Wholesale Banking and Bank Runs in
Macroeconomic Modelling of Financial Crises
Mark Gertler, Nobuhiro Kiyotaki and Andrea Prestipino∗
NYU, Princeton University and Federal Reserve Board of Governors
November 2015

Abstract
There has been considerable progress in developing macroeconomic models
of banking crises. However, most of this literature focuses on the retail sector
where banks obtain deposits from households. In fact, the recent financial crisis
that triggered the Great Recession featured a disruption of wholesale funding
markets, where banks lend to one another. Accordingly, to understand thefinancial crisis as well as to draw policy implications, it is essential to capture
the role of wholesale banking. The objective of this paper is to characterize amodel that can be seen as a natural extension of the existing literature, but
in which the analysis is focused on wholesale funding markets. The modelaccounts for both the buildup and collapse of wholesale banking, and also
sketches out the transmission of the crises to the real sector. We also drawout the implications of possible instability in the wholesale banking sector for
lender-of-last resort policy as well as for macroprudential policy.
Keywords: financial crises, wholesale banking, interbank markets, rolloverrisk
JEL Classsification: E44∗

Prepared for Handbook of Macroeconomics, edited by John B. Taylor and Harald Uhlig. Thanks
to Behzad Diba, Stavros Panageas, John B. Taylor, Harald Uhlig, Ivan Werning and Randall Wrightfor helpful comments

1

1

Introduction

One of the central challenges for contemporary macroeconomics is adapting the coremodels to account for why the recent financial crisis occurred and for why it thendevolved into the worst recession of the postwar period. On the eve of the crisis
the basic workhorse quantitative models used in practice largely abstracted fromfinancial market frictions. These models were thus largely silent on how the crisis
broke out and how the vast array of unconventional policy interventions undertakenby the Federal Reserve and Treasury could have worked to mitigate the effects ofthe financial turmoil. Similarly, these models could not provide guidance for the
regulatory adjustments needed to avoid another calamity.
From the start of the crisis there has been an explosion of literature aimed atmeeting this challenge. Much of the early wave of this literature builds on the financial accelerator and credit cycle framework developed in Bernanke and Gertler(1989) and Kiyotaki and Moore (1997). This approach stresses the role of balance
sheets in constraining borrower spending in a setting with financial market frictions.
Procyclical movement in balance sheet strength amplifies spending fluctuations andthus fluctuations in aggregate economic activity. A feedback loop emerges as conditions in the real economy affect the condition of balance sheets and vice-versa.
Critical to this mechanism is the role of leverage: The exposure of balance sheets tosystemic risk is increasing in the degree of borrower leverage.
The new vintage of macroeconomic models with financial frictions makes progress
in two directions: First, it adapts the framework to account for the distinctive features of the current crisis. In particular, during the recent crisis, it was highly leveraged financial institutions along with highly leveraged households that were mostimmediately vulnerable to financial distress1 . The conventional literature featuredbalance sheet constraints on non-financial firms. Accordingly, a number of recent
macroeconomic models have introduced balance sheet constraints on banks, whileothers have done so for households.2 The financial accelerator remains operative,
but the classes of agents most directly affected by the financial market disruption
differ from earlier work.
Another direction has involved improving the way financial crises are modeled.
For example, financial crises are inherently nonlinear events, often featuring a simul1

To be sure, the financial distress also directly affected the behavior of non-financial firms. See
Giroud and Mueller (2015) for evidence of firm balance sheet effects on employment during thecrisis.
2
See Gertler and Karadi (2011), Gertler and Kiyotaki (2010) and Curdia and Woodford (2010)
for papers that incorporate banking and Eggertsson and Krugman (2012), Geurreri and Lorenzoni(2011) and Midrigan and Philippon (2011) for papers that incuded household debt.

2

taneous sudden collapse in asset prices and rise in credit spreads.3 A sharp collapse
in output typically ensues. Then recovery occurs only slowly, as it is impeded by aslow process of develeraging. A number of papers have captured this nonlinearity byallowing for the possibility that the balance sheet constraints do not always bind.4
Financial crises are then periods where the constraints bind, causing an abrupt contraction in economic activity. Another approach to handling the nonlinearity is toallow for bank runs.5 Indeed, runs on the shadow banking system were a salientfeature of the crisis, culminating with the collapse in September 2008 of Lehman
Brothers, of some major money market funds and ultimately of the entire investment banking sector. Yet another literature captures the nonlinearity inherent infinancial crises by modeling network interactions (see, e.g., Garleanu, Panegeas, andYu, 2015).
One area the macroeconomics literature has yet to address adequately is thedistinctive role of the wholesale banking sector in the breakdown of the financial
system. Our notion of wholesale banks corresponds roughly, though not exactly,
to the shadow banking sector on the eve of the 2007-2009 financial crisis. Shadowbanking includes all financial intermediaries that operated outside the Federal Reserve’s regulatory framework. By wholesale banking, we mean the subset that (i)was highly leveraged, often with short term debt and (ii) relied heavily on borrowing
from other financial institutions in ”wholesale” markets, as opposed to borrowing
from households in ”retail” markets for bank credit.
When the crisis hit, the epicenter featured malfunctioning of the wholesale banking sector. Indeed, retail markets remained relatively stable while wholesale funding
markets experienced dry-ups and runs. By contrast, much of the macroeconomic
modeling of banking features traditional retail banking. In this respect it misses
some important dimensions of both the run-up to the crisis and how exactly thecrisis played out. In addition, by omitting wholesale banking, the literature may be
missing some important considerations for regulatory design.
In this Handbook chapter we present a simple canonical macroeconomic modelof banking crises that (i) is representative of the existing literature; and (ii) extends
this literature to feature a role for wholesale banking. The model will provide someinsight both into the growth of wholesale banking and into how this growth led toa build-up of financial vulnerabilities that ultimately led to a collapse. Because the3

See He and Krishnamurthy (2014) for evidence in support of the nonlinearity of financial crises.
See Brunnermeier and Sannikov (2014), He and Krishnamurthy (2013,2014) and Mendoza(2010).
5
See Gertler and Kiyotaki (2015), Ferrante (2015a), Robatto (2014) and Martin, Skeie and Von
Thadden (2014a,b).
4

3

model builds on existing literature, our exposition of the framework will permit us
to review the progress that is made. However, by turning attention to wholesalebanks and wholesale funding markets, we are able to chart a direction we believe theliterature should take.
In particular, the model is an extension of the framework developed in Gertler
and Kiyotaki (2011), which had a similar two-fold objective: first, present a canonicalframework to review progress that has been made and, second, chart a new direction.
That paper characterized how existing financial accelerator models that featured firmlevel balance sheet constraints could be extended to banking relationships in orderto capture the disruption of banking during the crisis. The model developed there
considered only retail banks which funded loans mainly from household deposits.
While it allowed for an inter-bank market for credit among retail banks, it did notfeature banks that relied primarily on wholesale funding, as was the case with shadowbanks.
For this Handbook chapter we modify the Gertler and Kiyotaki framework toincorporate wholesale banking alongside retail banking, where the amount credit
intermediated via wholesale funding markets arises endogenously. Another important difference is that we allow for the possibility of runs on wholesale banks. We
argue that both these modifications improve the ability of macroeconomic models
to capture how the crisis evolved. They also provide insight into how the financial
vulnerabilities built up in the first place.
As way to motivate our emphasis on wholesale banking, Section 2 presents descriptive evidence on the growth of this sector and the collapse it experienced during
the Great Recession. Section 3 presents the baseline macroeconomic model withbanking, where a wholesale banking sector arises endogenously. Sector 4 conducts
a set of numerical experiments. While the increased size of the wholesale banking
improves the efficiency of financial intermediation, it also raises the vulnerability ofthis sector to runs. Section 5 considers the case where runs in the wholesale sector might be anticipated. It illustrates how the model can capture some of the key
phases of the financial collapse, including the slow run period up to Lehman and theultimate ”fast run” collapse. In section 6 we introduce a second asset in which retailbanks have a comparative advantage in intermediating. We then show how a crisis
in wholesale banking can spill over and affect retail banking, consistent with what
happened during the crisis. Section 7 analyzes government policy to contain financial
crises, including both ex post lender of last resort activity and ex ante macroprudential regulation. Finally, we conclude in section 8 with some directions for future
research.

4

2

The Growth and Fragility of Wholesale Banking

In this section we provide some background motivation for the canonical macroeconomic model with wholesale funding markets that we develop in the following
section. We do so by presenting a brief description of the growth and ultimate collapse of wholesale funding markets during the Great Recession. We also describe
informally how the disruption of these markets contributed to the contraction of thereal economy.
Figure 1 illustrates how we consider the different roles of retail and wholesale
financial intermediaries, following the tradition of Gurley and Shaw (1960).6 The
arrows indicate the direction that credit is flowing. Funds can flow from households(ultimate lenders) to non-financial borrowers (ultimate borrowers) through threedifferent paths: they can be lent directly from households to borrowers K h ; they
can be intermediated by retail banks that raise deposits (D) from households and use
them to make loans to non-financial borrowers (K r ); alternatively, lenders’ deposits
can be further intermediated by specialized financial institutions that raise funds fromretail banks in wholesale funding markets (B) and, in turn, make loans to ultimate
borrowers (K w ). In what follows we refer to these specialized financial institutions
as wholesale banks. We think of wholesale banks as highly leveraged shadow banks
that rely heavily on credit from other financial institutions, particularly short term
credit. We place in this category institutions that financed long term assets, suchas mortgaged back securities, with short term money market instruments, includingcommercial paper and repurchase agreements. Examples of these kinds of financial
institutions are investment banks, hedge funds and conduits. We focus attention on
institutions that relied heavily on short term funding in wholesale markets to financelonger term assets because it was primarily these kinds of entities that experienced
financial turmoil.
Our retail banking sector, in turn, includes financial institutions that rely mainly
on household saving for external funding and provide a significant amount of short
6

Gurley and Shaw (1960) consider that there are two ways to transfer funds from ultimate lenders(with surplus funds) to ultimate borrowers (who need external funds to finance expenditure): direct
and indirect finance. In direct finance, ultimate borrowers sell their securities directly to ultimate
lenders to raise funds. In indirect finance, financial intermediaries sell their own securities to raise
funds from ultimate lenders in order to buy securities from ultimate borrowers. By doing so,
financial intermediaries transform relatively risky, illiquid and long maturity securities of ultimate
borrowers into relatively safe, liquid and short maturity securities of intermediaries. Here we divide
financial intermediaries into wholesale and retail financial intermediaries, while both involve asset
transformation of risk, liquidity and maturity. We refer to intermediaries as ”banks” and to ultimate
lenders as ”households” for short.

5Wholesale
Banks
Wholesale Holdings (Kw)

Wholesale Funding (B)RetailBanks
Retail Holdings (Kr)

Deposits (D)Productive
Asset

Households
Direct Holdings (Kh)

Figure 1: Modes of Financial Intermediation
term financing to the wholesale banks. Here we have in mind commercial banks,money market funds and mutual funds that raised funds mainly from households
and on net provided financing to wholesale banks.
Figure 1 treats wholesale banking as if it is homogenous. In order to understandhow the crisis spread, it is useful to point out that there are different layers within
the wholesale banking sector. While the intermediation process was rather complex,
conceptually we can reduce the number of layers to three basic ones: (1) origination;(2) securitization; (3) and funding. Figure 2 illustrates the chain. First there are”loan originators,” such as mortgage origination companies and finance companies,that made loans directly to non-financial borrowers. At the other end of the chain
were shadow banks that held securitized pools of the loans made by originators. In
between were brokers and conduits that assisted in the securitization process andprovided market liquidity. Dominant in this group were the major investment banks(e.g., Goldman Sachs, Morgan Stanley, Lehman Brothers, etc.). Each of these layers
relied on short term funding, including commercial paper, asset-backed commercial
paper and repurchase agreements. While there was considerable inter-bank lending among wholesale banks, retail banks (particularly money market funds) on net
provided short term credit in wholesale credit markets.
We next describe a set of facts about wholesale banking. We emphasize three6

ABS
Issuers
Holders

Funding (REPO/ABCP/ABS)

ABS
Issuers

Originators

Loan Origination

Wholesale Funding Markets

Ultimate
Borrower

RetailBanks

Figure 2: Wholesale Intermediation
sets of facts in particular: (1) wholesale banking grew in relative importance over thelast four decades; (2) leading up to the crisis wholesale banks were highly exposed
to systemic risk because they were highly leveraged and relied heavily on short term
debt; and (3) the subsequent disruption of wholesale funding markets raised credit
costs and contracted credit flows, likely contributing in a major way to the GreatRecession.
1. Growth in Wholesale Banking
We now present measures of the scale of wholesale banking relative to retail banking as well as to household’s direct asset holdings. Table 1 describes how we constructmeasures of assets held by wholesale versus retail banks. In particular it lists how wecategorized the various types of financial intermediaries into wholesale versus retailbanking.7,8 As the table indicates, the wholesale banking sector aggregates financial
institutions that originate loans, that help securitize them and that ultimately fund
7

The Appendix provides details about measurement of the time series shown in this section from
Flow of Funds data.
8
It is important to notice that the measures we report are broadly in line with analogous measurescomputed for shadow banking. See, e.g., Adrian and Ashcraft (2012), for an alternative definition of
shadow banking that yields very simlar conclusions and Pozsar et al (2013), for a detailed descriptionof shadow banking.

7them. A common feature of all these institutions, though, is that they relied heavily
on short term credit in wholesale funding markets.

Retail Sector

Wholesale Sector

Table 1
Private Depository Institutions
Money Market Mutual Funds
Mutual Funds
Finance Companies
Real Estate Investment Trusts
Origination
Government Sponsored Enterprises
Securitization
Security Brokers Dealers
ABS Issuers
GSE Mortgage Pools
F unding
Funding Corporations
Holding Companies

Figure 3 portrays the log level of credit to non-financial sector provided by wholesale banks, by retail banks, and directly by households from the early 1980s until
the present.9 The figure shows the rapid increase in wholesale banking relative tothe other means of credit supply to non-financial sector. Wholesale banks went fromholding under fifteen percent of total credit in the early 1980s to roughly forty percent on the eve of the Great Recession, an amount on par with credit provided byretail banks.
Two factors were likely key to the growth of wholesale banking. The first isregulatory arbitrage. Increased capital requirements on commercial banks raised the
incentive to transfer asset holding outside the commercial bank system. Second,financial innovation improved the liquidity of wholesale funding markets. The securitization process in particular improved the (perceived) safety of loans by diversifying idiosyncratic risks as well as by enhancing the liquidity of secondary markets
for bank assets. The net effect was to raise the borrowing capacity of the overall
financial intermediary sector.
2. Growth in Leverage and Short Term Debt in Wholesale Banking
Wholesale banking not only grew rapidly, it also became increasingly vulnerable
to systemic disturbances. Figure 4 presents evidence on the growth in leverage in
the investment banking sector. Specifically it plots the aggregate leverage multiple
9

The measure we present also include nonfinancial corporate equities. Excluding equities, households would become negligible but the relative size of wholesale and retail banks would evolve very
similarly. See the Appendix for details on how we construct the measures reported.

8

350

300

250

200

150

100
1975

1980

1985

Housheolds Intermediation

1990

1995

2000

Retail Banks Intermedaition

2005

2010

2015

2020

Wholesale Banks Intermediation

Figure 3: Intermediation by Sector
The graph shows the evolution of credit intermediated by the three different sectors. Nominal datafrom the flow of funds are deflated using the CPI and normalized so that the log of the normalized
value of real wholesale intermediation in 1980 is equal to 1. The resulting time series are thenmultiplied by 100

for broker dealers (primarily investment banks) from 1980 to the present. We define
the leverage multiple as the ratio of total assets held to equity.10 The greater is theleverage multiple, the higher is the reliance on debt finance relative to equity. The
key takeaway from Figure 4 is that the leverage multiple grew from under five in theearly 1980s to over forty at the beginning of the Great Recession, a nearly tenfold
increase.
Arguably, the way securitization contributed to the overall growth of wholesalebanking was by facilitating the use of leverage. By constructing assets that appeared
safe and liquid, securitization permitted wholesale banks to fund these assets by
issuing debt. At a minimum debt finance had the advantage of being cheaper due to
10

The data is from the Flow of Funds and equity is measured by book value. We exclude nonfinancial assets from measurement as they are not reported in the Flow of Funds.

9

45
40
35
30
25
20
15
10
5
0
1975

1980

1985

1990

Brokers Leverage

1995

2000

2005

2010

2015

2020

Brokers Leverage Net Repo and Security Credit

Figure 4: Brokers Leverage
Leverage is given by the ratio of total financial assets over equity. Equity is computed from theflow of funds by subtracting total financial liabilities from total financial assets. The net position
leverage computes assets by netting out long and short positions in REPO and Security Credit.
See the Appendix for details.

the tax treatment. Debt financing was also cheaper to the extent the liabilities wereliquid and thus offered a lower rate due to a liquidity premium.
Why were these assets funded in wholesale markets as opposed to retail markets? The sophistication of these assets required that creditors be highly informed toevaluate payoffs, especially given the absence of deposit insurance. The complicatedasset payoff structure also suggests that having a close working relationship withborrowers is advantageous. It served to reduce the possibility of any kind of financial
malfeasance. Given these considerations, it makes sense that wholesale banks obtainfunding in inter-bank markets. In these markets lenders are sophisticated financial
institutions as opposed to relatively unsophisticated households in the retail market.
Figure5 shows that much of the growth in leverage in wholesale banking involved
short term borrowing. The figure plots the levels of asset backed commercial pa10

250

200

150

100

50

0
2000

2002

2004

2006

ABCP

2008

2010

REPO

2012

2014

2016

Figure 5: Short Term Wholesale Funding
The graph shows the logarithm of the real value outstanding. Nominal values from Flow of FUnds
are deflated using the CPI

per (ABCP) and repurchase agreements (Repo). This growth reflected partly thegrowth in assets held by wholesale banks and partly innovation in loan securitization that made maturity transformation by wholesale banks more efficient. Also
relevant, however, was a shift in retail investors demand from longer term security
tranches towards short term credit instruments as the initial fall in housing prices
in 2006 raised concerns about the quality of existing securitized assets.11,12 As wediscuss next, the combination of high leverage and short term debt is what made thewholesale banking system extremely fragile.
11

See Brunnermeier and Oemke (2013) for a model in which investors prefer shorter maturities
when realease of information could lead them not to roll over debt.
12
It is not easy to gather direct evidence on this from the aggregate composition of liabilities ofwholesale banks since data from the Flow of Funds excludes the balance sheets of SIVs and CDOs
from the ABS Issuers category. Our narrative is based on indirect evidence coming from ABX
spreads as documented for example in Gorton (2009).

113. The Crisis: The Unraveling of Wholesale Bank Funding Markets
The losses suffered by mortgage originators due to falling housing prices in 2006
eventually created strains in wholesale funding markets. Short term wholesale funding markets started experiencing severe turbulence in the summer of 2007. In July
2007 two Bear Sterns investment funds that had invested in subprime related products declared bankruptcy. Shortly after, BNP Paribas had to suspend withdrawals
from investment funds with similar exposure. These two episodes led investors toreassess the risks associated with the collateral backing commercial paper offeredby asset backed securities issuers. In August 2007 a steady contraction of Asset
Backed Commercial Paper (ABCP) market began, something akin to a ”slow run”,
in Bernanke’s terminology.13 The value of Asset Backed Commercial Paper outstanding went from a peak of 1.2 trillion dollars in July 2007 to 800 billion dollars in
December of the same year and continued its descent to its current level of around200 billion dollars.
The second significant wave of distress to hit wholesale funding markets featured
the collapse of Lehman Brothers in September of 2008. Losses on short term debt
instruments issued by Lehman Brothers led the Reserve Primary Fund, a large Money
Market Mutual Fund (MMMF), to ”break the buck”: the market value of assets fell
below the value of its non-contingent liabilities. An incipient run on MMMFs wasaverted only by the extension of Deposit Insurance to these types of institutions.
Wholesale investors,14 however, reacted by pulling out of the Repo market, switching
off the main source of funding for Security Broker Dealers. Figure 5 shows the sharp
collapse in repo financing around the time of the Lehman collapse. Indeed if thefirst wave of distress hitting the ABCP market had the features of a ”slow run”, thesecond, which led to the dissolution of the entire investment banking system had thefeatures of a traditional ”fast run.”
We emphasize that a distinctive feature of these two significant waves of financial
distress is that they did not involve traditional banking institutions. In fact, theretail sector as a whole was shielded thanks to prompt government intervention that
halted the run on MMMFs in 2008 as well as the Troubled Asset Relief Program andother subsequent measures that supplemented the traditional safety net. In fact,
total short term liabilities of the retail sector were little affected overall (See Figure19). This allowed the retail banking sector to help absorb some of the intermediation
13

Covitz, Liang and Suarez (2013) provide a detailed description of the run on ABCP programsin 2007. A very clear description of the role of commercial paper during the 2007-2009 crisis ispresented by Kacperczyk and Schnabl (2010).
14
The poor quality of available data makes it difficult to exactly identify the identity of theinvestors running on Repo’s. See Gorton (2012) and Krishnamurthy Nagel and Orlov (2014).

12

Spreads and Investment
3
900

400

1

0

-100

-1

Billions of (2009) Dollars

Percentage Points

2

-600
-2

-3
2003

2005

2004

2006

ABCP Spread

FIN CP Spread

Residential

DurablesInvestment

Investment

2007

2008

2009

2010

-1100
2011

Excess Bond Premium
Business

Total

Investment

Investment

Figure 6: Credit Spreads and Investment
previously performed by wholesale banks.
Despite the unprecedented nature and size of government intervention and thepartial replacement of wholesale intermediation by retail bank lending, the distress inwholesale bank funding markets led to widespread deterioration in credit conditions.
Figure 6 plots the behavior of credit spreads and investment from 2004 to 2010. We
focus on three representative credit spreads: (1) The spread between the three month
ABCP rate and three month Treasury spread; (2) The financial company commercial
paper spread; and (3) The Gilchrist and Zakrajsek (2012) excess bond premium. In
each case the spread is the difference between the respective rate on the private
security and a similar maturity treasury security rate. The behavior of the spreads
lines up with the waves of financial distress that we described. The ABCP spreadjumps by 1.5% in August 2007, the beginning of the unraveling of this market. The
increase in this spread implies a direct increase in credit costs for borrowing fundedby ABCP including mortgages, car loans, and credit card borrowing. As problems
spread to broker dealers, the financial commercial paper spread increases reachinga peak at more than 1.5% at the time of the Lehman collapse. Increasing costs13

of credit for these intermediaries, in turn, helped fuel increasing borrowing costs for
non-financial borrowers. The Gilchrist and Zakrajsek’s corporate excess bond spreadjumps more than 2.5% from early 2007 to the peak in late 2008.
It is reasonable to infer that the borrowing costs implied by the increased credit
spreads contributed in an important way to the slowing of the economy at the onsetof the recession in 2007:Q4, as well as to the sharp collapse following the Lehman
failure. As shown in Figure 7, the contraction in business investment, residential
investment, durable consumption and their sum - total investment, moves inverselywith credit spreads.
In our view, there are three main conclusions to be drawn from the empiricalevidence presented in this section. First, the wholesale banking sector grew into avery important component of financial intermediation by relying on securitization to
reduce the risks of lending and expand the overall borrowing capacity of the financial
system. Second, higher borrowing capacity came at the cost of increased fragility
as high leverage made wholesale banks’ net worth very sensitive to corrections inasset prices. Third, the disruptions in wholesale funding markets that took place in2007 and 2008 seem to have played an important role in the unfolding of the GreatRecession. These observations motivate our modeling approach below and our focus
on interbank funding markets functioning and regulation.

3

Basic Model

3.1

Key Features

Our starting point is the infinite horizon macroeconomic model with banking andbank runs developed in Gertler and Kiyotaki (2015). In order to study recent financial
booms and crises, in this chapter we disaggregate banking into wholesale and retailbanks. Wholesale banks make loans to the non-financial sector funded primarily
by borrowing from retail banks. The latter use deposits from households to makeloans both to the non-financial sector and to the wholesale financial sector. Further,
the size of the wholesale banking market arises endogenously. It depends on two
key factors: (1) the relative advantage wholesale banks have in managing assets
over retail banks; and (2) the relative advantage of retail banks over households inover-coming an agency friction that impedes lending to wholesale banks.15
15

Our setup bears some resemblance to Holmstrom and Tirole (1997), which has non-financial
firms that face costs in raising external funds from banks that in turn face costs in raising deposits
from households. In our case it is constrained wholesale banks that raise funds from constrained
retail banks.

14

In the previous section we described the different layers of the wholesale sector,
including origination, securitization and funding. For tractability, in our model weconsolidate these various functions into a single type of wholesale bank. Overall, ourmodel permits capturing financial stress in wholesale funding markets which was a
key feature of the recent financial crisis.
There are three classes of agents: households, retail banks, and wholesale banks.
There are two goods, a nondurable good and a durable asset, ”capital.” Capital does
not depreciate and the total supply of capital stock is fixed at K. Wholesale andretail banks use borrowed funds and their own equity to finance the acquisition ofcapital. Households lend to banks and also hold capital directly. The sum of total
holdings of capital by each type of agent equals the total supply:
Ktw + Ktr + Kth = K,

(1)

where Ktw and Ktr are the total capital held by wholesale and retail bankers and Kth
is the amount held by households.
Agents of type j use capital and goods as inputs at t to produce output andcapital at t + 1, as follows:
date tKtj capitalF j (Ktj ) goods

date t+1→

Zt+1 Ktj output
Ktj capital(2)

where type j = w, r and h stands for wholesale banks, retail banks, and households,
respectively. Expenditure in terms of goods at date t reflects the management costof screening and monitoring investment projects. In the case of retail banks, themanagement costs might also reflect various regulatory constraints. We suppose thismanagement cost is increasing and convex in the total amount of capital, as givenby the following quadratic formulation:
F j (Ktj ) =αj j 2(Kt ) .
2

(3)

In addition we suppose the management cost is zero for wholesale banks and highest
for households (holding constant the level of capital):
αw = 0 < αr < αh .

(Assumption 1)

This assumption implies that wholesale bankers have an advantage over the other15

agents in managing capital.16 Retail banks in turn have a comparative advantage
over households. Finally, the convex cost implies that it is increasingly costly at themargin for retail banks and households to absorb capital directly. As we will see,this cost formulation provides a simple way to limit agents with wealth but lack ofexpertise from purchasing assets during a firesale.
In our decentralization of the economy, a representative household provides capital management services both for itself and for retail banks. For the latter, thehousehold charges retail banks a competitive price ftr per unit of capital managed,
where ftr corresponds to the marginal cost of providing the service:
ftr = F r0 (Ktr ) = αr Ktr .

(4)Households obtain the profit from this activity ftr Ktr − F r (Ktr ).

3.2

Households

Each household consumes and saves. Households save either by lending funds tobankers or by holding capital directly in the competitive market. They may deposit
funds in either retail or wholesale banks. In addition to the returns on portfolio investments, every period each household receives an endowment of nondurable goods,
Zt W h , that varies proportionately with the aggregate productivity shock Zt .
Deposits held in a bank from t to t + 1 are one period bonds that promise to pay
the non-contingent gross rate of return R̄t+1 in the absence of a run by depositors.
In the event of a deposit run, depositors only receive a fraction xrt+1 of the promised
return, where xrt+1 is the total liquidation value of retail banks assets17 per unit ofpromised deposit obligations. Accordingly, we can express the household’s return on
deposits, Rt+1 , as follows:
Rt+1 if no deposit run(5)
Rt+1 =
r
xt+1 Rt+1 if deposit run occurs
where 0 ≤ xrt < 1. Note that if a deposit run occurs all depositors receive the samepro rata share of liquidated assets.
16

In general we have in mind that wholesale and retail banks specialize in different types oflending and, as a consequence, each has developed relative expertise in managing the type of assets
they hold. We subsequently make this point clearer by introducing a second asset in which retailbanks have a comparative advantage in intermediating. Also relevant are regulatory distortions,though we view this as a factor that leads to specialization in the first place.
17
Under our calibration only retail banks choose to issue deposits. See below.

16Household utility Ut is given by
Ut = Et∞X!
hβ i ln Ct+ii=0

where Cth is household consumption and 0 < β < 1. Let Qt be the market priceof capital. The household then chooses consumption, bank deposits Dt and direct
capital holdings Kth to maximize expected utility subject to the budget constrainth
Cth + Dt + Qt Kth + F h (Kth ) = Zt W h + Rt Dt−1 + (Zt + Qt )Kt−1+ ftr Ktr − F r (Ktr ). (6)Here, consumption, saving and management costs are financed by the endowment,
the returns on savings, and the profits from providing management services to retailbankers.
For pedagogical purposes, we begin with a baseline model where bank runs arecompletely unanticipated events. Accordingly, in this instance the household chooses
consumption and saving with the expectation that the realized return on deposits,
Rt+i , equals the promised return, Rt+i , with certainty, and that asset prices, Qt+i ,are those at which capital is traded when no bank run happens. In a subsequent
section, we characterize the case where agents anticipate that a bank run may occur
with some likelihood.
Given that the household assigns probability zero to a bank run, the first ordercondition for deposits is given by
Et (Λt,t+1 )Rt+1 = 1(7)

where the stochastic discount factor Λt,τ satisfies
Λt,τ = β τ −t

Cth
.
Cτh

The first order condition for direct capital holdings is given byhEt Λt,t+1 Rkt+1
=1
with
h
Rkt+1
=

(8)

Qt+1 + Zt+1
Qt + F h0 (Kth )hwhere F h0 (Kth ) = αh Kth and Rt+1
is the household’s gross marginal rate of returnfrom direct capital holdings.

17

3.3

Banks

There are two types of bankers, retail and wholesale. Each type manages a financial
intermediary. Bankers fund capital investments (which we will refer to as ”nonfinancial loans”) by issuing deposits to households, borrowing from other banks in
an interbank market and using their own equity, or net worth. Banks can also lendin the interbank market.
As we describe below, bankers may be vulnerable to runs in the interbank market.
In this case, creditor banks suddenly decide to not rollover interbank loans. In theevent of an interbank run, the creditor banks receive a fraction xw
t+1 of the promisedw
return on the interbank credit, where xt+1 is the total liquidation value of debtor
bank assets per unit of debt obligations. Accordingly, we can express the creditor
bank’s return on interbank loans, Rbt+1 , as follows:
Rbt+1 if no interbank run(9)
Rbt+1 =w
xt+1 Rbt+1 if interbank run occurs
where 0 ≤ xw
t < 1. If an interbank run occurs, all creditor banks receive the samepro rata share of liquidated assets. As in the case of deposits, we continue to restrict
attention to the case where bank runs are completely unanticipated, before turning
in a subsequent section to the case of anticipated runs in wholesale funding markets.
Due to financial market frictions that we specify below, bankers may be constrained in their ability to raise external funds. To the extent they may be constrained, they will attempt to save their way out of the financing constraint byaccumulating retained earnings in order to move toward one hundred percent equity
financing. To limit this possibility, we assume that bankers have a finite expected
lifetime: Specifically, each banker of type j (where j = w and r for wholesale andretail bankers) has an i.i.d. probability σ j of surviving until the next period and aprobability 1 − σ j of exiting. This setup provides a simple way to motivate ”dividend payouts” from the banking system in order to ensure that banks use leveragein equilibrium.
Every period new bankers of type j enter with an endowment wj that is received
only in the first period of life. This initial endowment may be thought of as the startup equity for the new banker. The number of entering bankers equals the number
who exit, keeping the total constant.
We assume that bankers of either type are risk neutral and enjoy utility fromconsumption in the period they exit. The expected utility of a continuing banker at18

the end of period t is given by
Vtj = Et

"∞
X
i=1

#
β i (1 − σ j )(σ j )i−1 cjt+i ,where (1 − σ j )(σ j )i−1 is the probability of exiting at date t + i, and cjt+i is terminal
consumption if the banker of type j exits at t + i.
The aggregate shock Zt is realized at the start of t. Conditional on this shock,the net worth of ”surviving” bankers j is the gross return on non-financial loans net
the cost of deposits and borrowing from the other banks, as follows:jnjt = (Qt + Zt ) kt−1
− Rt djt−1 − Rbt bjt−1 ,(10)where djt−1 is deposit and bjt−1 is interbank borrowing at t − 1. Note that bjt−1 ispositive if bank j borrows and negative if j lends in the interbank market.
For new bankers at t, net worth simply equals the initial endowment:
njt = wj .

(11)

Meanwhile, exiting bankers no longer operate banks and simply use their net worth
to consume:
cjt = njt .
(12)
During each period t, a continuing bank j (either new or surviving) finances nonfinancial loans (Qt + ftj )ktj with net worth, deposit and interbank debt as follows:(Qt + ftj )ktj = njt + djt + bjt ,(13)

where ftr is given by (4) and ftw = 0. We assume that banks can only accumulate net
worth via retained earnings. While this assumption is a reasonable approximationof reality, we do not explicitly model the agency frictions that underpin it.18
To derive a limit on the bank’s ability to raise funds, we introduce the following
moral hazard problem: After raising funds and buying assets at the beginning oft, but still during the period, the banker decides whether to operate ”honestly” or
to divert assets for personal use. Operating honestly means holding assets until the
payoffs are realized in period t + 1 and then meeting obligations to depositors andinterbank creditors. To divert means to secretly channel funds away from investments
in order to consume personally.
18

See Bigio (2015) for a model that explains why banks might find it hard to raise external equity
during crises in the presence of adverse selection problems.

19

To motivate the use of wholesale funding markets along with retail markets, we
assume that the banker’s ability to divert funds depends on both the sources anduses of funds. The banker can divert the fraction θ of non-financial loans financed
by retained earnings or funds raised from households, where 0 < θ < 1. On theother hand, he/she can divert only the fraction θω of non-financial loans financed byinterbank borrowing, where 0 < ω < 1. Here we are capturing in a simple way thatbankers lending in the wholesale market are more effective at monitoring the banks
to which they lend than are households that supply deposits in the retail market.
Accordingly, the total amount of funds that can be diverted by a banker who is a
net borrower in the interbank market is given byθ[(Q + f j )ktj − bjt + ωbjt ]where (Q + f j )ktj − bjt equals the value of funds invested in non-financial loans that isfinanced by deposits and net worth and where bjt > 0 equals the value of non-financial
loans financed by inter-bank borrowing.
For bankers that lend to other banks, we suppose that it is more difficult to divert
interbank loans than non-financial loans. Specifically, we suppose that a banker candivert only a fraction θγ of its loans to other banks, where 0 < γ < 1. Here weappeal to the idea that interbank loans are much less idiosyncratic in nature than
non-financial loans and thus easier for outside depositors to monitor. Accordingly,
the total amount a bank that lends on the interbank market can divert is given byθ[(Qt + ftj )ktj + γ(−bjt )]with bjt < 0. As we will make clear shortly, key to operation of the inter-bank market
are the parameters that govern the moral hazard problem in this market, ω and γ.
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. For
this reason the banker must decide whether to divert at t, prior to the realization of
uncertainty at t + 1. The cost to the banker of the diversion is that the creditors canforce the intermediary into bankruptcy at the beginning of the next period.
The banker’s decision at t boils down to comparing the franchise value of the bankj
Vt , which measures the present discounted value of future payouts from operatinghonestly, with the gain from diverting funds. In this regard, rational lenders will
not supply funds to the banker if he has an incentive to cheat. Accordingly, any
financial arrangement between the bank and its lenders must satisfy the following
set of incentive constraints, which depend on whether the bank is a net borrower or20

lender in the interbank market:
Vtj ≥ θ[(Q + f j )ktj − bjt + ωbjt ], if bjt > 0
Vtj ≥ θ[(Qt + ftj )ktj + γ(−bjt )],(14)

if bjt < 0.

As will become clear shortly, each incentive constraint embeds the constraint that the
net worth njt must be positive for the bank to operate: This is because the franchisevalue Vtj will turn out to be proportional to njt .
Overall, there are two basic factors that govern the existence and relative sizeof the interbank market. The first is the cost advantage that wholesale banks havein managing non-financial loans, as described by Assumption 1. The second is thesize of the parameters ω and γ which govern the comparative advantage that retailbanks have over households in lending to wholesale banks . Observe that as ω and γ
decline, it becomes more attractive to channel funds through wholesale bank funding
markets relative to retail markets. As ω declines below unity, a bank borrowing in
the wholesale market can relax its incentive constraint by substituting inter-bankborrowing for deposits. Similarly, as γ declines below unity, a bank lending in thewholesale market can relax its incentive constraint by shifting its composition ofassets from non-financial loans to inter-bank loans.
In what follows, we restrict attention to the case in whichω + γ > 1.

(Assumption 2)

In this instance the parameters ω and γ can be sufficiently small to permit an empirically reasonable relative amount of inter-bank lending. However, the sum of theseparameters cannot be so small as to induce a situation of pure specialization by retailbanks, where these banks do not make non-financial loans directly but instead lend
all their funds to wholesale banks.19,20 Since in practice retail banks hold some ofthe same types of assets held by wholesale banks, we think it reasonable to restrict
attention to this case.
We now turn to the optimization problems for both wholesale and retail bankers.
Given that bankers simply consume their net worth when they exit, we can restate
the bank’s franchise value recursively as the expected discounted value of the sum of19

See Section 9.1 in the Appendix for the formal argument that shows that under Assumption 2
pure specialization of retail bankers cannot be an equilibrium.
20
Holmstrom and Tirole (1997) make similar assumptions on the levels and sum of the agency
distortions for banks and non-financial firms in order to explain why bank finance arises.

21

net worth conditional on exiting and the value conditional on continuing as:j
Vtj = βEt [(1 − σ j )njt+1 + σ j Vt+1
].

(15)

= Et [Ωjt+1 njt+1 ]whereΩjt+1 = β1 − σj + σjj Vt+1

njt+1

!
.

(16)

The stochastic discount factor Ωjt+1 ,which the bankers use to value njt+1 , is a probability weighted average of the discounted marginal values of net worth to exiting
and to continuing bankers at t+1. For an exiting banker at t + 1 (which occurs withprobability 1 − σ j ), the marginal value of an additional unit of net worth is simply unity, since he or she just consumes it. For a continuing banker (which occurs
with probability σ j ), the marginal value is the franchise value per unit of net worthjj
Vt+1/njt+1 (i.e., Tobin’s Q ratio). As we show shortly, Vt+1
/njt+1 depends only on
aggregate variables and is independent of bank specific factors.
We can express the banker’s evolution of net worth as:jnjt+1 = Rkt+1
Qt + ftj ktj − Rt+1 djt − Rbt+1 bjt(17)
jwhere Rkt+1
is the rate of return on non-financial loans, given by
j
Rkt+1
=

Qt+1 + Zt+1
Qt + ftj(18)

The banker’s optimization problem then is to choose ktj , djt , bjt each period to maximize the franchise value (15) subject to the incentive constraint (14) and the balance
sheet constraints (13) and (17).
We defer the details of the formal bank maximization problems to Appendix A.
Here we explain the decisions of wholesale and retail banks informally. Because
wholesale banks have a cost advantage over retail banks in making non-financial
loans, the rate of return on non-financial loans is higher for the former than for thelatter (see equation (18)). In turn, retail banks have an advantage over households
in lending to wholesale banks due to their relative advantage in recovering assets indefault. Therefore, if the interbank market is active in equilibrium, wholesale banks
borrow from retail banks in the interbank market to make non-financial loans. Indeed

22the only reason retail banks directly make non-financial loans is because wholesalebanks may be constrained in the amount of this type of loan they can make.21
In the text, we restrict attention to the case where the interbank market is active,
with wholesale banks borrowing from retail banks, and where both types of banks
are constrained in raising funds externally.
3.3.1

Wholesale banks

In general, wholesale banks may raise funds either from other banks or from households. Since the kinds of financial institutions we have in mind relied exclusively onwholesale markets for funding, we focus on this kind of equilibrium. In particular,
we restrict attention to model parameterization which generate an equilibrium where
the conditions for the following Lemma 1 are satisfied:w
Lemma 1 : dw
t = 0, bt > 0 and the incentive constraint is binding iff


w
w
w
0 < ωEt Ωw
t+1 (Rkt+1 − Rt+1 ) < Et [Ωt+1 (Rkt+1 − Rbt+1 )] < θωWe first explain why dw
t = 0 in this instance. The wholesale bank faces thefollowing trade-off in using retail deposits: If the deposit interest rate is lower than thew
w
w
interbank interest rate so that Et [Ωwt+1 (Rkt+1 −Rt+1 )] > Et [Ωt+1 (Rkt+1 −Rbt+1 )], thenthe bank gains from issuing deposits to reduce interbank loans. On the other hand,because households are less efficient in monitoring wholesale bank behavior, they will
apply a tighter limit on the amount they are willing to lend than will retail banks.
w
w
w
If ω is sufficiently low so that ωEt [Ωwt+1 (Rkt+1 − Rt+1 )] < Et [Ωt+1 (Rkt+1 − Rbt+1 )],the cost exceeds the benefit. In this instance the wholesale bank does not use retaildeposits, relying entirely on interbank borrowing for external finance. Everythingelse equal, by not issuing retail deposits, the wholesale bank is able to raise its overall
leverage in order to make more non-financial loans relative to its equity base. This
incentive consideration accounts for why the wholesale bank may prefer interbank
borrowing to issuing deposits, even if the interbank rate lies above the deposit rate.22
21

We do not mean to suggest that the only reason retail banks make non-financial loans inpractice is because wholesalse banks are constrained. Rather we focus on this case for simplicityof the basic model. Later we extend the model to allow for a second type of lending, which werefer to as commercial and industrial leanding, where retail banks have a comparative advantage.
In this instance, spillovers emerge where problems in wholesale banking can affect the degree ofintermediation of commercial and industrial loans.
22
Under our baseline parametrization, wholesale banks borrow exclusively from retail banks. Weview this as the case that best corresponds to the wholesale banking system on the eve of the GreatRecession. Circumstances do exist where wholesale banks will borrow from households as well as
retail banks. One might interpret his situation as corresponding to the consolidation of wholesale

23

w
Next we explain why the incentive constraint is binding. If Et [Ωwt+1 (Rkt+1 −Rbt+1 )] < θω, then at the margin the wholesale bank gains by borrowing on theinterbank market and then diverting funds to its own account. Accordingly, as the
incentive constraint (14) requires, rational creditor banks will restrict lending to thepoint where the gain from diverting equals the bank franchise value, which is what
the wholesale bank would lose if it cheated.
Given Lemma 1 we can simplify the evolution of bank net worth tow
w
w
nw
t+1 = [(Rkt+1 − Rbt+1 )φt + Rbt+1 ]nt

where φw
t is given byφwt ≡Qt ktw
.
nw
t

(19)

(20)

We refer to this ratio of assets to net worth as the leverage multiple.
In turn, we can simplify the wholesale banks optimization problem to choosing
the leverage multiple to solve:www
Vtw = maxEt {Ωw
t+1 [(Rkt+1 − Rbt+1 )φt + Rbt+1 ]nt }
w
φt

(21)

subject to the incentive constraint
w
w
θ[ωφw
t + (1 − ω)]nt ≤ Vt

(22)

Given the incentive constraint is binding under Lemma 1, we can combine theobjective with the binding incentive constraint to obtain the following solution forφwt :Et (Ωw
t+1 Rbt+1 ) − θ(1 − ω)
(23)
φw
t =
w
θω − Et [Ωwt+1 (Rkt+1 − Rbt+1 )]

23
w
w
w
InNote that φw
t is increasing in Et (Ωt+1 Rkt+1 ) and decreasing in Et (Ωt+1 Rbt+1 ).
w
tuitively, the franchise value Vt increases when returns on assets are higher anddecreases when the cost of funding asset purchases rises, as equation (21) indicates.
Increases in Vtw , in turn, relax the incentive constraint, making lenders will to supply
more credit.
Also, φw
t is a decreasing function of both θ, the diversion rate on non-financial
loans funded by net worth, and ω, the parameter that controls the relative ease of

and retail bank in the wake of the crisis, or perhaps the period before the rapid growth of wholesalebanking when retail banks were performing many of the same activities as we often observe incontinental Europe and Japan.
23w
This is because Et (Ωw
t+1 Rkt+1 ) > 1 > θ in equilibrium as shown in Appendix.

24

diverting nonfinancial loans funded by inter-bank borrowing relative to those fundedby the other means: Increases in either parameter tighten the incentive constraint,
inducing lenders to cut back on the amount of credit they supply. Later we will use
the inverse relationship between φwt and ω to help account for the growth in bothleverage and size of the wholesale banking sector.
Finally, from equation (21) we obtain an expression from the franchise value perunit of net worth
Vtw
ww= Et {Ωwt+1 [(Rkt+1 − Rbt+1 )φt + Rbt+1 ]}
nw
t

(24)

w
where φw
t is given by equation (23) and Ωt+1 is given by equation (16). It is straightw
Vforward to show that ntw exceeds unity: i.e., the shadow value of a unit of net wortht
is greater than one, since additional net worth permits the bank to borrow more andVw
invest in assets earning an excess return. In addition, as we conjectured earlier, ntw
tdepend only on aggregate variables and not on bank-specific ones.

3.3.2

Retail banks

As with wholesale banks, we choose a parametrization where the incentive constraint
binds. In addition, as discussed earlier, we restrict attention to the case where retailbanks are holding both non-financial and inter-bank loans. In particular, we consider
a parametrization where in equilibrium Lemma 2 is satisfied
Lemma 2 : brt < 0, ktr > 0 and the incentive constraint is binding iffr
0 < Et [Ωrt+1 (Rkt+1− Rt+1 )] =1
Et [Ωrt+1 (Rbt+1 − Rt+1 )] < θγFor the retail bank to be indifferent between holding non-financial loans versus interbank loans, the rate on interbank loans Rbt+1 must lie below the rate earnedr
on non-financial loans Rkt+1
in a way that satisfies the conditions for the lemma.
Intuitively, the advantage for the retail bank to making an interbank loan is thathouseholds are willing to lend more to the bank per unit of net worth than for a
non-financial loan. Thus to make the retail bank indifferent, Rbt+1 must be less than
r
Rkt+1
.
Let φrt be a retail bank’s effective leverage multiple, namely the ratio of assets to
net worth, where assets are weighted by the relative ease of diversion:
φrt ≡(Qt + ftr )ktr + γ(−brt )
.
nrt
25

(25)

The weight γ on (−brt ) is the ratio of how much a retail banker can divert frominterbank loans relative to non-financial loans.
Given the restrictions implied by Lemma 2, we can use the same procedure as in
the case of wholesale bankers to express the retail banker’s optimization problem aschoosing φrt to solve:r
Vtr = maxEt {Ωrt+1 [(Rkt+1− Rt+1 )φrt + Rt+1 ]nrt }
r
φt

(26)

subject toθφrt nrt ≤ Vtr
Given Lemma 2, we can impose that incentive constraint binds, which implies
φrt =Et (Ωrt+1 Rt+1 ).
rθ − Et [Ωrt+1 (Rkt+1− Rt+1 )](27)

As with the leverage multiple for wholesale bankers, φrt is increasing in expected asset
returns on the bank’s portfolio and decreasing in the diversion parameter.
Finally, from equation (26) we obtain an expression for the franchise value perunit of net worth
Vtrr= Et {Ωrt+1 [(Rkt+1− Rt+1 )φrt + Rt+1 ]}
nrt

(28)

As with wholesale banks, the shadow value of a unit of net worth exceeds unity anddepends only on aggregate variables.

3.4

Aggregation and Equilibrium without Bank Runs

Given that the ratio of assets and liabilities to net worth is independent of individual
bank-specific factors and given a parametrization where the conditions in Lemma 1
and 2 are satisfied, we can aggregate across banks to obtain relations between total
assets and net worth for both the wholesale and retail banking sectors. Let Qt Ktw
and Qt Ktr be total non-financial loans held by wholesale and retail banks, Dt be
retail bank deposits, Bt be total interbank debt, and Ntw and Ntr total net worth ineach respective banking sector. Then we have:w
Qt Ktw = φwt Nt ,(29)(Qt + ftr )Ktr + γBt = φrt Ntr ,

(30)

26

with
Qt Ktw = Ntw + Bt ,(31)

(Qt + ftr )Ktr + Bt = Dtr + Ntr ,

(32)

and
r
Et [Ωrt+1 (Rkt+1− Rt+1 )] =1
Et [Ωrt+1 (Rbt+1 − Rt+1 )].
γ(33)Equation (33) ensures that the retail bank is indifferent at the margin between holding non-financial loans versus interbank loans (see Lemma 2).
Summing across both surviving and entering bankers yields the following expression for the evolution of Nt :www
Ntw = σ w [(Rkt− Rbt )φw
t−1 + Rbt ]Nt−1 + W ,rr
Ntr = σ r [(Rkt− Rt )φrt−1 + Rt ]Nt−1+ Wrr+ σ r [Rbt − Rt − γ(Rkt− Rt )] Bt−1 ,

(34)
(35)

where W j = (1 − σ j )wj is the total endowment of entering bankers. The first termis the accumulated net worth of bankers that operated at t − 1 and survived to t,which is equal to the product of the survival rate σ j and the net earnings on bankassets.
Total consumption of bankers equals the sum of the net worth of exiting bankers
in each sector:
Nw − W w
Nr − W r
Ctb = (1 − σ w ) t w+ (1 − σ r ) t r(36)
σ
σ
Total gross output Y t is the sum of output from capital, household endowment
Zt W h and bank endowment W r and W i :Y t = Zt + Zt W h + W r + W i .

(37)

Net output Yt , which we will refer to simply as output, equals gross output minus
management costs
Yt = Y t − [F h (Kth ) + F r (Ktr )](38)

Equation (38) captures in a simple way how intermediation of assets by wholesalebanks improves aggregate efficiency. Finally, output is consumed by households andbankers:
Yt = Cth + Ctb .
(39)

27

The recursive competitive equilibrium without bank runs consists of aggregate
quantities
Ktw , Ktr , Kth , Bt , Dtr , Ntw , Ntr , Ctb , Cth , Y t , Yt
, prices(Qt , Rt+1 , Rbt+1 , ftr )and bankers’ variablesjj VtΩjt , Rkt
, j , φjt
nt

!
j=w,r


r
w
r
w
, Zt , which, Rt Dt−1, Rbt Bt−1 , Rt Dt−1
, Kt−1
as a function of the state variables Kt−1
satisfy equations (1, 4, 7, 8, 16, 18, 23, 24, 27 − 39).24

3.5

Unanticipated Bank Runs

In this section we consider unanticipated bank runs. We defer an analysis of anticipated bank runs to Section 5. In general three types of runs are conceivable: (i) arun on wholesale banks leaving retail banks intact; (ii) a run on just retail banks;and (iii) a run on both the wholesale and retail bank sectors. We restrict attention
to (i) because it corresponds most closely to what happened in practice.
3.5.1

Conditions for a Wholesale Bank Run Equilibrium

The runs we consider are runs on the entire wholesale banking system, not on individual wholesale banks. Indeed, so long as an asset firesale by an individual wholesalebank is not large enough to affect asset prices, it is only runs on the system thatwill be disruptive. Given the homogeneity of wholesale banks in our model, theconditions for a run on the wholesale banking system will apply to each individual
wholesale bank.
What we have in mind for a run is a spontaneous failure of the bank’s creditors toroll over their short term loans. In particular, at the beginning of period t, before therealization of returns on bank assets, retail banks lending to a wholesale bank decide
whether to roll over their loans with the bank. If they choose to ”run”, the wholesalebank liquidates its capital and turns the proceeds over to its retail bank creditors
who then either acquire the capital or sell it to households. Importantly, both theretail banks and households cannot seamlessly acquire the capital being liquidated
24

In total we have a system of 23 equations. Notice that (16, 18) have two equations. By Walras’law, the household budget constraint (6) is satisfied as long as deposit market clears as Dt = Dtr .

28

in the firesale by wholesale banks. The retail banks face a capital constraint which
limits asset acquisition and are also less efficient at managing the capital than arewholesale banks. Households can only hold the capital directly and are even lessefficient than retail banks in doing so.
Let Q∗t be the price of capital in the event of a forced liquidation of the wholesalebanking system. Then a run on the entire wholesale bank sector is possible if thewliquidation value of wholesale banks assets, (Zt + Q∗t )Kt−1, is smaller than theiroutstanding liability to interbank creditors, Rbt Bt−1 , so that liquidation would wipeout wholesale banks networth. In this instance the recovery rate in the event of aw∗
wholesale bank run, xw
t , is the ratio of (Zt + Qt )Kt−1 to Rbt Bt−1 and the condition
for a bank run equilibrium to exist is that the recovery rate is less than unity, i.e.
w
(Q∗t + Zt )Kt−1
w
xt =
< 1.
Rbt Bt−1(40)w∗
Let Rkt
be the return on bank assets conditional on a run at t :
w∗
Rkt
≡

Zt + Q∗t,
Qt−1

Then from (40) , we can obtain a simple condition for a wholesale bank run equilibw∗
rium in terms of just two endogenous variables: (i) the ratio of Rkt
to the interbank
wborrowing rate Rbt ; and (ii) the leverage multiple φt−1 :
xw
t =

w∗
φw
Rkt
· w t−1 < 1
Rbt φt−1 − 1(41)

A bank run equilibrium exists if the realized rate of return on bank assets conditionalw∗
on liquidation of assets Rkt
is sufficiently low relative to the gross interest rate oninterbank loans, Rbt , and the leverage multiple is sufficiently high to satisfy condition
φww(41). Note that the expression φwt−1−1 is the ratio of bank assets Qt−1 Kt−1
to interbank
t−1borrowing Bt−1 , which is decreasing in the leverage multiple. Also note that thew∗condition for a run does not depend on individual bank-specific factors since Rkt/Rbtw
and φt−1 are the same for all in equilibrium.
w∗
Since Rkt
, Rbt and φw
t−1 are all endogenous variables, the possibility of a bankrun may vary with macroeconomic conditions. The equilibrium absent bank runs(that we described earlier) determines the behavior of Rbt and φw
t−1 . The value ofw∗∗
Rkt , instead, depends on the liquidation price Qt , whose determination is describedin the next sub-section.
29

3.5.2

The Liquidation PriceTo determine Q∗t we proceed as follows. A run by interbank creditors at t induces allwholesale banks that carried assets from t − 1 to fully liquidate their asset positions
and go out of business.25 Accordingly they sell all their assets to retail banks andhouseholds, who hold them at t. The wholesale banking system then re-builds itself
over time as new banks enter. For the asset firesale during the panic run to bequantitatively significant, we need there to be at least a modest delay in the abilityof new banks to begin operating. Accordingly, we suppose that new wholesale banks
cannot begin operating until the period after the panic run.26
Accordingly, when wholesale banks liquidate, they sell all their assets to retailbanks and households in the wake of the run at date t, implying
K = Ktr + Kth .

(42)

The wholesale banking system then rebuilds its equity and assets as new banks enterat t+1 onwards. Given our timing assumptions and Equation (34) , bank net worth
evolves in the periods after the run according tow
Nt+1= (1 + σ w )W w ,w
w
Nt+i= σ w [(Zt+i + Qt+i )Kt+i−1− Rbt+i Bt+i−1 ] + W w , for all i ≥ 2.

Rearranging the Euler equation for the household’s capital holding (8) yields thefollowing expression for the liquidation price in terms of discounted dividends Zt+ih.
net the marginal management cost αh Kt+i"∞#XhQ∗t = Et
Λt,t+i (Zt+i − αh Kt+i
) − αh Kth .
(43)
i=1

Everything else equal, the longer it takes for the banking sector to recapitalize (meahto fall back to steady state), the lower will be thesured by the time it takes Kt+i
liquidation price. Note also that Q∗t will vary with cyclical conditions. In particular,a negative shock to Zt will reduce Q∗t , possibly moving the economy into a regime
where bank runs are possible.
25

See Uhlig (2010) for an alternative bank run model with endogenous liquidation prices.
Suppose for example that during the run it is not possible for retail banks to identify newwholesale banks that are financially independent of the wholesale banks being run on. New wholesalebanks accordingly wait for the dust to settle and then begin raising fund in the interbank market
in the subsequent period. The results are robust to alternative timing assumptions about the entryof new banks.
26

30

Table 3: Baseline
Steady State

Table 2: Baseline Parameters
β
αh
Wh
σr
αr
Wr
θ
γ
σw
αw
Ww
ω
Z

ρz

4

PARAMETERS
Households
discount rate
Intermediation cost
Endowment
Retail Banks
Survival Probability
Intermediation cost
Endowment
Divertable proportion of assets
Shrinkage of Divertable proportion of interbank loans
Wholesale Banks
Survival Probability
Intermediation cost
Endowment
Shrinkage of divertable proportion of assets
Production
Steady State productivity
Serial correlation of productivity shocks

.99
.03
.006
.96
.0074
.0008
.25
.67
.88
0
.0008
.46

Q
Kr
Kw
Rb
Rrk
R
Rwk
φw
φr
Y
Ch
Nr
Nw

STEADY STATE
price of capitalretail intermediation
wholesale intermediation
Annual interbank rate
Annual retail return on capital
Annual deposit rate
Annual wholesale return on capitalwholesale leverageretail leverage
outputconsumption
retail banks networthwholesale banks networth

.016
.9

Numerical Experiments

In this section we examine how the long-run properties of the model can account for
the growth of the wholesale banking sector and then turn to studying the cyclicalresponses to macroeconomic shocks that may or may not induce runs. Overall thesenumerical examples provide a description of the tradeoff between growth and stability
associated with an expansion of the shadow banking sector and illustrate the realeffects of bank runs in our model.

4.1

Calibration

Here we describe our baseline calibration. This is meant to capture the state of theeconomy at the onset of the financial crisis in 2007.
There are 13 parameters in the model:θ, ω, γ, β, αh , αr , σ r , σ w , W h , W r , W w , σz , ρz .
their values are reported in Table 2, while Table 3 shows the steady state values ofthe equilibrium allocation.

31

1
.4
.4
1.048
1.052
1.04
1.064
20
10
.0229
.0168
.0781
.02

We take the time interval in the model to be a quarter. We use conventional
values for households’ discount factor, β = .99, and the parameters governing thestochastic process for dividends, σz = .05 and ρz = .9. We set W h so that households
endowment income is twice as big as their capital income.
We calibrate managerial costs of intermediating capital for households and retailbankers, αh and αr , in order to obtain the spread between deposit and interbank
interest rates as well as the spread between interbank and non-financial loan rates
both to be 0.8% and 1.6% in annual in steady state.
The fraction of divertible assets purchased by raising deposits, θ, and interbank
loans, ωθ, are set in order to get leverage ratios for retail bankers and wholesalebankers of 10 and 20 respectively.
Our retail banking sector comprises of commercial banks, open end Mutual Funds
and Money Market Mutual Funds (MMMF). In the case of Mutual Funds and MMMF
the computation of leverage is complicated by the peculiar legal and economic detailsof the relationship between these institutions, their outside investors and sponsors.27
Hence, our choice of 10 quite closely reflects the actual leverage ratios of commercial
banks, which is the only sector for which a direct empirical counterpart of leveragecan be easily computed.
To set our target for wholesale leverage we decided to focus on private institutions within the wholesale banking sector that relied mostly on short term debt. A
reasonable range for the leverage multiple for such institutions goes from around 10
for some ABCP issuers 28 to values of around 40 for brokers dealers in 2007. Our
choice of 20 is a conservative target within this range.
The survival rates of wholesale and retail bankers, σ w and σ r , are set in orderfor the distribution of assets across sectors to match the actual distribution in 2007.
Finally, we set W r to make new entrants net worth being equal to 1% of total retailbanks net worth and W w to ensure that wholesale bankers are perfectly specialized.

4.2

Long Run Effects of Financial Innovation

As mentioned in Section 2, the role of wholesale banks in financial intermediation
has grown steadily from the 1980’s to the onset of the financial crisis. This growth
was largely accomplished through a series of financial innovations that enhanced theborrowing capacity of the system by relying on securitization to attract funds from27

On the relationship between MMFs and their sponsors see, for instance, Parlatore (2015) andMcCabe (2010).
28
The same caveat as in the case of MMFs applies here because it is very complicated to factor
in the various lines of credit that were provided by the sponsors of these programs.

32

Level

YSS
0.0235

0.6

0.023

0.5

0.6

1

0.5

0.95

0.4

0.9

0.4

0.02250.0215
0.2

0.2
0.4

1-

0.6

0.1
0.2

0.4

1-

0.6

0.4

0.6

1-
Rk- w,SS-R

0.1

80bps

0.04

70bps

0.02

60bps

1-

0.6

Rkr,SS-RSS

30.2bps

50bps
0.2

wSS

0.4

1-

0.6

0.2

rSS

25

11

20

10

0.4

1-

0.2

0.6

NwSS
0.08

0.02

0.078

0.018

0.076

9

0.016

0.074

10

8

0.014

0.072

5

7
0.2

0.012
0.6
0.2

0.4

0.6

0.4

0.4

1-
1-
1-
No Interbank
Active Interbank with Imperfect Specializaiotn

0.4

1-

0.6

NrSS

0.022

15

0.2

0.6

30bps

0
0.4

0.4

1-

30.4bps

0.06

0.3
0.2

0.85
0.2

SS

0.08

0.4

0.2

0.2

Dw,SS

0.5

0

1.05

0.3

BSS

Level

QSS

0.7

0.022

Level

Kr,SS

Kw,SS

0.6

0.07
0.2

0.4

1-
Baseline Equilibrium

0.6

Figure 7: Comparative Statics: a reduction in ω
institutional investors. While our model abstracts from the details of the securitization process, we capture its direct effects on wholesale banks’ ability of raising funds
in interbank markets with a reduction in the severity of the agency friction between
retail banks and wholesale banks, which is captured by parameter ω. Hence, in thissection we study the long run behavior of financial intermediation in response to a decrease in ω and compare it to the low frequency dynamics in financial intermediation
documented in Section 2.
The direct effect of ameliorating the agency problem between wholesale and retailbanks is a relaxation of wholesale banks’ incentive constraints. The improved abilityof retail banks to seize the assets of wholesale bankers in the case of cheating allows
wholesale bankers to borrow more aggressively from retail bankers.

33

Figure 7 shows how some key variables depend upon ω in the steady state. 29
The general equilibrium effects of a lower ω work through various channels. For an
economy with a lower interbank friction ω, the leverage multiple of the wholesalebanking sector is higher, with a larger capital K w and a larger amount interbank
borrowing B by wholesale banking sector. Conversely, capital intermediated byretail banks K r and households K h tends to be lower. In the absence of bank runs,
the relative shift of assets to the wholesale banking sector implies a more efficient
allocation of capital and consequently a higher capital price Qt . The flow of assets
into wholesale banking, further, reduces the spread between the return on capital forwholesale banks and the interbank rate, as well as the spread between interbank anddeposit rates. Despite lower spreads, both wholesale and retail banks enjoy higher
franchise values thanks to the positive effect of higher leverage on total returns onequity. A unique aspect of financial innovation due to a lower friction in the interbank
market is that the borrowing and lending among banks tends to be larger relative
to the flow-of-funds from ultimate lenders (households) to ultimate non-financial
borrowers. (See Appendix B).
Figure 8 compares the steady state effect of financial innovations on some key
measures of financial intermediation with the observed low frequency trends in theirempirical counterparts. In particular, we assume that the value of ω in our baseline
calibration results from a sequence of financial innovations that took place graduallyfrom the 1980’s to the financial crisis. For simplicity, we divide our sample into 2
periods of equal length and assign a value of ω to each subsample in order to matchthe observed percentage of intermediation of wholesale bankers over the period. Inorder to compute leverage of wholesale banks in Figure 8, we compute leverage of thethree sectors within the wholesale banking sector that were mainly responsible forthe growth of wholesale intermediation. Overall, the steady state comparative statics
capture quite well the actual low frequency dynamics in financial intermediation
observed over the past few decades.30
29

Notice that as ω increases above a certain threshold, two other types of equilibria arise: onein which wholesale bankers are imperfectly specialized and raise funds in both wholesale and retailmarkets; and one in which the interbank market shuts down completely. See the Appendix for
details.
30
The model overstatement of the role of retail intermediation relative to household direct holdingof assets can be rationalized by the lack of heterogeneity in ultimate borrowers’ funding sources
since, in the data, households mainly hold equities while intermediaries are responsible for mostdebt intermediation. Introducing a different type of asset for which intermediaries have a smaller
advantage would then help to reconcile the evolution of the distribution of capital across sectors
predicted by the model in response to financial innovation with the empirical one.

34

Proportion of total intermediation

Ratio between WS and Retail short term funding Proportion of total intermediation

kw
0.5

0.45
0.4

0.35
0.3

0.25
0.2

1980

1985

1990

1995

2000

2005

2010

kr
0.55

0.5

0.45

0.4

0.35
1980

1985

1990

Wholesale Sector's Leverage

0.7
0.6
0.5
0.4
0.3
0.2

1985

1990

1995

2000

2005

2010

2000

2005

2010

w

B/D
0.8

0.1
1980

1995

2000

30

25

20

15

10

2010

5
1980

 =0.61 =0.46

2005

1985

1990

1995

Data

Figure 8: Low Frequency Dynamics in Financial Intermediation

4.3

Recessions and Runs

We now turn to the cyclical behavior of our model economy. Figure 9 shows the
response of the economy to an unanticipated negative six percent shock to productivity Zt , assuming that a run does not happen.31 To capture the effects of financial
liberalization on the cyclical properties of the economy, we consider both our baseline parameterization and one with a higher ω which we set to be equal to the one
associated with the early 1980’s in Figure 10. In both cases the presence of financial
constraints activates the familiar financial accelerator mechanism of Bernanke and31

We choose the size of the shock to generate a fall in output similar to the one that occurred
during the Great Recession.

35

z

20

-0.06
-0.08
-0.1

40

0

20

Run on Wholesale

-0.4

40

-0.04

0

20

4

-0.01
-0.02
-0.03
-0.04

40

0

20

w

0.1

20

Quarters

40

0.1
0.05

0

20

40

Quarters
Recession

0.05

Rb-R
8

1

20

x 10

40

ERw
-R
k

6
4
2
0

40

0

20

40

Nr
0

-0.1
-0.2
-0.3
-0.4
-0.5

20

-3

2

0

0

Nw

0.15

0

0.1

0

0

% from ss% from ss

0.2

0.15

40

3

0

40

0.2

0.3

20

x 10

r

0.4

0

0

-3

Ann.  from ss% from ss

Ann.  from ss-0.02% from ss

-0.3

0

0

0

-0.2

Q

0.02

-0.06

-0.1

% from ss

0

-0.04

% from ss

-0.06

-0.02

0.2

Ann.  from ss

-0.04

kr

0

% from ss% from ss% from ss

0

-0.02

-0.08

kw

y

0

0

20

40

-0.05
-0.1
-0.15
-0.2

0

Quarters

20

40

Quarters

Recession Before Financial Innovation

Figure 9: A recession before and after financial innovation (NO RUN EQUILIBRIUM)
Gertler (1989) and Kiyotaki and Moore (1997). Leverage amplifies the effects of thedrop in Zt on bankers’ net worth, inducing a tightening of financial constraints, as
reflected by an increase in credit spreads. In turn, wholesale banks sell off loans,which reduces asset prices and feeds back into lower net worth. Higher exposure tovariations in Zt and higher leverage make this effect stronger for wholesale banks
that are forced into a firesale liquidation of their assets, which in turn leads them toreduce their demand for interbank loans. As a result, retail bankers increase theirasset holdings and absorb, together with households, the capital flowing out of thewholesale banking sector. However, the relative inefficiency of these agents in inter36

mediating assets makes this process costly as shown by the rise in the cost of bankcredit and the amplification in the drop in output. Under our baseline calibration,
spreads between gross borrowing costs for non financial borrowers and the risk freerate increase by sixty basis points and output drops by eight percent, which is two
percentage points greater than the drop in Zt .32
As we noted earlier, financial innovation makes the economy operate more efficiently in steady state. Figure 11 shows that, absent bank runs, it also makes the
economy more stable as the financial accelerator weakens. In response to the drop in
Zt , the economy with financial innovation features smaller increases in credit spreads
and a smaller drop in assets prices. Intuitively, with financial innovation, retail banks
provide a stronger buffer to absorb loan sales by wholesale banks, which helps stabilize asset prices. At the same time, the economy with financial innovation is more
vulnerable to a bank run.
This is illustrated by the panel titled ”Run on Wholesale” in Figure 9. In thispanel we plot a variable that indicates at each time t whether a run is possible attime t + 1. To construct this variable we definew
Runw
t = 1 − xt

where xw
t is the recovery rate on wholesale debt. Hence, in order for a run to exist
the run variable must be positive.
As shown by the Runw variable, a run on wholesale banks is not possible in thesteady state under both parameterization considered. With a six percent drop in Zt ,a run equilibrium remains impossible in the economy absent financial innovation, i.e.,the one with a high value of ω. However, for the economy with financial innovation(i.e. a low ω), the same drop in Zt is big enough to make a run on wholesale banking
possible. Intuitively, in the low ω economy, wholesale bank leverage ratios arehigher than would be otherwise, and asset liquidation values are lower, which raises
the likelihood that the conditions for a bank run equilibrium will be satisfied.
Figure 10 describes the effects of bank runs. In particular we assume that two
periods after the unanticipated drop in Zt , retail investors stop rolling over shortterm debt issued by wholesale banks, inducing them to liquidate all of their assets
and go bankrupt.
As explained in Section 3.5.1, the run on wholesale banks forces them into
bankruptcy and results in K w dropping to 0. Households and retail banks are forced
to absorb all of the wholesale banks’ assets, inducing asset prices to drop by about 7%
32

Observe also that in a production economy with investement and nominal rigidities, the dropin the asset price would reduce investment and thus aggregate demand, magnifying the overall dropin output.

37

-0.01
20

0
0

40

2
0
20
40
Quarters% from ss
20

0.02
0.01
0
0

40

-0.5
0

-0.5

-1
20
40
0
20
40
Quarters
Quarters
Recession and Ex-post Run
Recession

20

40

Nr

0

0

40

0.03

Nw

0.5
% from ss% from ss

20

20
ERkw-R

0.005

r

4

0
0

40

0.01

0

-0.1
0

40

20

0.5

Rb-R

0.1

w

-2
0

-1
0

40

% from ss

-0.02
0

-0.5

Q
% from ss

Ann.  from ss

Run on Wholesale
0.01

20

1

Ann.  from ss

-0.2
0

40

0

-0.1

Ann.  from ss

20

kr

0
% from ss% from ss% from ss

-0.05

-0.1
0

kw

y
0

0
% from ss

z
0

-0.2

-0.4
0

20
40
Quarters

Figure 10: A recession followed by a run on wholesale bankers
in total. The intermediation costs associated with the reallocation of assets to lessefficient agents leads to an additional contraction of output of around 7%, resulting
in an overall drop of about 15%.
As new wholesale bankers resume operations from the period after the run, highlevels of spreads for both retail and wholesale bankers allow them to increase theirleverage and recapitalize financial intermediaries thanks to above average retained
earnings. The re-intermediation process however is rather lengthy and output remains depressed for a prolonged period of time.

38

5

Anticipated Runs

So far, we have focused on the case in which runs are completely unexpected. In
this section we study how the equilibrium changes if agents anticipate that a run
will occur with positive probability in the future, focusing on the more realistic caseof a run on wholesale bankers only. The Appendix contains a detailed descriptionof the equilibrium in this case.33 Here we describe the key forces through whichanticipation of a run in the future affects financial intermediation. To keep theanalysis as simple as possible, we assume that once a negative shock to Zt hits, Zt
obeys perfect foresight path back to steady state.
The main difference from the unanticipated case is in the market for interbank
loans. In particular, once runs are anticipated, retail bankers internalize how wholesale bankers’ leverage affects returns on interbank loans in case of a run and they
adjust the required promised rate R̄bt+1 accordingly. We denote by pt the time t probability that retail banks will run on wholesale banks at time t + 1.34 The indifference
condition of the retail bank between making interbank loans and non-financial loans(33) becomes:w
Et [(1 − pt )Ωrt+1 (Rbt+1 − Rt+1 ) + pt Ωr∗
t+1 (xt+1 Rbt+1 − Rt+1 )]rr∗= γEt [(1 − pt )Ωrt+1 (Rkt+1− Rt+1 ) + pt Ωr∗
t+1 (Rkt+1 − Rt+1 )],

where
Ωr∗
t+1 = β

(44)



r∗
Vt+1
1 − σ + σ r∗
nt+1

is the value of the stochastic discount factor if a run occurs at t + 1.
Using equation (41) to substitute for xw
t+1 in (44) we obtain a menu of promised
35
rates:
r
Et Ωrt+1 Rkt+1w
R̄bt+1 (φt ) = (1 − γ)Rt+1 + γ
Et Ωrt+1



φw
pt
r∗
r∗
w∗
 Et Ωt+1 (1 − γ)Rt+1 + γRkt+1 − w
R(45)+
φ − 1 kt+1
(1 − pt )Et Ωrt+1
w
Notice that R̄bt+1 (φwt ) is an increasing function φt . This is because as leverageincreases, retail bankers suffer larger losses on interbank loans if a run occurs. This33

The analysis of anticipated runs draws heavily on Gertler and Kiyotaki (2015).
The determination of this probability of ”observing a sunspot” will be discussed below.
35
This is the relevant function for values of leverage high enough to induce bankruptcy in case of
a run.
34

39

induces them to require higher returns in the event of no run, to compensate for thelarger losses in the event of a run.
When choosing their portfolios, wholesale bankers will now have to factor in thatchanges in their leverage affect their cost of credit according to equation (45) . This
preserves homogeneity of the problem but the franchise value of the firm will change
to reflect that with probability pt the bank will be forced to liquidate assets at priceQ∗t+1 in the subsequent period. This will have the effect of reducing the franchisevalue of wholesale banks, hence tightening their financial constraints.
In particular the franchise value of a wholesale bank will be given by36
 w  w w
Vtw
w
w
)
.
)
+
R̄
(φ
=
(1
−
p
)
E
Ω
φ
R
−
R̄
(φ
bt+1
t
t
bt+1
t
t
t+1
t
t+1
nw
t

(46)

An increase in pt reduces the franchise value through two channels: First, it decreases
the likelihood that the bank will continue to operate next period. Second, it leads toan increase in the interbank loan rate each individual bank faces, R̄bt+1 (φwt ) , which
reduces the franchise value even if the bank continues to operate.
In order to pin down a state dependent probability of a run, we follow Gertler
and Kiyotaki (2015). In particular we assume that at each time t the probability oftransitioning to a state where a run on wholesale banks occurs is given by a reduced
form decreasing function of the expected recovery rate Et xw
t+1 as follows,δ
pt = 1 − Et (xw(47)
t+1 ) .
Although we don’t endogenize the functional dependence of pt on the state of
the economy, the above formulation allows us to capture the idea that as wholesalebalance sheet positions weaken, the likelihood of a run increases. This same qualitative conclusion would follow, for example, if the probability of a run was determined
endogenously by introducing imperfect information, as in the global games approachdeveloped by Morris and Shin (1998).
Figure 11 demonstrates how anticipation effects work to increase financial amplification of shocks in the model. The solid line is the response of the economy to an
unanticipated six percent shock to Zt when agents anticipate that a run can happen
at each time t + 1 with probability pt as determined in equation (47) .37 As we noted
earlier, we assume that after the shock Zt follows a perfect foresight path back to
36

Here we are already assuming that wholesale bankers will choose a leverage high enough toresult in bankruptcy when a run occurs. See the Appendix for a detailed description of hte wholesalebanker’s problem when runs are anticipated. There, we derive the conditions that ensure that it is
optimal for wholesale bankers to default in the event of a run.
37
In the numerical simulations below we pick δ to be 21 .

40

-0.2
0

40

-0.04

0.02

B

-0.01
0

40

20
40
Quarters

-0.1

-0.2
0

20

40

0.01
0
-0.01
0

Nw

40

0.5

-0.5

-1
0

20
40
Quarters

20
Nr

0% from ss

-0.5

40

0.02

0

0
% from ss% from ss

20

20
ERkw-R

0.01

D

0

0
0

40

0.02

0.04

0
0

40

20

0.2

Rb-R
Ann.  from ss

-0.02

-1
0

-1
0

40

0.06
Ann.  from ss% from ss

0

20

-0.5

Rkw-Rb

Q

-0.06
0

20

% from ss

-0.1

0.4

Ann.  from ss

20

kr

0
% from ss% from ss% from ss

0.1

0
0

kw

y
0

% from ss

p
0.2

20
40
Quarters

Recession with Positive Run Probability

0

-0.5
0

20
40
Quarters

Recession

Figure 11: A recession in the model with anticipated runs
steady state. To isolate the effect of the anticipation of the run, we suppose in thiscase that the run never actually occurs ex-post. For comparison, the dotted line
reports the responses of the baseline economy in which individuals assign probability
zero to a bank run.
While it is still the case that in steady state a run cannot occur, the shock to
Zt leads the probability of a run to increase to 15%. As wholesale bankers’ balance
sheets weaken and the liquidation price decreases, retail bankers expect more losses
on interbank loans in case of a run and the probability of coordinating on a run
equilibrium increases as a result. The increase in pt leads to a sharp contraction in
the supply of interbank credit and a further tightening of wholesale bankers financial
constraints. This, in turn, results in an overall reduction in their net worth of about
80% compared to a 50% in the baseline and to a spike in spreads between non-

41

-0.2
0

40

0.02

B

40

-0.05
0

20
40
Quarters

-0.2

-0.4
0

20

40

0.02
0
-0.02
0

-1
0

Recession with Anticipated Run Happening ex-post

40

0.5

-0.5

20
40
Quarters

20
Nr

0% from ss

-0.5

40

0.04

Nw

0
% from ss% from ss

20

20
ERkw-R

0

D

0

0
0

40

0.05

0.04

0
0

40

20

0.5

Rb-R
Ann.  from ss

-0.05

-1
0

-1
0

40

0.06
Ann.  from ss% from ss

0

20

-0.5

Rkw-Rb

Q

-0.1
0

20

% from ss

-0.1

1

Ann.  from ss

20

kr

0
% from ss% from ss% from ss

0.2

0
0

kw

y
0

% from ss

p
0.4

20
40
Quarters

0

-0.5
0

20
40
Quarters

Recession with Positive Run Probability

Figure 12: A recession followed by a run in the model with anticipated runs
financial loan and interbank loan rates that increase by 400 basis points compared toonly 30 in the baseline. As wholesale banks are forced to downsize their operations,
total interbank credit falls by about 70%, more than twice the percentage drop in thebaseline. These massive withdrawals of funds from wholesale markets is the model
counterpart to the ”slow runs” on the ABCP market in 2007. These disruptions inwholesale funding markets are then transmitted to the rest of the economy inducing
a drop in asset prices of five percent and a total contraction of output of thirteen
percent.
Figure 12 shows the case in which the run actually occurs two periods after therealization of the shock to Zt . There are two main differences with respect to theanalogous experiment performed in the case of unanticipated runs depicted in Figure
10 . First, the initial increase in the probability of a run that precedes the actual
42

ABCP Spread (Rb-R)

Total Finance Premium (ERk-R)

4

2.5

3.5

2

3
1.5
2.5
1
2
0.5
1.5
0
1

-0.5

0.5

0
2007

2008

2009

-1
2007

2010

Baseline Model

2008

2009

2010

Data

Figure 13: Total Credit Spreads and Interbank Spreads in the Model and in the Data
run allows the model to capture the ”slow runs” followed by ”fast runs” in wholesalefunding markets that was a central feature of the financial crisis, as discussed in
the Introduction. Second, the run induces a further increase in the probability ofadditional runs in the future, that goes back to about 20% the period after the
run occurs. This hampers wholesale bankers ability to increase their leverage andgenerates higher spreads in the interbank market preventing the relatively smoothincrease in asset prices that characterizes the recovery in the baseline model.
Figure 13 shows how the model with anticipated runs can reproduce some key
features of the financial disruptions that occurred in 2007 and 2008. In particular,b
we compare the model predicted path for interbank spreads, R̄t+1− Rt+1 , and excessw
finance premium, ERk,t+1 − Rt+1 , with their empirical counterparts over the period
going from 2007Q2 to 2009Q4. For the interbank spreads we choose the ABCP

43

spread, since the first ”slow runs” in wholesale funding markets in the third quarterof 2007 took place in the ABCP market. The measure of excess borrowing costs isthe Excess Bond Premium of Gilchrist and Zakrajsek (2012). We assume that theeconomy is in steady state in 2007Q2 and the unanticipated shock hits in 2007Q3
followed by a run on wholesale banks in 2008Q3.38 In the data excess borrowing costsw
lag financial spreads, so the model predicts a stronger initial increase in ERk,t+1−Rt+1
and attributes a slightly smaller proportion of the increase to interbank spreads,
probably due to the behavior of the risk free rate. On the other hand, the faster
decline in spreads in the data after 2009 can be attributed to the effects of government
intervention in this period. Overall, the experiment can capture the credit spreads
and bank equity dynamics reasonably well.

6

Two Productive Assets and Spillover Effects

In our baseline model there is only one type of capital. Wholesale banks have an
efficiency advantage in holding this capital. Retail banks exist mainly because wholesale banks may be constrained by their net worth; otherwise the latter would hold all
the capital. In this section we introduce a second type of capital which retail banks
have an efficiency advantage in intermediating. In addition to providing a stronger
motivation for the existence of retail banking, the second asset allows us to illustratespillover effects from a crisis in wholesale banking into retail banking.
In particular, one of the salient features of the recent crisis was the strong contagion effect through which the collapse in subprime mortgage related products within
the wholesale banking sector led to a deterioration in financial conditions within thecommercial banking sector, ultimately affecting the flow credit through these institutions. Even though on the eve of the crisis, much of the credit provided by the retailsector had no direct reliance on shadow banks, the collapse of the latter ultimately
disrupted commercial bank lending, enhancing the downturn.
As is the case with the first type of capital, we suppose the second type is fixed insupply and denote the total as L. We refer to bank loans made to finance this capitalas ”C&I” loans (for”commercial and industrial” loans). What we have in mind are
the kinds of information-intensive loans that are not easily securitized, which retailbanks have historically specialized in intermediating. This contrasts with the kindsof securitized assets, involving mortgages, car loans, credit card debt, trade credit
and so on, that were principally held by wholesale banks.
38
To be closer to the observed dynamics of spreads we resize the innovation to Zt to five percentage
points.

44

For simplicity, we assume that only retail banks and households fund the second
type of capital. Given Lrt and Lht are the amounts funded by retail banks andhouseholds, we have:
Lht + Lrt = L̄(48)
We model retail banks’ comparative advantage in making C&I loans by assuming thatmanagement costs of intermediating these loans are zero for these types of banks.
Conversely, we think of management costs for wholesale banks as being infinity.
Finally, we allow households to directly fund this asset, where claims on this capital
directly held by households may be thought of as corporate bonds. We suppose thathouseholds are at disadvantage to retail banks in funding the second type of capital,
though at an advantage relative to wholesale banks: They must pay the managementfee
αL L 2(Kt )F L (KtL ) =2with 0 < αL < ∞.
In analogy to the first type of capital, there is an exogenous dividend payoutL
Zt that obeys a stationary first order stochastic process. In addition, for simplicity
we restrict attention to the case where bank runs are completely unanticipated.
hAccordingly, let Rlt+1
be the household’s rate of return from funding the second
asset. Then the household’s first order condition for holding the second asset isgiven byhEt (Λt,t+1 Rlt+1
)=1
(49)
with
h
Rlt+1
=

L
Zt+1
+ QLt+1
QLt + αhL Lht

where QLt is the asset price and αhL controls the degree of inefficiency of households
in directly holding this asset.
The optimization problem of wholesale bankers is unchanged. Accordingly, we
focus on retail bankers. Given retail banks now have the option of intermediating
the second asset, we can rewrite the balance sheet and flow of funds constraints as(Qt + ftr )ktr + QLt ltr + (−brt ) = nrt + drtrr
QLt ltr + Rbt+1 (−brt ) − Rt+1 drt
nrt+1 = Rkt+1(Qt + ftr ) ktr + Rlt+1

rwhere Rlt+1
is the rate of return on the type L asset and is given by,
L
Zt+1+ QLt+1
r
Rlt+1 =.
QLt

45

Because the incentive constraint isθ[(Qt + ftr )ktr + QLt ltr + (−brt )] ≤ Vtr ,the effective leverage multiple for this case φrt now includes the holdings of the second
type of capital:(Qt + ftr )ktr + QLt ltr + γ(−brt )
φrt ≡
.
nrt
Proceeding as earlier to solve the retail bank’s maximization problem yields asolution for φrt which is the same as in the baseline case (see equation (27)). In
addition, at the margin the retail bank must be indifferent between holding thetypes of capital, which implies the following arbitrage condition:
rr
Et [Ωrt+1 Rlt+1− Rkt+1] = 0.
(50)
We now consider a numerical example designed to illustrate the contagion effect.
The real world phenomenon that motivates the experiment is the fall in housing prices
beginning in 2006 that led to the collapse of the wholesales banking sector that inturn disrupted commercial banking. In particular, we suppose that the dividend to
the L asset is fixed at its steady state value Z L . Then we consider a negative shockto the dividend on the type K asset and, as in our earlier baseline experiments, allow
for an unanticipated run two periods after the initial shock. Tables 4 and 5 describe
the changes in the calibration for this experiment.
Figure 14 reports the results from the experiment and demonstrates the spillover
effects of shocks to Zt on the market for L. The source of contagion in this environment is the balance sheet position of retail bankers.39 Losses on their capital
investment and, in case of a run, on their interbank loans, result in a decrease inretail bankers’ net worth and a tightening of their respective incentive constraints.
As long as there are incentive costs associated with intermediating asset L, the tightening of financial constraints leads retail bankers to increase required excess returns
in both markets, as shown by equation (50) . The negative shock to returns on capitaland the run on wholesale banks lead to a costly reallocation of assets to households
and to an increase in spreads between returns on Lrt and the deposit rate of about
60 basis points.
39
Other similar models of spillover are Bocola (2015) and Ferrante (2015b). An alternative
mechanism based on market fragmentation is developed by Garleanu, Panageas and Yu (2015).

46

Table 4: Parameters in 2 Assets Table 5: Steady Statein 2 Assets Model
Model
β
αh
αLh
Wh
σr
αr
αLr
Wr
θ
γ
σw
αw
αLw
Ww
ω
Z

ρz

7

PARAMETERS
Households
discount rate
Intermediation cost
Intermediation cost for CI loans
Endowment
Retail Banks
Survival Probability
Intermediation cost
Intermediation cost for CI loans
Endowment
Divertable proportion of assets
Shrinkage of Divertable proportion of interbank loans
Wholesale Banks
Survival Probability
Intermediation cost
Intermediation cost for CI loans
Endowment
Shrinkage of divertable proportion of assets
Production
Steady State productivity
Serial correlation of productivity shocks

.99
.06
.006
.016
.96
.01
0
.0014
.27
.67
.88
0
∞
.0012
.47

Q
QL
Kr
Kw
Lr
Lh
Rb
Rrk
RrL
R
Rwk
φw
φr
Y
Ch
Nr
Nw

STEADY STATE
price of capital
price of CI loans
retail intermediation
wholesale intermediation
retail holding of CI loans
household holding of CI loans
Annual interbank rate
Annual retail return on capital
Annual retail return on CI loans
Annual deposit rate
Annual wholesale return on capitalwholesale leverageretail leverage
outputconsumption
retail banks networthwholesale banks networth

.016
.9

Government Policy

In this section we study the effects of two types of policy interventions to combatbanking crises: first an ex-post intervention where the central bank acts as a lenderof last resort; second, an ex-ante macroprudential regulation that limits banks’ risk
exposure. Within the literature, these policies have largely been studied in the context of dampening negative financial accelerator effects on the economy. Here weemphasize a somewhat different perspective: How these policies might be useful in
reducing the likelihood of damaging bank runs? As we show, lender of last resort policy that is anticipated ex ante in the event of an ex post crisis reduces the likelihoodof a run by raising asset liquidation prices. Macroprudential does so by reducing
bank leverage.
A case for ex ante macroprudential regulation arises because banks tend to choose
an inefficiently high level of leverage in the laissez-faire economy. Roughly speaking,because individual banks ignore the consequences of their own borrowing decisions

47

1
1
.3
.6
.5
.5
1.048
1.052
1.052
1.04
1.064
20
10
.0466
.0363
.1371
.03

z

-0.06

-0.15
-0.2
0

40

Lr

20

40

0.05
0
-0.05
-0.1
0

l
-3
x 10ER r-R

4
2
0
0

6
Ann.  from ss

6

20
40
Quarters% from ss

-0.6

20

0.5

-0.05
0

40

20

0.02
0.01
0
0

40

Nw

2
20
40
Quarters

20

40

Nr
0

-0.5

-1
0

40

ERkw-Rb

0

4

20

0.03

0

b
-3
x 10R -R

0
0

1

0
0

40

0.05

% from ss% from ss

-0.4

-0.8
0

20

1.5

Ql

0.1

-0.2

Ann.  from ss

-1
0

40

Q

0
% from ss

20

-0.5

Ann.  from ss

20

-0.1

% from ss

-0.08
0

-0.05

2
% from ss

-0.04

kr

0
% from ss% from ss% from ss

0

-0.02

8

kw

y

0

20
40
Quarters

Ex-post run on wholesale

-0.1
-0.2
-0.3
-0.4
0

20
40
Quarters

Recession

Figure 14: Spilloveron the level of aggregate risk, their are prone to issue more debt than would besocially desirable.40 In addition, as Farhi and Tirole (2012) Chari and Kehoe (2014),
and Gertler, Kiyotaki and Queralto (2011) emphasize, the expectations of some typeof government interventions ex post will also encourage excessive leverage in thebanking system ex ante.
In this section we explore each of this kinds of policy’s within our framework of
Anticipated Runs of Section 5.
40

See Geanakoplos Polemarchakis (1986) for the original result of generic constrained inefficiency
in a model with incomplete markets. Lorenzoni (2008) and Bianchi (2011) are recent applications
to environments with financial frictions.

48

7.1

Ex-Post Intervention: Lender of the Last Resort

It is well known that if there are limits to arbitrage in private financial intermediation,then a central bank who plays as the lender of last resort during a financial crisis canenhance the flow of credit and in turn mitigate the economic downturn. What makes
the lender of last resort effective is that the central bank can elastically obtain funds
by issuing interest bearing reserves, while private financial intermediaries may beconstrained in their ability to obtain funds by the condition of their balance sheets(Gertler and Karadi, 2011, Gertler and Kiyotaki, 2011).
Following the onset of the recent financial crisis, the Federal Reserve introduced
a variety of lender of last resort programs. The most prominent involved large scaleasset purchases (LSAPs) of high grade long term debt, including primarily agency
mortgage backed securities (AMBS), instruments that were held primarily in the
shadow banking sector. The Fed announced this program in December 2008 following the collapse of the shadow banking system and began phasing it in the following
March. The objective of this kind of lender of last resort intervention was to reducethe cost and thereby increase the availability of credit to the nonfinancial sector.
There is evidence which suggests the Fed achieved this objective. Beyond these
considerations, however, by acting as buyers in the secondary market for AMBS,
the Fed raised the price and accordingly the liquidation value of these assets. As
we noted, the impact of these policies on liquidation prices has important implications for banking stability. (See equation (40), for the condition for a bank run
equilibrium.)
To model this type of intervention, we assume that the central bank can directly
undertake intermediation by borrowing from retail banks and then making nonfinancial loans. The way the central bank obtains funds from retail banks is to issue
interest bearing bank reserves. We assume that retail banks are unable to divertbank reserves, since they are held in an account at the Fed. Given retail banks
cannot divert reserves, they are not constrained in their ability to raise deposits tofund reserves. Because there are no limits to arbitrage for banks funding reserves,
the interest rate on reserves will equal the deposit rate. Therefore, when the central
bank supplies interest-rate bearing reserves to retail banks, it effectively raises funds
directly from households by issuing overnight government bond. What gives thecentral bank an advantage in intermediating assets is that, unlike retail and wholesalebanks, it is not balance sheet constrained.
We also assume, following Gertler and Karadi (2011) that the central bank isless efficient than the private sector. As with retail banks and households, the government faces quadratic managerial costs 12 αg (Ktg )2 , where Ktg is the size of central
bank’s intervention and where αh > αg > αr . To ensure that it is desirable for the49

central bank to intervene only in a crisis, we also allow for inefficiency in the average
performance of the government’s portfolio: In particular, we assume that the returnon government intermediated assets is:
g
Rkt+1
=ϕ

Zt+1 + Qt+1
Qt + αg Ktg(51)

where ϕ ∈ (0, 1) controls the relative inefficiency of central bank’s intermediation for
the average return on assets, independent of scale.
We assume that the central bank intervenes in credit markets whenever expectedasset returns exceed its cost of borrowing. That is we posit a policy rule for central
bank’s intervention given byg
Ktg = 0, 
if Et Rkt+1− Rgt+1 < 0(52)g
Et Rkt+1− Rgt+1 = 0,
if Ktg ≥ 0
where Rgt+1 is the interest paid on reserves issued to retail banks.
As we just noted, since there is no incentive problem associated with central bankintermediation, in equilibrium the interest rate on reserve Rgt+1 must equal to thedeposit rate:41
Rgt+1 = Rt+1 .
(53)
The key variable to which the central bank responds in determining credit market
intervention is the spread between the wholesale bank’s return on assets and thew
deposit rate, Rkt+1− Rt+1 , which can be thought of as a measure of the degreeof inefficiency in private financial markets. The central bank intervenes when thisexcess return is high.42 In particular, the policy rule (52) prescribes that the Fed
starts intermediating assets as soon as the ratio of the credit spread to the deposit41
To see formally, first notice that, since retail bankers cannot divert reserves, their incentive
constraint (14) is not affected by the amount of reserves held on their balance sheet. Hence theintroduction of interest bearing reserves only affects retail bankers’ optimization problem by modifying the objective function (26) , which becomesr
Vtr = Mr ax
Et Ωrt+1 φrt (Rkt+1− Rt+1 ) + Rt+1 + drgt (Rgt+1 − Rt+1 ) nrt
rφt ,dgt

where drgt is the amount of reserves per unit of networth held by retail bankers. The optimality
condition with respect to drgt is just given by Rgt+1 = Rt+1 . Covariance terms are zero since both
Rgt+1 and Rt+1 are known at date t.
42
Our policy rule, which has the central bank target credit spreads, is consistent with how thecentral bank behaved throughout the crisis. What motivated an unconventional intervention in agiven credit market was typically a sharp increase in the spread wthin that market.

50

20

-0.2
0

40

-0.04

0.02

B

-0.01
0

40

20
40
Quarters

40

0
-0.01
0

Nw

-0.1

-0.2
0

20

0.01

20
40
Quarters

Recession with Positive Run Probability

40

0.5

-0.5

-1
0

20
Nr

0
% from ss

-0.5

40

0.02

0

0
% from ss% from ss

20

20
ERkw-R

0.01

D

0

0
0

40

0.02

0.04

0
0

40

20

0.2

Rb-R
Ann.  from ss

-0.02

-1
0

-1
0

40

0.06
Ann.  from ss% from ss

0

20

-0.5

Rkw-Rb

Q

-0.06
0

20

% from ss

-0.1

0.4

Ann.  from ss

0.1

kr

0
% from ss

0% from ss% from ss

0.2

0
0

kw

y

% from ss

p

20
40
Quarters

0

-0.5
0

20
40
Quarters

Recession with Lender of Last Resort

Figure 15: Anticipation Effects of Government intervention
rate exceeds a given threshold that varies inversely with the inefficiency parameterϕ:w) − Rt+1
Et (Rkt+1
1−ϕ
Ktg > 0, iff
>
.
Rt+1
ϕ
From equation (52), the size of the intervention in the region where Ktg > 0 is thengoverned by:w
Et (Rkt+1) − Rt+1 1 − ϕϕg
Kt = g Q t
−
.
α
Rt+1
ϕ
We choose ϕ in order to ensure that the central bank only intervenes after arun happens: that is, the threshold for the credit spread to justify an intervention isreached only in the event of a run. We choose the management cost parameter αgin order for the intervention to be around 5 percent of total capital.
51

kg

p
0.4
% from ss

 from ss

0.06
0.04
0.02
0
0

10

20

30

0.2

0
0

40

10

Q

20

30

40

30

40

30

40

y

0

% from ss% from ss

0

-0.05
-0.1
0

10

20

30

-0.1

-0.2
0

40

10

ERkw-Rw
2% from ss

Ann.  from ss

0.04

0.02

0
0

20

10

20
Quarters

30
Ex-Post Run

1
0
-1
0

40

10

20
Quarters

Ex-Post Run with Lender of Last Resort

Figure 16: Government intervention when a Run happens at time 2
Figure 15 shows the response of the economy to a recession when agents anticipate that, if a run happens, the monetary authority intervenes with large scaleasset purchases according to (52). Even though in this experiment the run does not
happen and the central bank accordingly does not intervene, the anticipation of theintervention in the event of a run significantly dampens the downturn. It does so by
reducing the probability of a run: The central bank’s conditional intervention policy
increases the liquidation price of wholesale banks assets. In turn, by equation (47),the higher recovery rate associated with higher liquidation prices decreases the probability of a run. In the experiment the probability of a run decreases by 10 percent
in the first two periods and becomes zero thereafter. This drastic reduction in therun probability implies that, overall, anticipation of government intervention works
to stimulate the economy. Notice that, even though the reduction in the run proba52

bility relaxes the incentive constraint and hence allows wholesale bankers to increase
their leverage for any given level of spreads, the general equilibrium effects of asset
prices on their balance sheet results in better capitalization and lower leverage inboth the wholesale and retail bank sectors.
Figure 16 illustrates the effect of the intervention when a run happens one period
after the shock to Z . The intervention is around 5 percent of total capital andreduces the drop in asset prices and output by about 2.5 and 4 percent respectively.

7.2

Ex-Ante Intervention: Macroprudential Policy

One of the most important challenges facing policy makers in the aftermath of thefinancial crisis is the development of financial regulations that can help prevent therecurrence of similar episodes in the future. In this respect, the most relevant innovation in the policy landscape has been the introduction of various macroprudential
measures in the oversight of financial institutions, such as stress tests by central
banks and the revised provisions in Basel III. These measures are aimed at ensuring that financial institutions’ capital is sufficient to absorb losses during adverse
economic conditions.
There is now a significant literature that analyzes the impact of capital requirements on banks for macroeconomic stability (e.g. Christiano and Ikeda (2015),
Begenau (2015), Bianchi and Mendoza (2013), Chari and Kehoe (2015)), Gertler,
Kiyotaki and Queralto (2012). Most of this literature analyzes how the introductionof leverage restrictions can dampen financial accelerator effects by dampening fluctuations in bank capital. The need for leverage restrictions, or equivalently capitalrequirements, stems from an externality that leads individual banks to fail to take
into account their own borrowing on the stability as a whole.43
Our framework offers a somewhat different perspective on the potential benefitsof leverage restrictions. Not only can these restrictions dampen financial accelerator
effects: Importantly, they can also make the banking system less susceptible to runs.
As equation (41) makes clear, a bank run can only happen if the leverage ratio is highenough. Thus, by limiting the leverage ratio sufficiently, the regulatory authority
can in principle eliminate the possibility of a run. The question then is what are thetradeoffs. We turn to this issue next.
We capture macroprudential policies in our model economy by introducing lever43

Much of the literature, following Lorenzoni (2008), features a pecuniary externality stemmingfrom the presence of asset prices in the borrowing constraint. Fahri and Werning (2015), andKorinek and Simsek (2015) show that if aggregate demand is sensitive to aggregate leverage, asimilar kind of externality can emerge.

53age restrictions on wholesale banks. In particular, we assume that a financial regulator can impose an upper bound on wholesale banks’ leverage, φ̄w . This implies
that the effective limit to wholesale banks’ leverage will be given by the smaller between the market imposed limit and the regulatory limit. Accordingly, constraint(22) becomes)( 1 Vtw
−
(1
−
ω)
w
θ nt
, φ̄w
φw ≤ minωIn a fully stochastic simulation of the economy, leverage restrictions would tradeoff lower frequency of crises, resulting from reduced variation of bankers’ capital,
against lower average output, as the impaired ability of wholesale banks to increase
their leverage would induce a costly reallocation of capital to less efficient agents.
While our numerical experiments in Sections 7 and 4.3 provide an illustration of
the tradeoff between steady state output and fragility associated to changes in thelong run level of wholesale bankers’ leverage, here we focus on the conditional effects of leverage restrictions upon the occurrence of a recession that would leave thedecentralized economy vulnerable to bank runs.
We focus on two possible levels for φ̄w : the steady state level of wholesale banks’leverage and a level that is higher than steady state but still sufficiently low to preventa run. Permitting a leverage ratio above the steady state allows banks to issue moredebt in a recession, which has the overall effect of dampening the contraction in
financial intermediation and thus dampening the downturn in real activity. Indeed,
the more forgiving leverage restriction comes closer to mimicking the behavior of theleverage ratio in the decentralized economy, which moves countercylcially.
Figure 17 and 18 compare the response of the economy with anticipated runs toa negative Z innovation, with and without macroprudential regulation. In Figure17
the regulator imposes the tighter leverage restriction, i.e. φ̄w is set to the steadystate value of wholesale leverage, while in Figure 18 the restrictions are more lax andallow maximum regulatory leverage to exceed the steady state value by 15 percent.
As mentioned, in both cases, the leverage restrictions are sufficient to prevent a run
and hence avoid the recessionary effects associated to the endogenous increase in
the probability of a run that characterizes the unregulated economy. This results inhigher asset prices in the regulated economy throughout the recession. Under theless strict requirements the stimulative effect on asset prices is significantly higher,
reaching about 1.5 percent after the first three years of the recession. On the other
hand, by constraining the ability to leverage of the most efficient intermediaries,
macroprudential policies induces a costly reallocation of assets. The balance between
these two contrasting forces varies over time, in turn influencing output effects of the
54

-0.2
0

40

-0.04

0.02

w

40

-0.01
0

20

40

0.01
0
-0.01
0

Nw

0
0

-0.1
20
40
0
20
40
Quarters
Quarters
Recession with Positive Run Probability

-0.5

20

40

Nr

0
% from ss

0

40

0.02

0

0.1
% from ss% from ss

20

20
ERkw-R

0.01

r

1

0
0

40

0.02

0.04

0
0

40

20

0.5

Rb-R
Ann.  from ss

-0.02

0.5

-1
0

40

0.06
Ann.  from ss% from ss

0

20

-0.5

Rkw-Rb

Q

-0.06
0

20

% from ss

-0.1

1

Ann.  from ss

20

kr

0
% from ss% from ss% from ss

0.1

0
0

kw

y
0

0.5
% from ss

p
0.2

0

-1
0

-0.5
20
40
0
20
40
Quarters
Quarters
Recession with (tight) Leverage Restrictions

Figure 17: Macro Prudential Policy: φw = φss
policy.
During the early stages of the recession, the stimulative effects of macroprudential
policy are strongest because they eliminate the probability of a bank run, which in
the unregulated economy is highest at this time. Under the stricter policy, the impact
drop in output is very similar to the drop in the unregulated economy, while the morelax stance of policy dampens the drop in output by 2 percent and is stimulative
throughout the first year of the recession. As time passes, the probability of a run
becomes small in the unregulated economy, implying that the stimulative effects ofpolicy decreases. On the other hand, the slower recovery of financial institutions’equity in the regulated economy that results from their impaired ability to leverage,
implies a more persistent drag on output coming from financial misallocation. In
both cases output costs associated with the policy peak at around 10 quarters into

55

20

-0.1
-0.15
-0.2
0

40

-0.02
-0.04

0.02

w

20

-0.01
0

40

20
40
Quarters

0.05
0
-0.05
-0.1
0

20

20

40

0.01
0
-0.01
0

20

Nw

Recession with Positive Run Probability

40

Nr
0.1

-0.5

-1
0

20
40
Quarters

40

0.02

0
% from ss

0.2

0.1

ERkw-R0

r% from ss

0.4

0.2

0
0

40

0.01

0.1

0.6

20

0.02

0.04

0
0

40

0.8
% from ss

-0.6

0.3

Rb-R
Ann.  from ss

0

0
0

-0.4

-0.8
0

40

0.06
Ann.  from ss% from ss

0.02

20

-0.2

Rkw-Rb

Q

-0.06
0

20

% from ss

0.05

-0.05

0.4

Ann.  from ss

0.1

kr

0
% from ss% from ss% from ss

0.15

0
0

kw

y
0

% from ss

p
0.2

20
40
Quarters

0
-0.1
-0.2
-0.3
0

20
40
Quarters

Recession with (slack) Leverage Restrictions

Figure 18: Macro Prudential Policy: φw = 1.15φss
the recession and result in an additional drop in output of about 4 percent under thetighter requirements and 1.5 percent under the more lax stance.

8

Summary and Directions for Future Research

The financial crisis that triggered the Great Recession featured a disruption of wholesale funding markets, where banks lend to one another, as opposed to retail markets
where banks obtain funds from depositors. It is essential to capture the roles andpossible disruption of wholesale funding market to understand the financial crisis as
well as to draw policy implications. Our goal in this Handbook Paper was to sketcha model based on the existing literature that provides a step toward accomplishing56

this objective. The model first accounts for how, through innovation in the efficiency
of interbank loan markets, a wholesale banking sector emerges that intermediatesloans using funds borrowed from retail banks. This wholesale sector bears a close
resemblance to the shadow banking system featured in most descriptions of the crisis.
As we show, in ”normal” times, the growth of the wholesale banking sector improves both efficiency and stability. Improved efficiency stems from the comparativeadvantage that wholesale banks having in managing certain types of loans. Improved
stability arises because retail banks act as a buffer to absorb loans that wholesalebanks sell off, in effect improving the liquidity of secondary loan markets. On theother hand, the growth of wholesale banking system makes the economy more vulnerable to a crisis. As occurred in practice, the high leverage of wholesale banks
makes this sector susceptible to runs that can have highly disruptive effects on theeconomy. A contractionary disturbance that might otherwise lead to a moderaterecession, can induce a run on the wholesale banking sector with devastating effects
on the economy, as experienced during the Great Recession. We then describe howboth lender of last resort and macroprudential policies can help reduce the likelihoodof these kinds of banking crises.
Our framework also captures the buildup of safe assets prior to the crisis alongwith the subsequent collapse that a number of authors have emphasized (e.g. Gortonand Metrick (2012), Caballero and Farhi (2015)). The underlying mechanisms worka bit differently, in somewhat subtle ways: The ”safe asset” literature points to anincreased demand for safe assets as the driving force in the buildup of the shadowbanking system. By making assets riskier, the crisis then reduces the ability ofthe shadow banking sector to create safe assets. It is this reduction in safe assetsthat then leads a contraction in spending, essentially for liquidity reasons. Within
our framework, the increase in safe assets is a product of innovation in interbank
lending markets. Indeed, this is where much of the growth in safe assets occurred.
There is also a growth in households deposits as the overall banking system becomes
more efficient. The crisis similarly induces a contraction in safe assets: The exact
mechanism, though, is that, with an adverse shock to the net worth of banks, theprobability of runs on wholesale banks becomes positive, which constrains the abilityof both wholesale and retail banks to issue safe liabilities. In turn, a contraction inreal activity emerges because the costs of intermediation increase, as manifested by
the increase in credit spreads. In future work, it would be interesting to synthesize
the role of safe assets in our framework with that in the conventional literature on
this topic.
Another important area for further investigation involves the modeling of thegrowth of wholesale banking. Our approach was to treat this growth as the product57

of innovation as captured by a reduction in the agency friction in inter-bank lending
markets. Among the factors we had in mind that motivate this reduction is technological improvements that permit less costly monitoring, such as the development ofasset-backed securities and repo lending. Of course, more explicit modeling of thisphenomenon would be desirable. Also important is integrating regulatory considerations. While financial innovation was important for the development of shadowbanking, regulatory factors also played an important role. For example, tighteningof capital requirements on commercial banks in conjunction with innovation in asset securitization induced movement of a considerable amount of mortgage lending
from the retail to the wholesale banking sector. A careful integration of the roles ofregulation and innovation in the development of wholesale banking would be highly
desirable.
Finally, consistent with what occurred in the recent crisis, what makes the financial system within our model so vulnerable is high degree of leverage in the form ofshort term debt. Here we simply rule out a richer set of state-contingent financial
contracts that would permit banks to hedge against the systemic risk implied by thisliability structure. Why in practice we don’t seem to observe the kind of seemingly
desirable hedging is an important question for future research.44

9

Appendix

9.1

Appendix A: Details of the Equilibrium

From (13, 15, 16, 17) , we get
Vtj
= Et
njt

njt+1
j
Ωt+1 · j
nt
(

= Et

Ωjt+1

!

"
j
Rkt+1
+

j
Rkt+1
− Rt+1

 djt
njt

+

j
Rkt+1
− Rbt+1

 bjt

#)

njt

dj
bj
j
= νkt
+ µjdt tj + µjbt tj ,
nt
nt
44

Some efforts to address this issue include Krishnamurthy (2003), DiTella (2014), Gertler, Kiyotaki and Queralto (2012), and Dang, Gorton and Holmstorm (2012)).

58

wherejj)= Et (Ωjt+1 Rkt+1
νkt
 j

j
j
µdt = Et Ωt+1 Rkt+1 − Rt+1
jµjbt = Et Ωjt+1 Rkt+1− Rbt+1 .

(54)
(55)
(56)

From (13) , the incentive constraint (14) can be written ashi
Vtj ≥ θ njt + djt + ωbjt · Ibj >0 + (1 − γ)bjt · Ibj <0 ,ttwhere Ibj >0 = 1 if bjt > 0 and Ibj >0 = 0 otherwise, (and Ibj <0 = 1 if bjt < 0 andttt
Ibj <0 = 0 otherwise).
t

In order to save the notations, we normalize njt = 1 and suppress the suffix andtime subscript. The generic choice of a bank is given byψ = M ax(νk + µd d + µb b)(57)

θ [1 + d + ωb · Ib>0 + (1 − γ)b · Ib<0 ] ≤ νk + µd d + µb b,

(58)

b, d

subject tod ≥ 0,
1 + d + b ≥ 0.
Figure 20 and 21 depict the Feasible set and an Indifference Curve for Wholesale
Bankers and Retail Bankers under our baseline.
Defining λ and λk as Lagrangian multipliers of the incentive constraint and thenonnegativity constraint of capital, we have the Lagrangian as
L = (1 + λ)(νk + µd d + µb b) − λθ [1 + d + ωb · Ib>0 + (1 − γ)b · Ib<0 ] + λk (1 + d + b).
For the case of b ≥ 0, we know λk = 0 and the first order conditions are(1 + λ)µb ≤ λθω,where = holds if b > 0, and < implies b = 0.
(1 + λ)µd ≤ λθ,
where = holds if d > 0, and < implies d = 0.
59

In the following we restrict the attention to the case of µd > 0, and will verify theinequality later. Thus for the case of b > 0, we learn
µb
= ω,
µd
µb
> ω.
d = 0, ifµdd > 0, if

For the case of b ≤ 0, the first order conditions are(1 + λ)µb + λk ≥ λθ(1 − γ),
where = holds if b < 0, and > implies b = 0.
(1 + λ)µd + λk ≤ λθ,
where = holds if d > 0, and < implies d = 0.
Thus for the case of b < 0 and d > 0, we learn
k > 0, ifµb= 1 − γ,µdk = 0 and λk > 0, ifµb< 1 − γ.
µd

Therefore, under (Assumption 2): ω + γ > 1, we can summarize the bank’s
choice as:(i) b > 0, d = 0, k > 0, if µb > ωµd(ii) b > 0, d > 0, k > 0, implies µb = ωµd(iii) b = 0, d > 0, k > 0, if (1 − γ)µd < µb < ωµd(iv) b < 0, d > 0, k > 0, implies µb = (1 − γ)µd(v) b < 0, d > 0, k = 0, if µb < (1 − γ)µd .
In the steady state equilibrium, we knowµb
Rk − Rb=.
µd
Rk − R
Because we know Rkw ≥ Rkr and Rb ≥ R, we learn
µw
µrb
b
≥
.
µw
µrd
d
60

Therefore, market clearing for interbank loans implies that, if the interbank market isactive wholesale bankers’ choice can only be (i) or (ii) and retail banker’s choice (iv)or (v). Otherwise both types must choose according to (iii) and the interbank market
is inactive. That is, we have only the following possible patterns of equilibrium inthe neighborhood of the steady state.
(A) Perfect Specialization with active Interbank Market: dw = 0, k r = 0, bw >0 > br(B) Perfect Specialized Retail Banks with active Interbank Market: dw > 0, k r =0, bw > 0 > br
(C) Perfect Specialized Wholesale Banks with active Interbank Market: dw =0, k r > 0, bw > 0 > br
(D) Imperfect Specialization with active Interbank Market: dw > 0, k r > 0, bw >0 > br(E) Inactive Interbank Market: dw > 0, k r > 0, bw = 0 = br .
We can show that, under Assumption 2, there is no equilibrium of type (A) nor
(B):
w
r
r
Proof) Equilibrium of type (A) and (B) require µwb ≥ ωµd and (1 − γ)µd ≥ µb .
Thus

Rb ≤ ωR + (1 − ω)Rkw ,Rb ≥ (1 − γ)R + γRkr = (1 − γ)R + γRkw , as K r = 0 in (A) and (B).
This implies
ωR + (1 − ω)Rkw ≥ (1 − γ)R + γRkw ,or(ω + γ − 1) R ≥ (ω + γ − 1) Rkw .

But this is a contradiction as ω + γ > 1 and Rkw > R (as µw
d > 0 under our conjecture). Q.E.D.
Equilibrium C and D: Active Interbank Market
Suppose that 0 < µw
bt < θω. We will verify this numerically after we characterize
the equilibrium. Then the incentive constraint (58) holds with equality for wholesalebanks. Together with Bellman equation (57), we haveww
w w
ψtw = νkt
+ µw
dt dt + µbt btw= θ (1 + dw
t + ωbt ) ,

61

or
1
w
w
[νkt
− θ − (θ − µwdt )dt ] ,θω − µw
bt
θ
w w
w
w
− µw
ψtw =
[ωνkt
bt + (ωµdt − µbt )dt ] .
θω − µw
btbw
t =Maximizing Tobin’s Q, ψtw , with respect to dw
t ≥ 0, we learnwdw
t = 0, if µdt <

1 w
µ
ω bt

w
dw
t > 0 implies µdt =

1 w
µ .
ω bt

This provers Lemma 1 and the argument in the text follows for wholesale banks,
noting that we normalize nw
t = 1 above.
Suppose also that 0 < µrdt < θ. We will verify this numerically after we characterize the equilibrium. Then the incentive constraint (58) holds with equality for retailbanks. Together with Bellman equation (57), we haver
ψtr = νkt
+ µrdt drt + µrbt brt= θ[1 + drt + (1 − γ)brt ].

Then we get
1{ν r − θ + [θ(1 − γ) − µrbt ](−brt )},θ − µrdt ktθψtr =[ν r − µrdt + (µrdt − µrbt − γµrdt )(−brt )] .
θ − µrdt kt
drt =Maximizing Tobin’s Q, ψtr , with respect to ktr ≥ 0 and brt ≤ 0, we learn
ktr > 0 and brt < 0 imply µrdt − µrbt = γµrdt
ktr = 0 and brt < 0 if µrdt − µrbt > γµrdt .
This proves Lemma 2 and the argument in the text follows for retail banks, notingthat we normalize nrt = 1 above.
Therefore the argument in the text follows for the aggregate equilibrium.
r
Equilibrium E: No Active Interbank Market bw
t = bt = 062

From from Bellman
equation and the incentive constraint of each bank (57, 58)
 j
j jwith Qt + ft kt kt = 1+ djt , we havej
− µjdt + µjdt Qt + ftj ktj ktj ,ψtj = θ Qt + ftj ktj ktj = νktorν j − µjdt
Qt + ftj ktj ktj = kt,
θ − µjdt

ν j − µjdt
ψtj = θ kt
θ − µjdt(59)

The aggregate balance sheet conditions of wholesale and retail banking sectors are
Qt Ktw =

w
νkt
− µw
dt
Ntw = Ntw + Dtw
θ − µw
dt(Qt + ftr Ktr )Ktr =rνkt− µrdt r
N = Ntr + Dtr .
θ − µrdt t

(60)
(61)

The recursive competitive equilibrium without bank runs consistsof 24 variables aggregate quantities Ktw , Ktr , Kth , Dtw , Dtr , Ntw , Ntr , Ctb , Cth , Y t , Yt , prices (Qt , Rt+1 , ftr )jj
and bankers’ franchise values and leverage multiples Ωjt , Rkt, νkt, µjdt , ψtj j=w,r - as

w
r
w
r
a function of the state variables Kt−1
, Kt−1, Rt Dt−1, Rt Dt−1, Zt , which satisfy 24
equations (1, 4, 7, 8, 16, 18, 34 − 39, 54, 55, 59 − 61) where each of (16, 18, 54, 55, 59 −61) contain two equations.
After finding the equilibrium, we need to check the inequalitieswµw
bt < ωµdt ,µrbt > (1 − γ)µrdt .

In the neighborhood of the steady state, it is sufficient to show
Qt+1 + Zt+1
Qt+1 + Zt+1
+ ωRt+1 < γEt+ (1 − γ)Rt+1 .
(1 − ω)Et
Qt
Qt + αr Ktr

63

(62)

9.2

Appendix B: Steady State of the Economy without Run

In order to characterize the steady state of (C,D,E), define xj as the growth rate of
the net worth of continuing bank j in the steady state:
j

njt+1

(Q + f j )k j
bj
dj
−
R
−
R
b
nj
nj
nj
njt
 bj
 dj
= Rkj − Rb j + Rkj − R j + Rkj .
n
n

x =

= Rkj

Then we have the aggregate net worth of bank j as
N j = σ j xj N j + W j
Wj=≡ N j (xj ),1 − σ j xj
if σ j xj < 1, which we guess and verify later. Tobin’s Q of bank j is
ψ j = β(1 − σ j + σ j ψ j )xj
β(1 − σ j )xj=≡ ψ j (xj ).
jj
1 − βσ x
The ratio of bank loans to net worth isψ w (xw ) 1 − ω
dw
Qk w
=
−
1 + w , if bw > 0,
nw
θω
ω
n
w w
w
ψ (x )
Qk=, if bw = 0,
w
n
θ
 r
(Q + f r )k r
ψ r (xr )b=−γ − r .
rnθnCase of Active Interbank Market: C and D
From the condition for the retail banks, we have1−γ =Rkr − Rb
µrb
=
,
µrd
Rkr − Ror
Rb = γRkr + (1 − γ)R.
64

x

r

br
r
− R = (Rk − Rb ) r + (Rkr − R)


dr
1+ r
n
n


r
r
d
b
= (Rkr − R) 1 + r + (1 − γ) rnn r 
r r(Q + f )kb= (Rkr − R)
+γ − rrnnr r
ψ (x ).
= (Rkr − R)θThus from R = β −1 ,βxr − 1(βxr − 1) (1 − σ r βxr )=θ≡ ϕr (βxr ) ,ψ r (xr )(1 − σ r )βxr
βxr − 1β(Rb − R) = γθ r r = γϕr (βxr ) .
ψ (x )
Thus Rkr and Rb are functions of only xr :β(Rkr − R) = θ

Rkr = Rkr (xr ), Rb = Rb (xr ).
Differentiating log of the right hand side (RHS) of the above equation with respect
to xr , we learn
1σr1d ln ϕr (βxr )=−−d(βxr )
βxr − 1 1 − βσ r xr βxr∝ 1 − σ r (βxr )2> 0, iff σ r (βxr )2 < 1.
Thus if σ r (βxr )2 < 1, Rkr and Rb are increasing functions of only xr :Rkr = Rkr (xr ), Rkr0 (·) > 0,
Rb = Rb (xr ), Rb0 (·) > 0.
Similarly
x

w


bw
dw
w
1 + w + (Rk − R) w
n
n


w
b
1
dw
= (Rkw − Rb ) 1 + w + (Rkw − Rb ) w
n
ω
n

 w
w
1−ωd
Qk
= (Rkw − Rb )
+
w
n
ω nw


1 w 1−ωw= (Rk − Rb )
ψ −
.
ωθ
ω

− Rb = (Rkw − Rb )



65

Thus
xw − Rb,
ψ w − θ(1 − ω)1
Rkw − R = w[ωθ (xw − R) + (ψ w − θ) (Rb − R)] .
ψ − θ(1 − ω)

Rkw − Rb = ωθBecausedωθ (xw − R)
ln w
dxw
ψ − θ(1 − ω)
1
σw
∆
∝
−
, where ∆ = 1 − σ w + θ(1 − ω)σ w
−
w
w
w
w
βx − 1 1 − σ βx
∆βx − θ(1 − ω)
∝ (1 − σ w ) 1 − σ w (βxw )2 − θ(1 − ω)(1 − σ w βxw )2 ,Rkw is an increasing function of xw and xr
Rkw = Rkw (xw , xr ),if(1 − σ w ) 1 − σ w (βxw )2 > θ(1 − ω)(1 − σ w βxw )2 ,σ r (βxr )2 < 1.
In the following we assume these conditions to be satisfied.
In the steady state, we know the rates of returns on capital for wholesale andretail banks and households are
Z +Q
Rkw =
Q
Z +Q
Rkr =
Q + αr K r
Z +Q
= R.
Rkh =
Q + αh K hThus we haveQ=Z
Rkw − 1,Z − (Rkr − 1)Q
Rkw − Rkr=Z,
Rkr
Rkr (Rkw − 1)
Rkw − RZ − (R − 1)Q
αh K h ==Z,
R
R(Rkw − 1)
αr K r =66

and Q, K r and K w are functions of (xw , xr ) .
Equilibrium C: Dw = 0
Here, the market clearing condition of capital is given by
Qk w wNnw
ψ w (xw ) − θ (1 − ω) w w
N (x )=θω= Q (xw , xr ) K − K r (xw , xr ) − K h (xw , xr )QK w =The market clearing condition of interbank credit is given by
 w

Qk
B=
− 1 Nwwnw w
ψ (x ) − θ w w=
N (x ) θω
1 ψ r (xr ) r r
w
r
r r
w
r
r
w
r
=
N (x ) − [Q (x , x ) + α K (x , x )] · K (x , x )
γ
θ

(63)

(64)

The equilibrium value of (xw , xr ) is given by (xw , xr ) which satisfies (63, 64) simultaneously.
r
In order to verify µw
d > 0 and µd > 0, it is sufficient to check the inequalities
xw > xr > R = β −1 .
w
For the other inequality µwb > ωµd , it is sufficient to check

Rkw − Rb > ω (Rkw − R) ,or
(1 − ω)(Rkw − R) > Rb − R.
This is equivalent with
(1 − ω)βxw − 1βxr − 1>γ.
ψ w (xw )ψ r (xr )Equilibrium D: Dw > 0
67

(65)

w
For this type of equilibrium, we need µw
kb = ωµd , or

Rkw − Rb = ω (Rkw − R) .
Thus


dw
bw
x
1+ w +ω w
nnw
ψ,= (Rkw − R)θ
Thus being similar to the expression for β(Rkr − R), we getw

β(Rkw − R) = θ− R = (Rkw − R)

βxw − 1(βxw − 1) (1 − σ w βxw )=θ≡ ϕw (βxw ) .
ψ w (xw )(1 − σ w )βxw

Rkw is an increasing function of xw if σ w (βxw )2 < 1.
Also we learn

Rb − R = (1 − ω) (Rkw − R) = γ (Rkr − R) ,or
(1 − ω)ϕw (βxw ) = γϕr (βxr ) ,(66)

and thus xr is an increasing function of xw . We can solve Q and K h as functions of
xw as
Z
Q= w
Rk − 1βZ= w≡ Q(xw ),ϕ (βxw ) + 1 − β
1[βZ − (1 − β)Q]αh
1
βZϕw (βxw )= h w≡ K h (xw ).
α ϕ (βxw ) + 1 − βKh =We also get

1 Z − (Rkr − 1)Q
Z Rkw − Rkr=αr
Rkr
αr Rkr (Rkw − 1)βZϕw (βxw )
1
γ+ω−1
= r w
w
α ϕ (βx ) + 1 − β γ + (1 − ω)ϕw (βxw )
γ+ω−1
αh h
=
K ≡ K r (xw )γ + (1 − ω)ϕw (βxw ) αr

Kr =

68

The capital market equilibrium is given by
1 w w 1−ω w
ψ N −(N + Dw )θωω
1 w w 1−ωψ N −(QK w − B)=θωω1
= ψ w N w + (1 − ω)Bθ
1 w w 1 − ω ψr r
r rr
= ψ N +N − (Q + α K )Kθγθ
= Q K − Kh − Kr .

QK w =Thus
ψw w 1 − ω ψr r
N +
N
θ
γ θ


ψw
βxr − 1 rw
N , ( ∵ (66))
=
N + w
θ
βx − 1
1 − ω Q + αr K r rhr=Q K −K −K +
K
γ
Q


1 − ω Rkw r
hr
K=Q K −K −K +
γ Rkr


γ+ω−1
h
r
=Q K −K −K ,γ + (1 − ω)ϕw (βxw )or
βxr − 1 r r
ψ w (xw )w w
N (x ) + w
N (x )
θβx − 1


γ+ω−1
w
h w
r
w
= Q(x ) K − K (x ) −K (x ) .
γ + (1 − ω)ϕw (βxw )(67)

The equilibrium is given by (xr , xw ) which satisfies (66, 67) .
We need to check Dw > 0, or
 w



ψ
1−ω
1 w w 1 − ω ψr r
w
r r
r
0<
−
N − ψ N −N − (Q + α K )K ,
θω
ω
θ
γ
θ
or

 r r

ψ w (xw )ψ (x ) r r
w w
w
r r
w
r
w
γ
− 1 N (x ) > ωN (x ) − [Q (x ) + α K (x )] · K (x ) .
θ
θ


69

Equilibrium E: No Active Interbank Market
We have for j = w, r thator

(Q + f j )k j
ψ j (xj ),=njθ (Q + f j )k j
 ψ j (xj )j,xj − R = Rkj − R
=
R
−
R
k
nj
θ
Rkj − R = θ

xj − R,ψ j (xj )or
Rkj = Rkj xj , Rkj0 (·) > 0
if σ w (βxj )2 < 1. Thus
Q = Q (xw ) , Q0 (·) < 0
K h = K h (xw ) , K h0 (·) > 0.
The aggregate capital of retail banks satisfiesww
rr
Z − (Rkr − 1)Qw Z Rk (x ) − Rk (x )=Q
(x)αr Rkrαr Rkr (xr )[Rkw (xw ) − 1]ψ r (xr ) r r=
N (x )
θThe capital market clearing condition is

QK r = Q

ψ w (xw ) w w
N (x )θ = Q (xw ) K − K r (xr , xw ) − K h (xw )(68)

QK w =(69)

The equilibrium is given by (xr , xw ) which satisfies (68, 69) .
9.2.1

Appendix C: Anticipated Bank Run Case

Here we describe the conditions determining agents policy functions in the case of
anticipated runs. As in the text, we focus on the case in which variation in Zt+1 is
negligible. Moreover, we follow the notation by which for any given variable ξ˜t∗Et∗ ξet+1 = (1 − pt ) ξt+1 + pt ξt+1∗
where ξt+1
is the value taken by ξ˜t+1 when a run occurs.

709.2.2

Households

Households optimal choices of capital holdings and deposits are given byEt∗ Λ̃t,t+1 Rt+1 = 1hEt∗ Λ̃t,t+1 R̃kt+1
=1
9.2.3

Retail Bankers

The conditions in Lemma 2 that guarantee that retail banks are constrained are now
modified as follows:
Lemma 3 brt < 0 ,ktr > 0 and the incentive constraint is binding iffihi 1 hr
0 < Et∗ Ω̃rt+1 R̃kt+1− Rt+1 = Et∗ Ω̃rt+1 R̃bt+1 − Rt+1 < θ.
γ
The optimal choice of leverage isEt∗ Ω̃rt+1 Rt+1
h

i .
φrt =
r
θ − Et∗ Ω̃rt+1 R̃kt+1− Rt+1
9.2.4

Wholesale Bankers

The optimization problem of wholesale banks when bank runs are anticipated iscomplicated by the fact that the banker can avoid bankruptcy by reducing its leveragein case a run materializes. Here we derive conditions under which he does not wish
to do this. For simplicity, we focus on the problem of a wholesale banker that only
funds himself in the interbank market.
In this case we can derive a threshold level for leverage, φwM, under which thetbanker will survive a bank run, which is given byrEt∗ Ω̃rt+1 R̃γ,t+1
φwM
t
w∗


Rbt+1 = Rf,t+1 ≡= Rkt+1
wM
φ
−1
∗
t
E Ω̃
t

t+1

where
r
r
R̃γ,t+1≡ γ R̃kt+1+ (1 − γ) Rt+1

and Rf,t+1 is the risk free interbank rate that satisfies equation (44) with xw
t+1 = 1.
71

The objective function of wholesale bankers displays a kink at φwM, so that intorder to derive their optimal leverage choice we need to study separately the optimal
choice in the region where leverage is high enough to induce bankruptcy when a run, ∞),happens, [φwMt and in the region where bankruptcy is avoided even if a run
wM
happens, 0, φt. As long as wholesale bankers objective is strictly increasing inleverage in both of these regions, the incentive constraint holds with equality.
In the bankruptcy region, [φwM
, ∞), (45) with deterministic Zt+1 is simplified tot
φw
pt Ωr∗
t+1
t
r∗
w
r
w∗
Rγ,t+1 − w
Rbt+1 (φt ) = Rγ,t+1 +R.
1 − pt Ωrt+1
φt − 1 t+1
Then the objective function of a wholesale bank with one unit of networth is givenby
 w  w w
w
w
R̄
(φ
)
+
−
R̄
(φ
)
Ω
φ
R
)
=
(1
−
p
)
ψ w (φw
bt+1
bt+1
t
t
t+1
t
 w t w kt+1 r  t r w= (1 − pt ) Ωt+1 φt Rt+1 − Rγ,t+1 + Rγ,t+1
 w w∗


Ωr∗
t+1
r∗
r∗
+ pt Ω w
φ
R
−
R
+
R
t+1 r
k,t+1
γ,t+1
γ,t+1
Ωt+1 twhich is strictly increasing in φwt if and only if


Ωr∗
w∗
r∗
w
r
− Rγ,t+1>0(1 − pt ) Rkt+1− Rγ,t+1+ pt rt+1 Rkt+1
Ωt+1

(70)

Notice that condition (70) ish impliedby the condition
that guarantees that retail
i

r
r
∗
> 0, together with the fact thatbankers are constrained, Et Ω̃t R̃kt+1 − Rt+1
retail bankers are less efficient at intermediating capital than wholesale bankers αr >
0:


Ωr∗
w
r
w∗
r∗
− Rγ,t+1(1 − pt ) Rkt+1− Rγ,t+1+ pt rt+1 Rkt+1
Ωt+1


Ωr∗
r∗
r∗
r
r
− Rγ,t+1> (1 − pt ) Rkt+1− Rγ,t+1+ pt rt+1 Rk,t+1
Ωt+1
n

o
(1 − γ) ∗
r
r
=
E
Ω̃
R̃
−
R
>0
t+1
t
kt+1
Ωrt+1 tIn the region where the banker is able to avoid bankruptcy even when a run
happens, 0, φwM, the objective is instead
t
h 

n
io
w
w,n
w
∗
w
w
ψ (φt ) = Et Ω̃t+1 φt R̃kt+1 − Rf,t+1 + Rf,t+1
 w  w w
(1 − p
φt Rkt+1 − Rf,t+1+ Rf,t+1t ) Ωt+1



=
w
w∗
+pt Ωw∗
t+1 φt Rkt+1 − Rf,t+1 + Rf,t+1
72

and the condition that guarantees that the objective is strictly increasing in φwt inthis region ishiw
Et∗ Ω̃w
R̃
−
R
> 0.
(71)
f,t+1
t+1
kt+1
Given this we can modify the conditions in Lemma 1 as follows:
Lemma 4 : Under the conditions of Lemma 3, the incentive constraint is binding
iff
h
i
w
0 < Et∗ Ω̃w
(
R̃
−
R
)
f,t+1
t+1
kt+1


Ωr∗
w
r
r∗
w∗
.
− Rγ,t+1
Rkt+1
θω > (1 − pt ) Rkt+1− Rγ,t+1+ pt t+1
r
Ωt+1

9.3

Appendix D:Measurement

We use data from the Flow of Funds in order to construct empirical counterpartsof the financial flows in the simplified intermediation process described in Figure 1.
The first step in constructing our time series is a definition of the wholesale and retailsector within the broad financial business sector.
Our classification is based on the sectors and instruments reported in the Flowof Funds. We use the liability structure of the different sectors included in the”Financial Business” sector of the Flow of Funds in order to aggregate them into a
Retail sector, a Wholesale sector and Others. To do this, we proceed in two steps:
we first classify the funding instruments in the flow of funds into four categories
that we name Retail Funding, Wholesale Funding, Intermediated Assets and Other
Instruments; then we assign financial intermediaries to the Retail/Wholesale sector if
the funding instruments they mostly rely on belong to the Retail/Wholesale category.
Table 6 describes the four categories of funding we use. The labels in parentheses73are the identifiers in the Flow of Funds.
Table 6: Classification of Instruments in the Flow of Funds
Checkable Deposits and Currency (L.204)
Time and Saving Deposits (L.205)
Retail Funding
Money Market Mutual Fund Shares (L.206)
Mutual Fund Shares (L.214)
Repurchase Agreements (L.207)
Security Credit (L.224)
Short T erm
Financial Open Market Paper (L.208)
Wholesale Funding
Agency/GSE backed Securities (L.210)
Financial Corporate Bonds (L.212).
Long T erm
Retail Loans to Wholesale (L.215)
Non Financial Corporate Bonds (L.212)
Non Financial Equity (L.213)
Non Financial Open Market Paper (L.208)
Retail loans to non financial (L.215)
Intermediated Assets
Mortgages (L.217)
Consumer Credit (L.222)
Other Loans (L.216)
Other Types of Funding
All other instruments in the Flow of Funds
The criterion we use to define the above categories is the composition of demandand supply for each instrument. Instruments that are supplied by financial intermediaries and demanded by households fall in the Retail category, while instrumentsthat are mainly traded among financial intermediaries are included in Wholesale
Funding. Intermediated Assets consist of all of the claims issued by domestic non
financial business and households. Others is a residual category.
To define our Retail and Wholesale sectors, we start by excluding some typesof intermediaries from the ones that we are trying to study in our model economy.
These are the intermediaries listed in the ”Others” category in Table 7 below. The
remaining financial intermediaries appearing in the Flow of Funds are included in
the Retail/Wholesale sector if they mostly rely on Retail/Wholesale funding. The

74

resulting aggregation is described in Table 7.
Table 7:Aggregation of Financial Sectors in the Flow of Funds
Private Depository Institutions (L.110)
Money Market Mutual Funds (L.121)
Retail Sector
Mutual Funds (L.122)
Security Brokers Dealers (L.129)
ABS Issuers (L.126)
GSE and GSE Mortgage Pools (L.124-125)
Real Estate Investment Trusts (L.128)
Wholesale Sector
Finance Companies (L.127)
Funding Corporations (L.131).
Holding Companies (L.130)
Monetary Authority (L.109)
Private and Public Pension Funds (L.117)
Closed end and Exchange Traded Funds (L.123)
Other Intermediaries
Insurance Companies (L.115-116)
Government (L.105-106)
Rest of the World (L.132)
Households
L.101
Firms
L.102
Given this we construct the following measures:
1. Kth , Ktr , Ktw
The intermediation shares are constructed by computing aggregate short andlong positions of Households, Retail Banks and Wholesale banks in the markets
that make up the Intermediated Assets category in Table 6. The matrix below
describes each sectors’ activity in each market. If sector J has a long/shortposition in market X the corresponding entry is given by X+J /X−J . If sector J
has both long and short positions in market X, the corresponding entry alsoJJ
displays its net position, Xnet(+) /Xnet
(−).

75

Comm

Markets

Bonds

Equity
Paper

L.212

L.213

R
BO+
R
BO−
R
BOnet (+)
W
BO+
W
BO−
W
BOnet (−)
O
BO+
O
BO−
O
BOnet (+)

R
EQ+
NA

H
BO+

H
EQ+

0
0

F
EQ−

F
CP+
F
CP−
F
CPnet (−)

L.208

Consumer
Loans

Mortgages

L.215

L.208

CreditL.222

Sectors
RetailBanks

Wholesale
Banks

Other Item

Households

Firms

F
BO−

?
W
EQ+
NA

?
O
EQ+
NA

?

R
CP+
R
CP−
R
CPnet (+)
W
CP+
W
CP−
W
CPnet (−)
O
CP+
O
CP−
O
CPnet (+)

R
L+

R
M+

R
CC+

W
L−

W
M+

W
CC+

O
L−

O
M+

O
CC+

H
L−

H
M−

H
CC−

F
L−

F
M+
F
M−
F
Mnet (−)

FCC+

We make several assumptions in order to conduct our measures.
First, in the markets for bonds and commercial paper, some positions are potentially inconsistent with our intermediation model. This is because some sectors
within the retail category are short in these markets and some in wholesale areRWW
long, BOR
− > 0 , CP− > 0, BO+ > 0 and CP+ > 0. This allows for the possibility
that retail banks were borrowing from wholesale in these markets. However, we
rule out this possibility in constructing our measures for two reasons: given theheavy reliance on these types of instruments in financial transactions amongindustries within the respective categories and among financial firms within thesame industry, it is reasonable to assume that the vast majority of these offsetting positions were actually arising from cross holdings among firms withinR
the same category; moreover, the actual size of BOR− and CP− with respect to
W
BO−

CP R

CP R

−
−
and CPW− was very small, i.e. CP W '.1% and CP W '3% in 2007. This−−implies that we can safely work with the net positions for wholesalers andretailers.

Given the assumptions we make in these markets we can construct model consistent measures from bonds and commercial paper data by assuming thathouseholds lend to non financial firms, which is part of K h , while retail banks(and Other intermediaries) lend to both Wholesale banks, which is part of B,76

and firms, which is part of K r .45 We also assume that portfolio weights on
non financial and financial issued instruments in these markets are the same
i,F
for retail banks and other intermediaries.46 That is, letting zi,Fbo and zcp be
the proportions of lender’s i0 s holdings of bonds and commercial paper thatare issued by non financial firms, we haveBOF− − BOH
H,F
R,F
+
zbo = 1; zbo =
; 47
F
W
H
BO− + BOnet − BO+
CP F

−
R,F
Similarly for commercial paper: zH,F
cp = 0; zcp = CP F +CP W−

net

Second, for corporate equities the Flow of Funds does not report a disaggregated measure of equity issued by individual industries or the type of equity
held by the various industries. Since we use this market only in measuring K i ,we simply assume that each sector holds a scaled version of the same equity
portfolio consisting of the three sectors for which we have issuance data: Foreign equities, Financial Business equities and Non Financial Business Equities,
denoted by EQROW , EQF IN and EQN F I respectively. That is, in order tocompute how many funds flow to non financial firms from each other sector we
simply scale their total equity holdings by
η=

EQN F IEQN F I + EQF IN + EQROWGiven this we can computeH
Kth = ηEQH + BO+
R
R,FR
Ktr = ηEQR + zR,Fbo BOnet + zcp CPnet
R
R+ LF− + LH
− + M+ + CC+
45

The Households’ sector in the Flow of Funds is a residual category that includes Hedge Funds,
private equity funds and personal trusts, which are interemediaries that our model does not directly
capture. In any case, households’ intermediation in bonds and commercial paper market is a small
component of houseold intermediation so that very little would change if we instead made different
assumptions about households positions in these markets.
46
We include long positions of non-financial firms in the commercial paper within intermediation
performed by ”Others”.
47
H
Notice that we attribute all household’s lending in this market, BO+, to ”nonfinancial loans”hK ; we then allocate retail bankers supply of funds in this market to non financial loans, K r
proportionally to the weight of non financial firms demand for funds that is not met by households,
H
F
W
H
BOF
− − BO + , in the total demand for funds that is not met by housheolds, BO − + BO net − BO +77

KtW = ηEQW + M+W + CC+W
2. B, D
B is simply computed as wholesale net borrowing in all of the short termwholesale instruments: Repo, Commercial Paper, Agency Debt and Security
credit. D is given by Households and non financial Business holdings of retailfunding instruments.
3. Leverage multiple for broker dealers, finance companies and GSE
We compute financial leverage multiple for these three sectors by dividing total financial assets by financial assets minus financial liabilities plus equity
investment by holding companies. We do not have a measure of nonfinancial
assets in the flow of funds so the leverage multiple reported here overstates
financial leverage multiple that would include non financial assets in the computation. We compute average leverage multiple by using time varying weights
corresponding to the relative sizes of these three sectors as measured by total
financial assets.

9.4

Appendix E: Computation

It is convenient for computations to introduce the ex-ante optimal values of surviving
bankers at time t in the two sectors:
Ntw − W w=
(72)
V̄tw = [1 − σ + σθ (1 − ω + ωφw)]
tσw
Ntw − W w
= Ωw
tσw
Nr − W r
V̄tr = [1 − σ + σθφrt ] t r(73)
σNr − W r= Ωrt t rσ
Let the state of the economy if a run has not happened be denoted by x =(N w , N r , Z) , and the state in case a run has happened be denoted by x∗ = (0, N r , Z) .
We use time iteration in order to approximate the functions


 

Q (x) , Ch (x) , V̄r (x) , V̄w (x) , Γ (x)x ∈ W w , N̄ w × W r , N̄ r × [(.95) Z, Z]andQ∗ (x) , Ch∗ (x∗ ) , V̄r∗ (x∗ ) , Γ∗ (x∗ )x∗ ∈ {0} × W r , N̄ r × [(.95) Z, Z]
78

where Γ (x) and Γ∗ (x∗ ) are the laws determining the stochastic evolution of the state(See below).
The computational algorithm proceeds as follows:
1. Determine a functional space to use for approximating equilibrium functions.
(We use piecewise linear).

2. Fix a gridof values for the state in case no run happens G ⊂ W w , N̄ w ×rr∗
W r , N̄ r  × [.95, 1] and for the state in case a run happens G ⊂ {0} ×
W , N̄ × [.95, 1] .
3. Set j = 0 and guess initial values forhrw(x) , V̄t,j(x) , V̄t,j(x) , Γt,j (x) x∈G
N RP olt,j = Qt,j (x) , Ct,j
and

h∗
r∗
RP olt,j = Q∗t,j (x) , Ct,j(x∗ ) , V̄t,j(x∗ ) , Γ∗t,j (x∗ ) x∗ ∈G∗
.

r0
w0
r0∗
The guess for Γt,j (x) involves guessing pt,j (x) , Nt,j(x) , Nt,j(x) , Nt,j(x) , Z 0 (x)which implies


w0
r0
(x) , Nt,j(x) , Z 0 (Z) w.p. 1 − pt,j (x) Nt,j.
Γt,j (x) =


r0∗
0
0, Nt,j (x) , Z (Z)w.p. pt,j (x)

0
r0
w0
R
if there
We denote by x0N
t,j (x) = Nt,j (x) , Nt,j (x) , Z (Z) the state evolution

0R
r0∗
0
is no run in the following period and xt,j (x) = 0, Nt,j (x) , Z (Z) the evolution if a run happens in the following period.
onr0(x∗ ) , Z 0 (Z) which imSimilarly the guess for Γ∗t,j (x∗ ) involves guessing N̂t,j
plies


r0
Γ∗t,j (x∗ ) = (1 + σ w ) W w , N̂t,j(x∗ ) , Z 0 (Z)4. Assume that N RP olt,j and RP olt,j have been found for j ≤ i < M where M is
set to 10000. To find N RP olt,i+1 and RP olt,i+1 first use N RP olt,i and RP olt,ito find functions
 win theapproximating r r  space that take on these values on thewgrid, e.g. Qi : W , N̄ × W , N̄ × [.95, 1] → R is the price function thatsatisfies Qi (x) = Qt,i (x) for each x ∈ G.
79

5. Derive N RP olt,i+1 and RP olt,i+1 by assuming that from time t + 1 onwards
equilibrium outcomes are determined according to the functions associated to
N RP olt,i and RP olt,i found in step 4 :• NO RUN SYSTEM

r
h
h
r
At any point xt = (Ntw , Ntr , Zt ) ∈ G the system determining φw
t , φt , Bt , Qt , Ct , Kt , Kt
is given by

w
w
0N R
θ [1 − ω + ωφw(xt )t ] Nt = β (1 − pi (xt )) V̄i xi
w(φw
t − 1) Nt = Bt


w
r
h
φw
t Nt = Qt 1 − Kt − KtRθφrt Ntr = β (1 − pi (xt )) V̄ir x0N(x) + pi (xt ) V̄ir∗ x0Rii (x)φrt Ntr = (Qt + αr Ktr ) Ktr + (1 − γ) Bt
)
(


Cth
βEi
Z0 (Zt ) + Q̃i (Γi (x))= Qt + αh Kth
C̃hi (Γi (x))

2Kthαr (Ktr )2
+
=
2
2

Zt 1 + W h + W r + W w =(1 − σw ) (Ntr − W w ) (1 − σr ) (Ntr − W r ) αh
h
Ct +
+
+
σw
σr

where Ei is the expectation operator associated with the stochastic realizationof a run according to pi and tildes denote random variables whose values dependon the realization of the sunspot. For instance,
 h w00 Ci (Ni (x) , Nr0
i (x) , Z (Z)) w.p. 1 − pi (x)hC̃i (Γi (x)) =

r0∗
0
Ch∗
w.p. pi (x)i (Ni (x) , Z (Z))

One can then find Rt , R̄tb from
1

Rt =
βEi

80

n

Cth
h
C̃i (Γi (x))

o




(Z0 (Zt )+Q̃i (Γi (x)))
Ei Ω̃r (Γi (x)) γ+(1−γ)Rtr
Q +αr Ktt

R (x )(1−pi (xt ))Ωr (x0Nt )i

R̄tb =−whereΩ̃r (Γi (x)) =−pi Ωr∗ (x0R
i (xt ))

(Z0 (Zt )+Q̃i (Γi (x)))
Qt

φw
t
φw
t −1



R (x )(1−pi (xt ))Ωr (x0N
t )
i


r
w0
r0
0
r V̄i (Ni (x),Ni (x),Z (Z))
w.p. 1 − pi (x) σ
Nw0 (x)−W
i





σr

0
V̄ir∗ (Nr0∗i (x),Z (Z))
Nw0
(x)−W
i

w.p. pi (x)
and finally V̄tr , V̄tw , Γt are given byV̄tw = [1 − σ + σθ (1 − ω + ωφw
t )]

Γt =

Ntw − W w
σw

Nr − W r
V̄tr = [1 − σ + σθφrt ] t r
σ


r
w
, Z 0 (Z) w.p. 1 − pt
, Nt+1
 Nt+1r∗0, Nt+1, Z 0 (Z)w.p. pt

where
"
w
Nt+1
= σ w Ntw φw
t

!
#

R
Z0 (Zt ) + Qi x0N(x)i− R̄tb + R̄tb + W w
Qt

 0


R
Z (Zt ) + Qi x0N(x) Ktr + Bt R̄tb − Dt Rt + W w
i

 r

r∗
Nt+1
= σ r Z 0 (Zt ) + Q∗i x0R(Kt + Ktw ) − Dt Rt + W w
i (x)
δ

Z 0 (Zt )+Q∗i (x0R
i (x))w
φQt
pt = 1 −· wt 
φt − 1
R̄bt

r
Nt+1
= σr

• RUN SYSTEM
Analogously at a point x∗t = (0, Ntr , Zt ) ∈ G∗ the system determining

∗
h∗
h∗
φr∗
is given byt , Qt , Ct , Kt
r
r
∗
∗
θφr∗
t Nt = β V̄i (Γi (xt ))

81

r
∗
r r∗
r∗
φr∗
t Nt = (Qt + α Kt ) Kt


Cth∗
0
∗
∗
β
(Z (Zt ) + Qi (Γi (xt ))) = Q∗t + αh Kth∗
Chi (Γ∗i (x∗t ))

2 αr
2

αh
(1 − σr )(Ntr − W r )+Kth∗ +
1 − Kth∗ = Zt 1 + W h +W r
Cth∗ +σr22
 ∗ r∗ ∗and Rt , V̄t , Γt are given by
1

Rt∗ =


βEi

Cth∗
∗ ∗
Ch
(
i Γi (xt ))



r
r
r∗
r∗ Nt − WV̄t = [1 − σ + σθφt ]rσΓ∗i (x∗ ) =w(1 + σ ) Wwr, N̂t+1, Z 0 (Z)
Z 0 (Zt ) + Qi (Γ∗i (x∗t ))∗∗− Rt + Rt + W r
Qt

r∗
w
r
, Nt+1, Nt+1
6. Compute the maximum distance between N RP olt = Qt , V̄tr , V̄tw , Cth , pt , Nt+1
and N RP olt,idN R = max max |N RP olt − N RP olt,i |
xt ∈G
o
n
r
and RP olt,iand similarly for RP olt = Q∗t , V̄tr∗ , Cth∗ , N̂t+1
r
N̂t+1= σ r Nrr



φr∗
t



dR = max∗ max |RP olt − RP olt,i |xt ∈G

if dN R and dR are small enough, in our case e − 6, set
N RP olt,i+1 = N RP olt,iRP olt,i+1 = RP olt,iOtherwise set
N RP olt,i+1 = αN RP olt,i + (1 − α) N RP olt
where α ∈ (0, 1) .

RP olt,i+1 = αRP olt,i + (1 − α) RP olt

82

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Retail Short Term Funding
180

160

140

120

100

80

60

40

20

0
2000

2002

2004

2006

2008

2010

2012

2014

2016

Figure 19: Retail short term Funding
The graph shows the logarithm of the real value outstanding. Nominal values from Flow of Fnds
are deflated using the CPI and normalized so that the log of the normalized value of retail shortterm funding in 2001 is equal to 10089bIndifference Curve
Wholesale Bank

Incentive Constraint
Interbank Borrower

Incentive Constraint
Interbank Lenderand K>0d
Figure 20: Wholesale90
Banker’s Optimizationb

Indifference Curve
Retail Bank

Incentive Constraint
Interbank Borrower

Incentive Constraint
Interbank Lenderand K>091
Figure 21: Retail Banker’s
Optimizationd