate β −1 . Rt captures the effect of variation in the riskless interest
rate on the present value stream.30 Finally, ϒt captures the effect of excess returns in discounting
future cash flows.
We now compare the response of Qt and its three components in the wake of a run in the New
Keynesian economy versus the flex-price economy. Figure 4 shows the response of these threecomponents during a panic in our baseline economy (solid line) and in the flex-price economy(dashed line). We start each economy from the same initial state (Kt ,ξt ) of the run in our baseline
experiment above.
The drop in Q in the baseline New Keynesian model is 10%. What mainly accounts forthis drop is a 12% decline in the “excess returns” component of the price, ϒt , that reflects thetightening of financial constraints following the run. Working in the other direction is a modest
0.5% increase in the “dividend stream” component !t , due to higher future rental rates on capital(that stem from the investment decline), and also a small boost of 1.5% from the “real interestrate” component, Rt . The rise in Rt reflects that the central bank easing of monetary policy
causes real rates to decline and hence boosts asset valuations.
In contrast, the drop in Q in the flex-price economy is 5%, half the drop in the baseline. Eachof the three components of Q contributes to the difference. Two of the five percentage points are
explained by a smaller increase in the excess returns component: ϒt contributes to a 10% decline
in Qt in the flex-price model versus 12% in the New Keynesian case. Intuitively, given the extra
amplification that stems from nominal rigidities, in the New Keynesian framework there is greater
likelihood the economy can be pushed in the crisis zone for a given distribution of shocks. Hence,as the bottom left panel in Figure 4 shows, following the panic, the probability of another run iselevated in the New Keynesian model relative to the flexible-price model, and remains elevatedfor a period of time. The higher run probability, in turn, leads to a tighter leverage constraint that
limits banks ability to expand assets, leading to a slower recovery of the banking system. The net
effect is that excess returns are both higher and slower to return to normal in the New Keynesiancase.
The dividend stream component, !t , is roughly a percentage point higher in the flex-pricecase as the smaller employment contraction keeps the path of dividends higher.31 Finally, the realinterest rate component adds another 2% more in value than in the New Keynesian case reflecting
the much sharper drop in real interest rates in the flex-price economy.
Overall, while the banking panic is disruptive even within the flexible-price model, the New
Keynesian features help the framework account better for the depth and persistence of the banking
crisis. We elaborate this point within the next two subsections.
30. Rt will also capture the effects of variations in the covariance between consumption and the rental rate ofcapital. These effects however are quantitatively small.
31. We thank one of the referees for pointing out this channel.

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Banking panics: the role of nominal rigidities

4.2.4. Crisis experiment: model versus data. Figure 5 illustrates how the model canreplicate some salient features of the recent financial crisis. We hit the economy with a series ofmodest-sized capital quality shocks over the period 2007Q4 until 2008Q3. The starting point isthe beginning of the recession, which roughly coincides with the time credit markets first cameunder stress following Bear Stearns’ losses on its MBS portfolios. We pick the size of the capitalquality shocks to match the observed decline in investment during this period, in panel 1. We thenhit the economy with another small shock in 2008Q4 that is just sufficient to move the economy
into the region where a run equilibrium exists. We then assume that a run happens in 2008Q4,
the quarter in which Lehman failed and the shadow banking system collapsed.

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ArticleREVIEW OF ECONOMIC STUDIES
Shocks* :-0.2 %-0.5 %-0.4 %-0.6 %-0.6 %

Threshold :-0.9 %-0.8 %-0.7 %-0.7 %-0.6 %

2007q4

2008q1

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-50
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Data
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Model No Run
2004

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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.

Figure 5
Banking panics: the role of nominal rigidities

The solid line in Figure 5 shows the observed response of some key macroeconomic variables.
For output, investment, and consumption, we show the deviation from a trend computed by using
CBO estimates of potential output and similarly for hours worked we let the CBO estimate ofpotential labour represent the trend. To measure bank equity we use the XLF index, which is an[13:11 11/12/2019 OP-REST190033.tex]

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index of S&P 500 financial stocks. For excess returns, we use the spread between the rates on
AAA corporate bonds and 10-year government bonds. An advantage of the AAA spread is thatprivate sector forecasts of this variable are available, which we can subsequently make use of in
our analysis of belief driven credit booms in Section 4.3. We use the AAA rather than the BAA
spread, because the latter varies considerably with costly default risk, something our model does
not fully address.32
The dashed line in Figure 5 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 matchthe observed contraction in investment leads to a gradual decline in banks net worth that matchesclosely the observed decline in financial sector equity as measured by the XLF index, which is anindex of S&P 500 financial stocks. Given that banks net worth is already depleted by poor asset
returns, a modest innovation in 2008Q4 pushes the economy into the run region. When the run
occurs, the model economy generates a sudden spike in excess returns and a drop in investment,
output, consumption, and employment of similar magnitudes as those observed during the crisis,
as panels 3 to 6 show.33 The dotted line shows how, absent a run, the same shocks would generatea much less severe downturn.
The model economy also predicts a rather slow recovery following the financial crisis, althoughfaster 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. In practice, there were a series ofcontractionary disturbances that followed the banking crisis, including the tightening of fiscal
policy, the European debt crisis, and the tightening of bank regulations. Another factor we haveabstracted from is the deleveraging of household balance sheets, which likely also contributed tothe prolonged stagnation. Incorporating these factors could help the model account for the slow
recovery. However, we leave this extension for future research.
A news driven credit boom that leads to a bustIn the baseline experiment depicted in Figure 5, we assumed for simplicity that the economy isin steady state in 2007q3. We then fed in a sequence of modest fundamental shocks that moved
the economy steadily from a safe zone to a crisis zone where a banking panic is possible. This
exercise is useful for illustrating both the non-linear dimension of the banking crisis and how thecrisis can spill over into real activity. However, it is silent on the pre-crisis buildup of vulnerabilityof the banking system. As many have argued, the pre-crisis credit boom laid the seeds of the crisisby increasing banks’ fragility.
In this section, we extend the model to capture the pre-crisis credit boom. Here, we follow
the lead of several authors by introducing optimistic beliefs about the return on capital that are
eventually disappointed. That investors were overly optimistic pre-crisis is a widespread theme inthe literature. As emphasized by Bordalo Gennaioli and Schleifer (2017), survey data on investor
expectations confirm this view. For our purposes, the advantage of appealing to investor optimismis 2-fold: first, we can generate a pre-crisis leverage buildup without having a counterfactuallylarge increase in real activity. Second, having a leverage buildup that is not eventually supported
by fundamentals can raise the probability the economy enters into a crisis zone. It is then possible,
32. To make a proper comparison between the model and the data, we construct a model measure of bank excess
returns that has a maturity similar to that of the AAA spread. For simplicity, we use the expectations hypothesis of theterm structure to compute long rates. Details are in the Appendix Section 6.8.
33. The behaviour of the nominal interest rate is very similar to that portrayed during the bank run illustrated by
Figure 2: it drops slightly below zero during the panic before recovering.

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as we demonstrate, that a banking panic can arise within a given period, absent even modest new
aggregate disturbances.
To model optimistic beliefs, we consider a variant of a “news” shock.34 Under the standard
formulation, at time t, individuals suddenly learn that a fundamental disturbance of a given size
will occur j periods in the future. We make two modifications. First, we assume that there is a
probability the shock may not occur. Second, we assume that rather than having a single date in
the future when the shock can occur, there is a probability distribution over a number of possible
dates. As time passes without the occurrence of the shock, individuals update their priors on thesevarious possibilities. We also assume that only bankers, who are the experts at managing assets,
have optimistic beliefs. In fact, it is the relative optimism of bankers with respect to households
that generates the vulnerability of the financial system.35
In particular, at time t N bankers receive news that there may be a high return on capital in theform of a large capital quality shock. But they do not know for sure (1) whether the shock will
occur and (2) conditional on being realized, when it will occur. If the shock is realized at sometime τ > t N , it takes the form of a one-time impulse to the capital quality shock process of size
B. Formally, the news bankers receive is that the capital quality will follow the processBτ for τ > t N ,ξτ = 1−ρξ +ρξ ξτ −1 + τ +Bτ = 0, otherwise. Given the capital quality
where Bτ = B if the large shock realizes at τ and 
shock is serially correlated, there will be a persistent effect of B. However, given it is a one-time
shock, if it occurs, there will be no subsequent realizations of this impulse.
When they receive the news at t N , bankers’ prior probability that a shock will eventuallyoccur is given by P̄. Conditional on the shock happening, the future date when it will happen,τ ∈ {t N +1,t N +2,..., t N +T }, is distributed according to a probability mass function ζτ which
we assume to be a discrete approximation of a normal with mean μB and standard deviation σ B .
Thus at date t N , the probability that the shock happens at τ is given byBτ = B) =probt N (P̄ ·ζτ , for τ = t N +1,t N +2,...,t N +T.
0, for τ > t N +T

As long as no shock is observed, bankers update their beliefs using Bayes rule:Bτ = B) =probt (1−P̄ ·ζτ
*t*tN +T=

j=t N +1 P̄ ·ζjj=t+1 P̄ ·ζj1−*tj=t N +1 P̄ ·ζjζτ· *t +T

j=t+1 ζjNfor τ = tN +1,...,tN +T , and probt (Bτ = B) = 0 for τ > tN +T . The first term in the RHS is theposterior probability of the shock ever happening, which we denote by Pt and which is decreasing
34. See Christiano Motto and Rostagno (2014) for a medium scale estimated DSGE model in which news shocks
play an important role in explaining fluctuations in financial variables.
35. Technically, we assume that households are aware that bankers became optimistic but do not change theirbeliefs about the quality of capital, i.e. they do not believe the news. This allows us to have diverse beliefs without having
households extract information from prices. A similar assumption is made for the same reason, for instance, in Cogley
and Sargent (2009). Because households know bankers are more optimistic, they understand that there is less danger forbankers to divert their assets and loose their franchise. This allows bankers to raise their leverage multiple.

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with t. The second term is the probability that the shock realizes at τ conditional on the shock
eventually happening. The latter is increasing with t until t = tN +T , before becoming zero.
Observe that the process will generate a burst of optimism that will eventually fade if the goodnews is not realized. Early on, bankers will steadily raise their forecasts of the near term return tocapital as they approach the date where, a priori, the shock is most likely to occur. As time passes
without the realization of the shock, bankers’ become less certain it will ever occur, especiallyas they move past the peak time of when it was most likely to occur: the optimism proceeds tovanish.
We now illustrate how with the belief mechanism just described, the model generates aboom/bust scenario. We assume that the time the bankers receive the optimistic news, t N , isthe first quarter of 2005, in line with the description of events in Bordalo et al. We pick the meanof the conditional distribution for τ, μB , so that the prior on when the shock is most likely to occur
is 2007Q2, which coincides roughly with the peak of house prices. We pick the standard deviationσ B to ensure that by the end of 2008, if the shock has not occurred, bankers’ will completely giveup hope that it will ever occur.36 Next, we set the size of the impulse B to equal a two standard
deviation shock, that is, a shock which is unusually large but not beyond the realm of possibility.
Finally, we pick the prior probability that the shock will even occur P̄, to ensure that economy
reaches the crisis zone in 2008Q4 without any fundamental shocks. As external validation, we
check whether the forecast errors generated by our belief process (in conjunction with the model)
generates forecast errors consistent with the survey evidence in Bordalo et al.
Figure 6 characterizes the dynamics of beliefs and the credit boom that can emerge absent
any fundamental shocks. The top left panel gives the prior distribution for the time the shock willT
happen, conditional on it happening for sure, i.e. ζt N +i i=1 . The middle panel then illustrates
the ingredients bankers use to forecast the shock. The blue solid line in the top middle panel gives
the probability the shock will eventually happen, Pt . When the news is received in 2005Q1, theprobability jumps to its prior value near unity. Time passing without the shock occurring leadsbankers’ to reduce this probability. The optimism fades rapidly as time passes 2007Q2, the most
likely time the shock was expected to occur. The dashed red line then gives the probability theshock will occur in the subsequent period, conditional on it eventually happening. The estimatethat the shock will occur in the subsequent period is then the product of the blue solid and red
dashed lines.
To illustrate the boom/bust nature of beliefs, the right panel portrays the year ahead forecastof the capital quality shock (the dashed red line). After receiving the news in 2005Q1, optimism
steadily builds, peaking just before 2007Q2. However, as time continues to pass beyond the most
likely time, the optimism fades quickly, effectively vanishing by 2008Q4. Note that throughout
the boom and bust in beliefs, the true fundamental shock (the blue solid line), is unchanged. Thus,
there is serial correlation in the forecast errors of the capital quality. Shortly, we show that thisgenerates a behaviour of forecast errors of excess returns that is consistent with the evidence.
The bottom left panel shows the response of output to the news. The optimism leads to anincrease in asset values, which boosts investment, in turn leading to a modest increase in output
of about 0.3%.37 There is however a non-trivial debt buildup of 10%. Because bankers become
optimistic relative to households, they increase their relative holdings of capital and they fund thisincrease by issuing debt. In turn, the increase in leverage significantly raises the likelihood the

36. Given our discrete approximation of the normal distribution, a choice of σ B translates into a maximum numbersof periods within which the shock can occur.
37. Note that asset values depend on the expected discount forecast of the shock over the entire future. While thenear term estimates of the shock increase slowly, longer horizon estimates will jump more rapidly.

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ArticleREVIEW OF ECONOMIC STUDIES
Prior cond. prob. of shock
happening at time t
0.2

Beliefs Evolution

1

Capital Quality

0.6t

E t t+4

0.5

0.8

from SS

0.15
0.6
0.1

%

0.4

0.4

0.3

0.2

0.2

0.1

Time

Time

Output

1

20
05

Q
1
20
07
20 Q2
08
Q
4

20
05

08

Q

2
20

Q
20
07

1
Q
20
05

0
Q
1
20
07
20 Q2
08
Q
4

0
4

0

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0.05

Debt

12

Probability of being in crisis zone

0.3

10

0.25

0.2
6

Level

from SS

from SS

8
0.5

%

%

4
0

0.15

0.1
2
0.05

0

1
Q

07
20 Q2
08
Q
4

20

05
20

07
20 Q2
08
Q
4

20

Q

1

0

05
20

20

Q
05
20

07
20 Q2
08
Q
4

-2
1

-0.5

Figure 6
Belief dynamics and credit boom

economy moves into a crisis zone in the subsequent period, peaking at close to 30% by 2008Q4.38
In this respect, the boom lays the seeds of the bust.
We now repeat the crisis experiment described in Section 4.2.4, this time allowing for thewave of optimism beginning in 2005Q1 that we have just characterized. As before, we allow fora series of modest capital quality shocks from 2007Q4 to 2008Q3 so that the model tracks the
behaviour of investment over this period. But unlike before, we do not introduce a fundamental
shock in 2008Q4. Due to the debt buildup in this case, the economy reaches the crisis zone in
2008Q4 absent an additional shock.

38. As discussed by Gertler Kiyotaki and Prestipino (2016), there were additional factors contributing to theleverage buildup, including financial innovation. For simplicity we abstract from these factors and note only that including
them would increase the debt buildup further and the resulting degree of fragility.

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ArticleA MACRO MODEL WITH FINANCIAL PANICS

Shocks* :-0.2 %-0.4 %-0.3 %-0.5 %-0.0 %

Threshold :-0.1 %-0.1 %

0.0 %-0.0 %-0.0 %

2007q4

2008q1

2008q2

2008q3

2008q4

1. Investment
Bear Stearns

271

2. XLF Index and Net Worth
10

Lehman Brothers

25

0

0

-10
-25
-20
-50
Data
Model
Model No Run
2004

2007q3

2008q4

2016

-75

-40
-50

3. Spreads (AAA-Risk Free)

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-30

2004

2007q3

2008q42016-100

4. GDP
5

3

2
0
1
-5
0

2004

2007q3

2008q42016-1

5. Labor (hours)

2004

2007q3

2008q42016-10

6. Consumption
5

5

0
0
-5

2004

2007q3

2008q4

2016

-10

2004

2007q3

2008q4

2016-5

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.

Figure 7
Financial crisis after credit boom: model versus data

Figure 7 illustrates the experiment. The wave of optimism induces a rise in investment prior
to the recession along with the debt buildup portrayed in Figure 6. The latter helps move theeconomy into the crisis zone as the recession hits. This can be seen from the top two lines thatlist the sequence of shocks that we feed along with the threshold values of the shocks needed
to push the economy into the crisis zone. In contrast to the previous case, the economy is in the[13:11 11/12/2019 OP-REST190033.tex]

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ArticleREVIEW OF ECONOMIC STUDIESForecast Errors: AAA-Treasury (4-Quarters Ahead)
100
Lehman Brothers
Data (Bordalo-Gennaioli-Shleifer)
Model

75

50

25

0

-50

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-25

Error (Next 4Q Average) = Actual - Forecast

Bear Stearns-75

2004q3

2007q3

2008q42013q4-100

Figure 8
Forecast errors in excess returns: model versus data

crisis zone throughout the recession. In 2007Q4 and 2008Q1, modest negative shocks push theeconomy into the crisis zone. From 2008Q2 to 2008Q4, the economy is in the crisis zone absent
any new shocks.
Given the economy is in the region where a panic is feasible, we again suppose the rollovercrisis takes place in 2008Q4. Again, this time we do not require a fundamental shock to movethe economy to a crisis zone, so we do without it. Overall, the effect of the banking crisis is very
similar to the case without the debt boom. The contractions in output, investment, employment
and consumption in terms of both amplitude and persistence are all close to what happens in theprevious case. There is also a similar increase in the credit spread and contraction in net worth.
In the end, what the credit boom does is to raise the likelihood of the subsequent crisis.
Finally, as a check on our belief mechanism, we compare the forecast errors for excess returns
generated by the model with evidence from the Survey of Professional Forecasters. Following
Bordalo Gennaioli and Schleifer (2017), the dark line in Figure 8 reports errors in the mean
forecast over the year ahead of the spread between the AAA bond rate and the 10-year government
bond rate. Specifically, a point at time t is the date-t consensus forecast of AAA-Treasury yieldspreads averaged across forecast horizons going from t +1 to t +4.39 The shaded area is a twostandard deviation error band, based on the dispersion of the individual forecasters expectations.
As the survey data shows, investors were overly optimistic about credit spreads in the near futurefrom mid 2007 to the Lehman collapse: Over this period, they persistently underestimated thecredit spread over the following one year horizon. Following the Lehman collapse, there is amodest swing towards excess pessimism, though not as significant or persistent as the optimism
wave.
The dashed line represents the analogous forecast error in the excess return that the model
generates. As bankers become optimistic and expect a boom that fails to materialize, their forecast
39. Consensus forecasts are cross-sectional averages of individual respondents forecasts.

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ArticleA MACRO MODEL WITH FINANCIAL PANICS

273

errors about the yield spreads over 1 year horizon follow a pattern that is similar in duration andmagnitude to the data in the pre-Lehman period. The model forecast stays within the standard
error bands for the most part, though it creeps slightly above a few quarters before the collapse.
There is also very modest period of pessimism following the collapse. This is due to the fact that
following the run in the model, the probability of a second run increases (and this second run
does not materialize). The standard error bands are sufficiently wide following the collapse, sothat the difference between the data and model simulations are not significant. Overall, the belief
mechanism we used is roughly in line with the evidence.
5. CONCLUSION
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We have developed a macroeconomic model with a banking sector where costly financial panics
can arise. A panic or run in our model is a self-fulfilling failure of creditors to roll over their shortterm credit to banks. When the economy is close to the steady state a self-fulfilling rollover crises
cannot happen because banks have sufficiently strong balance sheets. In this situation, “normalsize” business cycle shocks do not lead to financial crises. However, in a recession, banks may
have sufficiently weak balance sheet so as to open up the possibility of a run. Depending on thecircumstances either a small shock or no further shock can generate a run that has devastatingconsequences for the real economy. We show that our model generates the highly nonlinear
contraction in economic activity associated with financial crises. It also captures how crises may
occur even in the absence of large exogenous shocks to the economy. We then illustrate that themodel is broadly consistent with the recent financial crisis. Finally, we extend the framework tomodel the credit boom that leads to the bust. We do so by introducing a wave of optimism that
leads to a leverage buildup that increases the fragility of the economy, measured explicitly by the
exposure to a banking panic.
One issue we save for further work is the role of macroprudential policy. As with other models
of macroprudential policy, externalities are present that lead banks to take more risk than is sociallyefficient. Much of the literature is based on the pecuniary analysed by Lorenzoni (2008), whereindividual banks do not properly internalize the exposure of the system to asset price fluctuations
that generate inefficient volatility, but not runs. A distinctive feature of our model is that the key
externality works through the effect 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 thepossibility of a financial collapse in the most efficient way. Our model will permit us to explore
the optimal design of policies qualitatively and quantitatively.
A. APPENDIXThis Appendix describes the details of the equilibrium.
The aggregate state of the economy is summarized by the vector of state variables M̃t =b ,R D(St−1 ,St−1
t t−1 ,ξt ), with sunspot realization ιt at time t, where St−1 = capital stock at theb
end of t −1; St−1
= bank capital holdings in t −1; Rt Dt−1 = bank deposit obligation at the
beginning of t; and ξt = capital quality shock realized in t.
A.1. Producers
As described in the text, the capital stock for production in t is given by
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ArticleREVIEW OF ECONOMIC STUDIESThe capital quality shock is serially correlated as follows
ξt+1 ∼ F ξt+1 | ξt = Ft ξt+1
with a continuous density:Ft ξt+1 = ft ξt+1 , for ξt+1 ∈ (0,∞).
Capital at the end of period is
St = (It )Kt +(1−δ)Kt .

(A.61)
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As we described in the text, capital goods producer’s first order condition for investment is
Qt   (It ) = 1.

(A.62)

A final goods firms chooses intermediate goods {Yt (f )} to minimize the cost
$ 1
0

Pt (f )Yt (f )df

subject to the production function:
$
Yt =

 ε

1

Yt (f )

0

ε−1
ε

ε−1

df

.

(A.63)

The cost minimization then yields a demand function for each intermediate good f :%Yt (f ) =&Pt (f ) −εYt ,Pt

(A.64)

where Pt is the price index, given by
$
Pt =

 1

1

1−ε

Pt (f )

1−ε

df

0

.

Conversely, an intermediate goods producer f chooses input to minimize the production cost
wt Lt (f )+Zt Kt (f )subject toAt [Kt (f )]α [Lt (f )]1−α = Yt (f ).

The first-order conditions yield
Kt (f )
α w t Kt
= ,=Lt (f ) 1−α Zt Lt(A.65)and the following relation for marginal cost:MCt =[13:11 11/12/2019 OP-REST190033.tex]

1
At

'

( '
(
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ArticleA MACRO MODEL WITH FINANCIAL PANICS275

Each period, the intermediate goods producer chooses Pt (f ) and Yt (f ) to maximize the expecteddiscounted value of profits:
∞
'
(
'
(2 
r

P
P
ρ
(f
)
(f
)
τ
τ
t,τ
,

−MCτ Yτ (f )− Yτ
Et
−1
Pτ
2
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t,τ = β τ −t (Cτ /Ct )−γh is the discount factor of thesubject to the demand curve (A.64), where 
representative household. Taking the firm’s first order condition for price adjustment and imposing
symmetry implies the following forward looking Phillip’s curve:
'
%
&
(
Yt+1
ε
ε −1

πt+1 −1 πt+1 ,(A.67)+Et t,t+1(πt −1)πt = r MCt −ρεYttis the realized gross inflation rate at date t. The cost minimization conditions
where πt = PPt−1
with symmetry also imply that aggregate production is simply

Yt = At Kt α Lt 1−α .
A.2.

(A.68)

Households

We modify the household’s maximization problem in the text by allowing for a riskless nominalbond which will be in zero supply. We do so to be able the pin down the riskless nominal rate
Rtn . Let Bt be real value of this riskless bond. The household then chooses Ct ,Lt ,Bt ,Dt , and Sth
to maximize expected discounted utility Ut :
∞


1−γh
1+ϕ

(C
)
)
(L
τ
τ
Ut = Etβ τ −t−−ς (Sτh ,Sτ ) ,
1−γ
1+ϕ
h
τ =t
subject to the budget constraint
Rnh.
Ct +Dt +Qt Sth +Bt = wt Lt −Tt + t +Rt Dt−1 + t−1 Bt−1 +ξt [Zt +(1−δ)Qt ]St−1πt
As explained in the text, the rate of return on deposits is given byb
ξt [Zt +(1−δ)Qt ]St−1
Rt = Max Rt ,Dt−1
b
Qt−1 St−1
ξt [Zt +(1−δ)Qt ]= Max Rt ,
b −N
Qt−1
Qt−1 St−1
t−1
'
(
φt−1
= Max Rt ,Rtb,
φt−1 −1t]and where φt = Qt Stb /Nt is the bank leverage multiple.
where Rtb = ξt [Zt +(1−δ)Q
Qt−1
We obtain the first order conditions for labour, riskless bonds, deposits, and direct capital
holding, as follows:

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ArticleREVIEW OF ECONOMIC STUDIES'(
Rtn
Et t+1
=1
πt+1
%
(&
'
φt
b
=1
Et t+1 Max Rt+1 ,Rt+1
φt −1
⎧
⎨

Zt+1 +(1−δ)Qt+1(A.70)(A.71)⎫⎬= 1,
Et t+1 ξt+1⎩
Qt + ∂ h ς (Sth ,St )·Ct γh ⎭(A.72)∂St

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where
(
'
Ct+1 −γh, andt+1 = t,t+1 = β
Ct
 
 
∂
Sth
h
ς(St ,St ) = Max χ−γ , 0 .
St
∂Sth
A.3.

Bankers

For ease of exposition, the description of the banker’s problem in the text does not specify how
the individual choice of bank’s leverage affects its own probability of default. This was possible
because, as argued in footnote 10, the indirect marginal effect of leverage on the objective of
the firm, Vt , through the change in pt is zero. Therefore the first order conditions for the bank’s
problem, equations (2.27) and (2.28), can be derived irrespectively of how the individual choiceof bank’s leverage affects its own probability of default.
We now formalize the argument in footnote 10 and describe how the default thresholds forindividual banks vary with individual bank leverage. As will become clear in Section A.5 below,
this analysis is key in order to study global optimality of the leverage choice selected by using
the first order conditions in the text, equations (2.27) and (2.28).
As in the text, ι is a sunspot which takes on values of either unity or zero. We can then expressbthe rate of return on bank capital Rt+1
b
Rt+1
= ξt+1

Zt+1 +(1−δ)Qt+1b= Rt+1(ξt+1 ,ιt+1 ),Qt

The individual bank defaults at date t +1 if and only if1>

ξt+1 [Zt+1 +(1−δ)Qt+1 ]stb
Rt+1 dt

Rb (ξt+1 ,ιt+1 ) Qt stb
= t+1
,
Rt+1
Qt stb −nt

or
b
Rt+1
(ξt+1 ,ιt+1 ) < Rt+1

φt
.
φt −1

Let !Dt+1 (φ) be the set of capital quality shocks and sunspot realizations which make theindividual bank with a leverage multiple of φ default and conversely let !N
t+1 (φ) be the set that
leads to non-default at date t +1:

+
φ −1
b
!D
(φ)
=
(ξ
,ι
)
|
R
(ξ
,,ι
)
<
R
(φ)
,
t+1 t+1
t+1
t+1
t+1 t+1 t+1
φ[13:11 11/12/2019 OP-REST190033.tex]

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ArticleA MACRO MODEL WITH FINANCIAL PANICS
+
φ −1
N
b
!t+1 (φ) = (ξt+1 ,ιt+1 ) | Rt+1 (ξt+1 ,ιt+1 ) ≥Rt+1 (φ) ,
φ


277

where Rt+1 (φ) is the promised deposit interest rate when the individual bank chooses φ which
satisfies the condition for the household to hold deposits:
$
$
φ
b

1 = Rt+1 (φ)
t+1 d Ft +
t+1 Rt+1(ξt+1 ,ιt+1 )d
Ft .
(A.73)
D (φ)
φ
−1
!N
(φ)
!
t+1
t+1

I (φ) and ιξt+1 < ξt+1
t+1 = 0where
I (φ)Rb (ξ I (φ),0) = φ−1 R
ξt+1t+1 (φ)
φt+1 t+1orR (φ) = sup
ξt+1

R (φ) and ιξt+1 < ξt+1
t+1 = 1
where,

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Here 
Ft (ξt+1 ,,ιt+1 ) denotes the distribution function of (ξt+1 ,ιt+1 ) conditional on date t
information.
Assume that the aggregate leverage multiple is given by φ t . When the individual banker
chooses the leverage multiple φ, which can be different from φ t , the individual bank defaults at
date t +1 if and only if(A.74)-bξt+1 s.t. ξt+1 Rt+1
ξt+1 ,1 < φ−1
φ Rt+1 (φ)(A.75).

Thus, the set of capital quality shocks and sunspots which make the individual bank default!D
t+1 (φ) is⎫⎧.
. ξt+1 < ξ I (φ) and ιt+1 = 0 ⎬
⎨
t+1
.
.
or
!D
.
(A.76)
t+1 (φ) = ⎩(ξt+1 ,ιt+1 ) .
. ξt+1 < ξ R (φ) and ιt+1 = 1 ⎭
t+1
⎫
⎧
.
I (φ) and ι.
ξt+1 ≥ ξt+1⎨
t+1 = 0 ⎬.
.
or.
(A.77)!N
t+1 (φ) = ⎩(ξt+1 ,ιt+1 ) .
. ξt+1 ≥ ξ R (φ) and ιt+1 = 1 ⎭t+1
I (φ) is straightforward and can be easily characterized from (A.74) under
The behaviour of ξt+1
b is increasing in the quality of capital at t +1. This gives:
the natural assumption that Rt+1
I (φ)
dξt+1
> 0, for φ ∈ (1,∞)
dφlim ξ I (φ) = 0.
φ↓1 t+1(A.78)R (φ) is more complicated because, when a sunspot is observed, theThe behaviour of ξt+1bfunction Rt+1
ξt+1 ,1 that determines returns on bank’s assets as a function of the capital quality
R = ξ R (φ̄ ) : at the threshold ξ R asset
is discontinuous around the aggregate run threshold ξt+1
t+1 t
t+1
prices jump from liquidation prices up to their normal value (see Figure 5):


b
b
R
b
,0 > lim Rt+1
(A.79)
ξt+1
ξt+1 ,1 = Rt+1
ξt+1 ,1 .
lim Rt+1
Rξt+1 ↓ξt+1

R
ξt+1 ↑ξt+1

R , an increase inThis implies that, if the capital quality shock is at the aggregate run threshold ξt+1
leverage from the value that makes the recovery rate equal 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.

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ArticleREVIEW OF ECONOMIC STUDIES

R , the value of leverage that makes the recovery rate at
By definition of the run threshold ξt+1
liquidation prices equal to unity is exactly the aggregate leverage φ̄t , that isφ̄t −1bξt+1 ,1 .
Rt+1 φ̄t = lim Rt+1
R
φ̄t
ξt+1 ↑ξt+1
On the other hand, we let φ̂t denote the value above which the bank defaults at the aggregate run
R even at normal prices. This value satisfies
threshold ξt+1

φ̂t


 

b
R
Rt+1 φ̂t = Rt+1
,0
ξt+1

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φ̂t −1and (A.79) implies that φ̂t > φ̄t .
For any value of leverage above the aggregate level φ̄t but below φ̂t , when a sunspot is observed,
R (φ) is insensitive tothe bank defaults if and only if a system wide run happens. That is, ξt+1
variation in individual bank’s leverage in this region:
R
R
/t ].
(φ t ) for φ ∈ [φ t , φ
ξt+1(φ) = ξt+1

/t the bank is always insolvent even at non-liquidation prices
For values of leverage above φ
R (φ) = ξ I (φ) for φ > φ̂ . When φ is smaller than aggregate φ , thewhenever defaults, i.e. ξt+1
t
t
t+1
R (φ) < ξ R . In the extreme when the leverage multiple
bank is less vulnerable to the run so that ξt+1
t+1
R (1) = 0.
equals unity, the individual bank is not vulnerable to a run so that ξt+1
R
To summarize, the behaviour of ξt+1 (φ) can be characterized as follows:lim ξ R (φ) = 0
φ↓1 t+1R (φ)
dξt+1> 0, for φ ∈ 1,φ tdφ

R (φ) = ξ R , for φ ∈ [φ , φ
I
R
/
/
ξt+1
t t ] where ξt+1 (φt ) = ξt+1
t+1
R (φ) = ξ I (φ), for φ ∈ [φ/t ,∞).
ξt+1
t+1(A.80)

See Figure A1.
We can now rewrite the problem of the bank as in the text, but incorporating explicitly thedependence of the default and non-default sets on the individual choice of leverage, as capturedNby !D
t+1 (φ) and !t+1 (φ) :(A.81)max (μt φ +νt ),φsubject to the incentive constraint:θφ ≤ μt φ +νt ,(A.82)

the deposit rate constraint obtained from (A.73) :
0

2
φ 1
b (ξ

1− φ−1

R
,ι
)d
F
D
t
!t+1 (φ) t+1 t+1 t+1 t+1
1
Rt+1 (φ) =.
!N (φ) t+1 d Ft

(A.83)

t+1[13:11 11/12/2019 OP-REST190033.tex]

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279

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Figure A1

μt and νt given by

$
μt =
$
νt =

!N
t+1 (φ)!Nt+1 (φ)bt+1 [Rt+1−Rt+1 (φ)]dFt(A.84)

t+1 Rt+1 (φ)dFt(A.85)N
I
R
and where !D
t+1 (φ) and !t+1 (φ) are given by (A.76)−(A.77), ξt+1 (φ) and ξt+1 (φ) satisfy(A.74)–(A.75).
Using (A.84)–(A.85) in the objective, we can write the objective function as$b#t (φ) =t+1 {[Rt+1(ξt+1 ,ιt+1 )−Rt+1 (φ)]φ +Rt+1 (φ)}dFt .
(A.86)!N
t+1 (φ)Before proceeding with differentiation of the objective above, we introduce some notationthat will be helpful in what follows. For any function G φ,ξt+1 ,ιt+1 and for any φ different
from φ̄t or φ̂t we let
 $
$
∂G
d
∗
G(φ,ξ,ι)d
Ft (ξ,ι) −(φ,ξ,ι)dFt (ξ,ι)(G)φt ≡
dφ !Dt+1 (φ)∂φ!D
t+1 (φ)  dξ I (φ)
  dξ R (φ)
t+1
t+1
I
I
R
R
= (1−)G φ,ξt+1+G φ,ξt+1(φ),0 ft ξt+1
(φ)(φ),1 ft ξt+1
(φ)
dφ
dφ
(A.87)
denote the marginal effect of φ on G only through its effect on the default probability. Then, we
I (φ) and ξ R (φ) we haveknow that as long as G(·) is continuous at ξt+1
t+1
 $
$
∂G
d
G(φ,ξ,ι)d
Ft (ξ,ι) −(φ,ξ,ι)dFt (ξ,ι) = −(G)∗φt .
N
dφ !Nt+1 (φ)∂φ!t+1 (φ)
Notice that we have not defined (G)∗φt for φt = φ̄t or φt = φ̂t becausethat point.

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ArticleREVIEW OF ECONOMIC STUDIES

Differentiation of (A.86) at any value different from φ̄t and φ̂t yields#t (φ) = μt −(φ −1),-∗νt dRt+1 (φ) b(ξt+1 ,ιt+1 )−Rt+1 (φ)]φ +Rt+1 (φ).
− t+1 [Rt+1
φt
dφ
Rt+1

Now notice that for φ ∈ [1, φ̄t ) and φ > φ̂t we have that the bank networth is zero at both thresholds,
that is
b
I
[Rt+1
(ξt+1
(φ),0)−Rt+1 (φ)]φ +Rt+1 (φ) = 0
b
R
(ξt+1
[Rt+1
(φ),1)−Rt+1 (φ)]φ +Rt+1 (φ) = 0
-∗
b (ξ
= 0.
implying t+1 [Rt+1
t+1 ,ιt+1 )−Rt+1 (φ)]φ +Rt+1 (φ)φtFor φ ∈ φ̄t , φ̂t , we have that at the insolvency threshold net worth is still zero,Downloaded from https://academic.oup.com/restud/article-abstract/87/1/240/5505450 by guest on 14 May 2020

b
I
[Rt+1
(ξt+1
(φ),0)−Rt+1 (φ)]φ +Rt+1 (φ) = 0

while the run threshold is fixed at the aggregate level
R (φ)
dξt+1

dφ

=0

,
-∗

b (ξ
= 0.
so that again t+1 [Rt+1
t+1 ,ιt+1 )−Rt+1 (φ)]φ +Rt+1 (φ)
Therefore, we have that for all φ different from φ̄t and φ̂t#t (φ) = μt −(φ −1)

φtνt dRt+1 (φ)
dφ
Rt+1

(φ)
, it can be extended to φ̄t and φ̂t as well.
and by continuity of #t (φ) and μt −(φ −1) νt dRt+1
dφ
Rt+1Then, as reported in the text, the first-order condition isνt, if μrt > 0, and
φt = θ −μtνtrμt = 0, if φt < θ −μ,t(A.88)

νt dRt+1 (φt ).
(A.89)dφt
Rt+1(Here, we assume μt < θ which we will verify later).
As explained below in Section A.5, we make assumptions such that conditions (A.88)–(A.89)
characterize the unique global optimum for the bank’s choice of leverage. Since these conditions
don’t depend on the individual net worth of a banker, every banker chooses the same leverage
multiple and has the same Tobin’s Q
μrt = μt −(φt −1)

ψt = μt φt +νt .

(A.90)

Thus from the discussion in the text, it follows that there is a system wide default if and only ifb
Rt+1(ξt+1 ,0) = ξt+1

Zt+1 +(1−δ)Qt+1 φt −1<
Rt+1 (φt ), orQtφt∗
Z ∗ +(1−δ)Qt+1
φt −1
b
Rt+1
(ξt+1 ,1) = ξt+1 t+1<
Rt+1 (φt ),Qt
φtwhere Rt+1 (φt ) is the aggregate promised deposit interest rate.

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281

A systemic default occurs if and only ifII, where ξt+1
ξt+1 < ξt+1

Zt+1 +(1−δ)Qt+1 φt −1=
Rt+1 (φt ),Qt
φt(A.91)

or

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∗ +(1−δ)Q∗
Zt+1
φt −1
t+1
R
R
(A.92)
ξt+1 < ξt+1
and ιt+1 = 1, where ξt+1=
Rt+1 (φt ).
Qt
φt
It follows that the probability of default at date t +1 conditional on date t information in thesymmetric equilibrium is given by
0
2
I
R
I
pt = Ft (ξt+1)+ Ft (ξt+1)−Ft (ξt+1) .
(A.93)

The aggregate capital holding of the banking sector is proportional to the aggregate net worthas
Qt Stb = φt Nt .
The aggregate net worth of banks evolves as⎧,b −R D⎪⎨ σ max ξt (Zt +(1−δ)Qt )St−1
t t−1 , 0 +ζ St−1 if no run at t
.
Nt =
⎪
⎩
0
otherwise(A.94)(A.95)

Banks finance capital holdings by net worth and deposit, which implies
Dt = (φt −1)Nt .

(A.96)

St = Stb +Sth .

(A.97)

A.4. Market clearing
The market for capital holding implies

The final goods market clearing condition implies
Yt = Ct +It +ρr 2
πt Yt +G.
2

As is explained in the text, the monetary policy rule is given by

ϕ y
MC
1
t
Rtn = (πt )ϕπ ε−1.
β(A.98)(A.99)εThe recursive equilibrium is given by a set of ten quantity variablesh b(Kt ,St ,It ,Lt ,Yt ,C(wt ,Zt ,MCt ,πt ,Rt+1 , Qt ,Rtn ), and eightt ,St ,St ,Dt ,Nt ), seven price variablest =as a function of the four state variables Mbank coefficients ψt ,μt ,νt ,μrt ,φt ,pt ,ξ R ,ξ I
t+1

t+1

b ,R D(St−1 ,St−1
t t−1 ,ξt ) and a sunspot variable ιt , which satisfies twenty five equations, given by:(A.60,A.61,A.62,A.65,A.66,A.67,A.68,A.69,A.70,A.71,A.72,A.74,A.75,A.84,A.85,A.88,A.89,
A.90,A.93,A.94,A.95,A.96,A.97,A.98,A.99). Here, the capital quality shocks follow a Markov
process ξt+1 ∼ F ξt+1 | ξt and the sunspot is iid. with ιt = 1 with probability .

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A.5. On the global optimum for individual bank’s choice
To study global optimality of the individual leverage choice selected by the first order conditionsin (A.88), we need to analyse the curvature of the objective function #t (φ) in (A.86).
d R̄
To do so, we use (A.83) to derive an expression for dφand substitute it into (A.89) to obtain#t (φ) =$!N
t+1 (φ)bt+1 Rt+1d
Ft − 1−1$!D
t+1 (φ)!N (φ) t+1 d Ftbt+1 Rt+1
d
Ft 1 t+1

t+1 d Ft!N
t+1 (φ),(A.100)Downloaded from https://academic.oup.com/restud/article-abstract/87/1/240/5505450 by guest on 14 May 2020

Proceeding as in Section A.3 to differentiate (A.100) for any value of φ different from φ̄tand φ̂t , we get−#t "(φ) =1∗ ∗ !N (φ) t+1 dFtbb
t+1 Rt+1 + t+1 Rt+1 · 1 t+1φt
φt !N (φ) t+1 d Ft



1

t+1

+

0 1
2 !N (φ) t+1 dFtb d
1− !D (φ) t+1 Rt+1
Ft 1 t+1
t+1!N
t+1 (φ)t+1 d Ft(t+1 )∗φt
(t+1 )∗φt1
−1
 d Ft dFt!N (φ) t+1!N (φ) t+1
t+1


.

t+1

(A.101)
Note that for φ ∈ [1, φ̄t ) φ −1
b
I
b
R
Rt+1
ξt+1
ξt+1
(φ),ι = Rt+1(φ),1 =
Rt+1 (φ).
φ
dξ R (φ)
For φ ∈ φ̄t , φ̂t , we have t+1= 0 which implies that for any function G ξt+1 ,ιt+1dφ
 dξ I (φ)



t+1
I
I
(φ),0)ft φ,ξt+1(φ)
for φ ∈ φ̄t , φ̂t(G)∗φt = (1−)G(φ,ξt+1
dφ(A.102)and also
 φ −1bI(φ),0 =
Rt+1 (φ)
ξt+1
Rt+1
φ
Then, we learn∗∗ φ −1b= t+1 φt ·Rt+1 (φ)
t+1 Rt+1
φtφbt+1 Rt+1[13:11 11/12/2019 OP-REST190033.tex]

∗

∗

φt

= t+1 φt ·φ −1
Rt+1 (φ).
φ

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283

Substituting this back into (A.101) and using (A.83) to substitute for Rt+1 (φ), we get
⎡
⎤
1
∗
∗

t+1 φtt+1 φt
1 !Nt+1 (φ) t+1 d Ft ⎣⎦1#t "(φ) = 1−1

φ !N (φ) t+1 d
Ft

d
F

d
F
N
N
t
t
t+1
t+1
! (φ)
! (φ)
t+1

t+1

(A.103)

t+1!t+1 (φ)

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for any φ different from φ̄t and φ̂t .40
We assume that a bank that individually survives a systemic bank run by choosing its own
leverage below the aggregate level φ t behaves just like new entrants during the panic: it storesits net worth and starts operating the period right after the crisis. Given that both leverage andspreads increase dramatically after a crisis, new banker’s Tobin’s Q is very high during a crisis
so thatt+1 ξt+1 ,1
t+1 ξt+1 ,1RR1>1for ξt+1 = ξt+1(φ) < ξt+1.
t+1 dFtt+1 dFtNN!t+1 (φ)

By the same argument, we also have that1

t+1 ξt+1 ,0!N
t+1 (φ)t+1 dFt>1

t+1 ξt+1 ,0!N
t+1 (φ)t+1 dFtI
Ifor ξt+1 = ξt+1.
(φ) < ξt+1

Given this, equation (A.103) and (A.87) imply that the objective function of the banker is strictly
convex in the region where leverage is below the aggregate level φ̄t , thatis #t "(φ) > 0 for φ ∈R (φ) > 0. If, on the other[1, φ̄t ), as long as the probability of a run is still positive, i.e. ft ξt+1R (φ) = f ξ I (φ) = 0, the second
hand, leverage is so low that default is not possible, i.e. ft ξt+1
t t+1
derivative iszero. For φ ∈ φ̄t , φ̂t , equations (A.102) and (A.103) imply that #t "(φ) depends on the relative
increase 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, #t "(φ) > 0,
I (φ) > 0.
as long as ft ξt+1
Summing up, we have:
⎧




R (φ) = 0 = f ξ I (φ)
⎪
)
and
f
ξ
=
0
if
φ
∈
[1,
φ̄
⎪
t
t
t
⎪
t+1 
⎪
 t+1
⎪
⎪
R
⎨ >0
if φ ∈ [1, φ̄t ) and ft ξt+1 (φ) > 0#t "(φ).
I (φ) = 0
⎪
and
f
ξ
=
0
if
φ
∈
φ̄
,
φ̂
⎪
t
t
t
⎪
t+1
⎪




⎪
⎪
I (φ) > 0
⎩ >0if φ ∈ φ̄t , φ̂t and ft ξt+1
40. Notice that (t+1 )∗φt and (t+1 )∗φt are not continuous at φ̄t since, for instanceI,0)ft
lim (t+1 )∗φt = (1−)t+1 (ξt+1
φ↑φ̄t

I
ξt+1

I
I
> (1−)t+1 (ξt+1,0)ft ξt+1

%
where

R (φ)
dξt+1
dφ

[13:11 11/12/2019 OP-REST190033.tex]

&
−

I (φ)
dξt+1

dφ
I (φ)
dξt+1

dφ

R
+t+1 (ξt+1
,1)ft

R
ξt+1



R (φ)
dξt+1

dφ

(A.104)−= lim (t+1 )∗φt ,φ↓φ̄t

R (φ) at φ̄ . This implies that #"(φ) does not exist at φ̄ .
is the left derivative of ξt+1
t
t

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Equation (A.104) implies that the objective of the bank is weakly convex. Thus, to study
global optimality it is sufficient to compare the equilibrium choice of leverage, φ̄t , to deviations
to corner solutions.
νtWhen the incentive constraint is binding, i.e. #t φ t = μrt > 0 at φ t = θ −μ
, a bank cannottincrease its own leverage above φ̄t so that the only deviation that we need to check is φ = 1.
Therefore, the condition for global optimality in this case is:'(νt#t (1) < #t.
θ −μt

(A.105)
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νt
When the constraint is not binding, i.e. #t φ t = μrt = 0 and φ t < θ −μ
, an individual bankt

could deviate to either φ = 1 or φ = φtIC , where φtIC is the maximum level of leverage compatible
with incentive constraints, i.e. #t (φ IC ) = θφ IC . In this case, given weak convexity of the objective,
the global optimality condition is satisfied if and only if#t (1) = #t φ̄t = #t φtIC .

(A.106)

Notice that equation (A.104) implies that the above equality is satisfied
 if and only if the probabilityof default is zero for any feasible choice of leverage φ ∈ 1,φtIC which would result in a flat
objective function.
We verify numerically that condition (A.105) is satisfied in the neighbourhood of the riskadjusted steady state, where the constraint is binding. Moreover, in our calibration, whenever the
incentive constraint is not binding in equilibrium the probability
is zero for any of insolvencyI (φ) = 0 for φ ∈ (φ̄ ,φ IC ], so
feasible choice of leverage above the equilibrium level, i.e. ft ξt+1
t t
that a deviation by an individual bank to a higher level of leverage is never strictly preferred.
However, the economy does occasionally transit to extreme states in which the constraint isbinding but the probability of the run is high enough that equation (A.105) is violated and to statesin which the constraint is slack and the probability of the run is positive thus violating (A.106).
In such states, a bank would gain by a deviation to φ = 1, see Figure A2. The only equilibrium inthese cases would then be one in which a fraction of banks decrease their leverage in anticipationof 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 φ = 1. This cost could reflect expenses involved in a major
restructuring of the bank’s portfolio. It could also reflect reputation costs associated with thebank’s refusal to accept deposits in a given period in order to survive a run in the subsequent
period. In particular, we posit that the objective of the bank is given by
Vt (nt ) = #t (φ)nt 1−τ φ̄t for φ ∈ [1, φ̄t ).
That is, a deviation of a bank that reduces leverage below the aggregate value φ̄t entails a fixed
cost τ φ̄t per unit of net worth. We check computationally that the deviation is never profitable,
i.e. #t φ̄t > #t (1), in all of our experiments for values of τ which are greater than or equal to
0.77%.41 Examining asymmetric equilibrium without such reputation cost is a topic of future
research.
41. The value of deviating can increase in very extreme cases but in a simulation of 100 thousands periods it is still
below 1.7% for 99% of the times.

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A.6.

ArticleA MACRO MODEL WITH FINANCIAL PANICS

285

Computation

It is convenient for computations to let the aggregate state of the economy be given by
Mt =(St−1 ,Nt ,ξt ,ιt ).

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Notice that bank net worth replaces the specific asset and liability position of banks in the naturalb ,Dstate that we have used so far M̃t =(St−1 ,St−1
t−1 R̄t ,ξt ). To see that this state is sufficient
to compute the equilibrium we rewrite the evolution of net worth, equation (A.95), forward.
Using the definition of the leverage multiple and the budget constraint of the banker, we get thatwhenever there is no run at time t, so that Nt > 0, the evolution of net worth is given by⎧, 

if there is no default :Zt+1 +(1−δ)Qt+1
⎪
⎪
σ
N
φ
ξ
+R
+ζ St
−R
t
t
t
t+1
t+1
⎪
Qt
⎪
ξt+1 ∈ !N⎪
t+1 (φt )
⎪
⎪
⎪
⎪
⎪
⎨
if there is a run :Nt+1 =.
0
R and ι< ξt+1ξ⎪
t+1
t+1 = 1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
if banks are insolvent:
⎪
⎩
ζ St
I
and ιt+1 = 0
ξt+1 < ξt+1
(A.107)
Otherwise, if a run has happened at time t so that Nt = 0, the evolution of net worth is given byequation (2.34), which we report for convenience:
'
(
St−1
.
(A.108)Nt+1 = ζ St 1+σSt
We can then look for a recursive equilibrium in which each equilibrium variable is a function of
Mt and the evolution of net worth is given by a function Nt+1 Mt ; ξt+1 ,ιt+1 that depends on
the realization of the exogenous shocks ξt+1 ,ιt+1 and satisfies equations (A.107) and (A.108)
above.
We use time iteration in order to approximate the functions,
RIϑ = Q(M),C(M),ψ (M),ξt+1(M),ξt+1(M),T M;  ,ι ,where T Mt ;  ,ι is the transition law determining the stochastic evolution 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. Fix a grid of values for the state G ⊂ S m ,S M × 0,N M × 1−4σ ξ ,1+4σ ξ × {0,1}
3. Set j = 0 and guess initial values for the equilibrium objects of interest on the grid,RIϑj = Qj (M),Cj (M),ψj (M),ξt+1,j(M),ξt+1,j(M),Tj M;ξ  ,ιM∈G

4. Assume that ϑi has been found for i < M where M is set to 10,000. Use ϑi to find associatedfunctions ϑi in the approximating space, e.g. Qi is the price function that satisfies Qi (M) =Qi (M) for each M ∈ G.
5. Compute all time t +1 variables in the system of equilibrium equations by using the functions
ϑi from the previous step, e.g. for each M ∈ G let Qt+1 = Qi Ti M;  ,ι , and then solve thesystem to get the implied ϑi+1
6. Repeat 4 and 5 until convergence of ϑi[13:11 11/12/2019 OP-REST190033.tex]

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A.7. Impulse response functions
The risk-adjusted steady state is M̄ =(S̄, N̄,1,0) which satisfies:
M̄ =T M̄;0,0,
We compute responses to a sequence of n shocks

irfs irfs n,ι
by starting the economy in thett
t=1

risk-adjusted steady state, M0 = M̄, and computing the evolution of the state given the assumed
shocks from time 1 to n and setting all future shocks to 0, i.e. t = ιt = 0 for t ≥ n+1 :Downloaded from https://academic.oup.com/restud/article-abstract/87/1/240/5505450 by guest on 14 May 2020
 irfs irfs
if t ≤ n
T Mt ; t ,ιt
Mt+1 =
T(Mt ;0,0) if t > n
We then plot for each variable, the values of the associated policy function computed along thispath for the state, e.g. Qt = Q(Mt ). Notice that, given our nonlinear policy,functions,-these values
are different from conditional expectations given the sequence of shocks

irfs irfs n
.
t ,ιt
t=1

A.8. Long rates and forecast errors of credit spreads
We compute long rates as the average of future short rates. In particular, we let rtAAA denote the
10 years expected rate of return on bank assets
1  b
Rt+i −1
Et
40
40

rtAAA =

i=1

and rtT the 10 years risk free rate
rtT =

40

1  f
Rt+i −1 .
Et
40
i=1

We approximate expectations by simulating the model and then measure the spread rtAAA −rtT by
the difference between the AAA-rated corporate bonds and the 10-year Treasuries.
We follow Bordalo et al. (2017) to compute averages of the forecast errors over the year ahead.
That is, letting SPt = rtAAA −rtT , the forecast errors that we report at time t are given by1
SPt+i −Et SPt+i .
4
4

FEt =

i=1



b −Rf
Notice that SPt will be an average of the quarterly excess returns, Et Rt+1
t+1 which, asshown e.g. in Figure 1, are countercyclical. This implies that periods of optimistic beliefs about
the capital quality will translate into periods in which forecasts of the spread over the following 1
year (period from t +1 to t +4) will be low. As long as no boom is realized, the realization of theaverage spreads between periods t +1 and t +4 will then tend to be higher than the date-t forecastof the average spreads of the same periods so that positive date-t forecast errors are associatedwith periods of optimistic expectations.

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Acknowledgments. The views expressed in this article are solely those of the authors and do not necessarily reflect
those of the Board of Governors of the Federal Reserve or the Federal Reserve System. We thank for their helpful comments
Frederic Boissay, Larry Christiano, Pat Kehoe, John Moore, Dominik Thaler, Alejandro Van der Ghote, participants invarious seminars and conferences, as well as anonymous referees and the editor. Financial support from the National
Science Foundation and the Macro Financial Modeling group at the University of Chicago is gratefully acknowledged.

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