Discussion Paper No. 951

MONEY IN THE EQUILIBRIUM
OF BANKING

Jin Cao
Gerhard Illing

December 2015

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Money in the Equilibrium of Banking 1
Jin Cao
Research Department, Norges Bank, Bankplassen 2, P. O. Box 1179 Sentrum, N-0107 Oslo, Norway
and CESifo, Germany
jin.cao@norges-bank.no
Gerhard Illing
Seminar für Makroökonomie, Ludwig-Maximilians-Universität München, Ludwigstrasse 28/012(Rgb.), D-80539 Munich, Germany and CESifo, Germany
illing@lmu.de

Abstract
In most banking models, money is merely modeled as medium for transaction, but in reality, money isalso the most liquid asset for banks. Central banks do not only passively supply money to meetdemand for transaction, as often assumed in these models, instead they also actively inject liquidity
into market, taking banks’ illiquid assets as collateral. We examine both roles of money in an
integrated framework, in which banks are subject to aggregate illiquidity risk. With fixed nominal
deposit contracts, the monetary economy with active central bank can replicate constrained efficient
allocation. This allocation, however, cannot be implemented in market equilibrium without additional
regulation: Due to moral hazard problems, banks invest excessively in illiquid assets, forcing thecentral bank to provide liquidity at low interest rates. We show that interest rate policy to reduce
systemic liquidity risk on its own is dynamically inconsistent. Instead, the constrained efficientsolution can be achieved by imposing ex ante liquidity coverage requirement.
Keywords: Central banking; liquidity facility; systemic liquidity risk
JEL classification: G21; G28

1 Introduction
After the financial crisis in 2008, concerns about financial stability and the disruption of financial
intermediation have become a key focus for central banks. Unconventional monetary policy measures
such as credit easing try to prevent systemic banking crises. At the same time, there are increasing
concerns that accommodative monetary policy with ample liquidity provision may encourage
excessive risk taking, resulting in a rapid increase in leverage of financial intermediation and soendangering financial stability in the future. Our paper tries to analyze the feedback mechanism
between central bank actions and bank’s incentives to cope with liquidity risk in a fiat money
economy.
Traditional banking models are real models without fiat money. Even though they provide useful
insights on the sources of financial instability, they are silent on the question what impact liquidity
provision via fiat money has on financial stability in reality. Recently, several studies, such as Allen etal. (2014) and Skeie (2008), try to fill in this gap. In these models, money is mostly used only as the1

The views expressed in this paper are those of the authors and do not necessarily reflect those of Norges Bank.

medium for transactions. The central bank is merely issuing bank notes passively to meet the money
demand in transactions (in Allen et al., 2014 even the banks are passive: they pass deposits to firms
which make decisions on investment portfolios). As a result, the (constrained) efficient equilibrium
can always be achieved as long as depositors’ nominal claims are met.
However, in reality fiat money is not only the medium for transaction, but also the most liquid asset inbanking sector. Central banks take a much more active role by providing liquidity to the economy.
Traditionally, the quantity theory of money focuses on the role of money as medium for transaction,with the price level being determined by the aggregate supply of money. The new Keynesian(“Woodfordian”) monetary theory focuses on interest rate adjustments as mechanism for stabilizing
the price level. However, in both perspectives banks are essentially doing nothing more than passivelytransmitting money into the real economy. The contribution of this paper is to capture both
perspectives in one framework, analyzing the impact of banks’ strategies on money demand, inflation,and financial stability in a purely monetary economy.
Apart from monetizing transactions, central banks also conduct active monetary policy, using shortterm interest rate to affect banks’ refinancing cost in the market for liquidity. Here, banks use their
illiquid assets – which provide return in the future – as collateral to raise money as liquid asset in order
to meet depositors’ demand for liquidity today. Money as liquidity and central bank’s monetary policy
practice have profound implications for financial stability: banks react actively to the path of central
bank’s policy rate, exploiting central bank’s liquidity facilities to reshuffle resources across periods,
whenever possible, by increasing the share of high-yield illiquid assets in their investment portfolios.
Such strategic feedback between central bank’s monetary policy and banks’ investment decisions, as
this paper shows, may become a source of systemic risk such that the economy gets stuck in an
inferior equilibrium.
1.1 Related literature
Freixas, Martin and Skeie (2011) analyze the efficiency of the interbank lending market in allocating
funds and the optimal policy of a central bank in response to liquidity shocks. They show that, during
an aggregate liquidity crisis, central banks need to manage both interest rates and liquidity injection(the aggregate volume of liquidity). In their model, failure to cut interest rates during a crisis erodes
financial stability by increasing the probability of bank runs. They do not, however, address the
feedback mechanism between central bank’s policy and the incentives of the banking system to invest
in private provision of liquid assets.
Allen, Carletti and Gale (2014) introduce nominal contracts in a banking model with idiosyncratic andaggregate liquidity risk. They show that the first-best efficient allocation can be achieved in adecentralized banking system when the central bank accommodates the demands of the private sector
for fiat money. In their model, variations in the price level allow full sharing of aggregate risks. Their
paper, however, does not analyze how financial intermediaries react ex ante to anticipated liquidity
injections by the central bank. The probability of an aggregate liquidity shock is not affected by theamount of private investment in liquid assets. In contrast, our paper focuses explicitly on theendogenous response of private banks to liquidity injections. We show that they engage in activities
creating systemic risk by investing in less liquid assets.
Our paper extends Cao and Illing (2011). That paper characterizes incentives for private liquidity
provision in a static framewok when the central bank mitigates fragility of the financial system arisingfrom banks liquidity transformation via nominal deposit contracts. Here, we analyze a repeated settingof Cao and Illing (2011), focusing on the case that systemic shocks are extremely rare events. In caseof a systemic shock, the central bank, trying to prevent costly bank runs, provides additional paper
money to the banks. Lender of last resort policy prevents interest rates from shooting up in case of
systemic risk, ensuring that banks are always able to pay out their nominal commitments. We showthat the central bank may be forced to keep interest rates low for an extended period, crowding out
private liquidity provision.
1.2 Structure of the paper
Section 2 presents the structure of the model in a real economy in the absence of central bank activity
and fiat money. It characterizes the constrained efficient solution for the pure market case. Banks
engaged in maturity transformation have an incentive to hold liquid assets to meet depositors’ demandfor idiosyncratic risks But negative aggregate liquidity shocks trigger system-wide bank runs,
generating huge social costs. Section 3 shows that in a decentralized monetary economy with thecentral bank providing money both as medium of transaction as well as bank liquidity, banks can get
access to liquidity through central bank’s interday loans, using their long maturity assets as collateral.
Costly bank runs can be eliminated and the constrained efficient allocation can be replicated. However,section 4 shows that this allocation cannot be implemented as dynamic consistent equilibrium due tothe moral hazard arising from the central bank’s commitment to provide standing liquidity facilities,
so the monetary economy ends up in an inferior equilibrium. Section 5 further shows that the same
result holds in the long-run equilibrium. In the absence of aggregate liquidity risk, banks can perfectly
hedge against idiosyncratic risks in the real economy by holding liquidity buffer, however, section 6.1
shows that the incentive for private liquidity provision will be crowded out in the monetary economy.
To achieve constrained efficiency, section 6.2 suggests that due to the time-inconsistency problem
central bank’s liquidity facility should not condition on the banks’ liquidity; instead, liquidity supportshould be made contingent on the banks’ fulfilling liquidity constraints as entry requirement. Section 7
concludes.

2 The real economy
In this section we consider the real economy. Following Cao & Illing (2011), we describe the sourceof liquidity risk and, as the baseline result for the rest of the paper, characterize the constrained
efficient allocation.
2.1 The model setup
Consider the following economy with three types of risk neutral agents: depositors, banks (run bybank managers), and entrepreneurs. The economy extends over three periods, 𝑡 = 0, 1, 2, and thedetails of timing will be explained later. We assume that:(i) There is a continuum of depositors, each being endowed with one unit of resources at 𝑡 = 0. They
will be willing to deposit in banks as long as the real return is (weakly) larger than 1. Depositors are
impatient: they want to withdraw and consume at 𝑡 = 1; in contrast, banks and entrepreneurs are
indifferent between consuming at 𝑡 = 1 and 𝑡 = 2;(ii) There is a finite number 𝑁 of active banks participating Bertrand competition, competing fordepositors’ deposits at 𝑡 = 0. Using these deposits, banks as financial intermediaries can fund projectsof entrepreneurs;

(iii) There is a continuum of entrepreneurs, and there are sufficiently many entrepreneurs competingfor funds so that bank deposits are scarce. Each of the entrepreneurs runs one of two types of the
projects:• Safe projects, which are realized early at 𝑡 = 1 with a safe return 𝑅1 > 1.

• Risky projects, which give a higher return 𝑅2 > 𝑅1 > 1. With probability 𝑝, these projects will be
realized at 𝑡 = 1, but the return may be delayed (with probability 1 − 𝑝) until 𝑡 = 2. Therefore, in
the aggregate, the share 𝑝 of type 2 projects will be realized early. The value of 𝑝, however, is not
1

known at 𝑡 = 0. It will only be revealed between period 0 and 1 at some intermediate period, 𝑡 = 2.

In the following, we are interested in the case of aggregate shocks to all risky projects (but we will
discuss idiosyncratic shocks later): The value of 𝑝 can be either 𝑝𝐻 or 𝑝𝐿 with 𝑝𝐻 > 𝑝𝐿 . The “good”
state with 𝑝 = 𝑝𝐻 will be realized with probability 𝜋. Since the “bad” state, or the “crisis” state, is rare,
we assume that 𝜋 is almost 1 2. In the following, we further assume that 1 < 𝑝𝑠 𝑅2 < 𝑅1 (𝑠 ∈ {𝐻, 𝐿})to focus on the relevant case. 3

At 𝑡 = 0 banks compete for depositors by offering them fixed deposit contracts that promise a return𝑑0 at 𝑡 = 1. At the same time, banks decide the proportion 𝛼 of deposits to be invested in the safe
assets. Depositors have rational expectations: they deposit in banks that offer them the highest
expected return.
1
2

At 𝑡 = the value of 𝑝 is revealed, and this is public information. Given the value of 𝑝, if one bankwill not be able to meet its depositors’ claims at 𝑡 = 1, the depositors will run on the bank at 𝑡 =1
2

because of first-come-first-served rule. If a bank experiences a run at this date, it has to liquidate all
unmatured assets, i.e., both safe and risky projects. Each unit of liquidated asset yields a poor return𝑐 < 1.

If there is no bank run, banks collect a proportion 𝛾 from the return of early projects (safe projects plus
those risk projects that return early) and early entrepreneurs retain the rest (the implication of 𝛾 will be
explained later). To maximize depositors’ return, banks can raise more funds from early entrepreneursin liquidity market: banks borrow from early entrepreneurs, promising a borrowing rate 𝑟 and using
their late projects as collateral. Since entrepreneurs are indifferent between consuming at 𝑡 = 1 and𝑡 = 2, they will be willing to lend to banks as long as 𝑟 ≥ 1. Banks make payout to depositors using
the return collected from early projects and the liquidity borrowed from liquidity market.
At 𝑡 = 2 banks collect return from late projects and pay back early entrepreneurs.

If there were no financial friction, depositors would contract directly with entrepreneurs, investing
entirely in the safe projects, and receive all the return at 𝑡 = 1. However, due to hold up problems asmodeled in Hart & Moore (1994), entrepreneurs can only commit to pay a fraction 𝛾 of their return(assume that 𝛾𝑅𝑖 > 1). Banks as financial intermediaries are assumed to have better collection skills(higher 𝛾) than depositors, so that depositors become better off by depositing their endowments in
2

As long as 𝜋 ∈ �𝛾𝛾[𝑅𝐿 ]−𝑐𝛾𝛾[𝑅𝐻 ]−𝑐, 1� with 𝐸[𝑅𝑠 ] = 𝛼𝑠 𝑅1 + (1 − 𝛼𝑠 )𝑅2 (𝑠 ∈ {𝐿, 𝐻}) and 𝛼𝑠 =

𝛾−𝑝𝑠

𝑅
𝑅2

𝛾−𝑝𝑠 +(1−𝛾) 1. Thisallows us to focus on the case that the cost of bank failure is small enough that banks can take more
liquidity risk instead of investing higher share on liquid assets to avoid bank runs completely.
3
In that case, depositors care about the share invested in liquid projects. On the other hand, due tofinancial frictions captured by the hold-up problem as characterized at the end of this section,
depositors gain from banks also investing some share in illiquid projects.

banks. However, as is shown in Diamond & Rajan (2001), banks then have the incentive to abuse theircollection skills and force depositors to renegotiate at 𝑡 = 1, making deposit contracts break down. To
avoid this, banks need to offer fixed deposit contracts, as is assumed in this paper, and depositors are
privileged to run once they perceive that banks cannot meet the contracts, leaving all banks’ assets
destroyed. The threat of bank run is thus a device to discipline banks to respect deposit contracts.
2.2 The constrained efficient allocation
The baseline of the model is the constrained efficient allocation, achieved from the solution of thesocial planer’s problem. This will be the reference point for the market equilibrium characterized in
the next section, when banks serve as financial intermediaries. Assume that the social planner has thesame collection skill (the same 𝛾) as banks, and she maximizes the depositors’ real return by choosing
a portfolio of safe and risky assets. The result is characterized in the following proposition:
Proposition 1 The constrained efficient allocation is featured by(1) At 𝑡 = 0 the planner invests a share 𝛼𝐻 =

𝛾−𝑝𝐻

𝑅
𝑅2

𝛾−𝑝𝐻 +(1−𝛾) 1

on the safe assets, and the depositors’

expected return at 𝑡 = 1 is 𝐸[𝑅(𝛼𝐻 )] = 𝜋𝜋𝜋[𝑅𝐻 ] + (1 − 𝜋)𝛾𝛾[𝑅𝐿 ];12(2) When 𝑝𝐿 is realized at 𝑡 = , depositors’ return at 𝑡 = 1 is 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 <𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑅2 ].

Proof See Appendix.

Since the social planner cannot reshuffle resources between periods and depositors only value
consumption at 𝑡 = 1, a share of the funds should be invested in safe projects. However, depositors
will also benefit from social planner’s holding of risky assets, since social planner can raise additionalliquidity from early entrepreneurs and maximize depositors’ return, using the delayed risky projects as
collateral. Given that the likelihood of a bad state is low, the social planner should choose theinvestment portfolio such that it maximizes depositors’ return in the good state; while in the bad state,
the abundant delayed projects allow social planner to use all the funds held by the early entrepreneurs
at 𝑡 = 1 and return to depositors, even though depositors’ return is lower than that in the good state.
2.3 Market equilibrium

In this section we consider a decentralized economy and characterize market equilibrium with banks.
∗) , ∀𝑖 ∈ {1, … , 𝑁} , that satisfy the
The equilibrium consists of banks’ strategic profiles (𝛼𝑖∗ , 𝑑0𝑖
following conditions:
Profit maximization of banks At 𝑡 = 0 every bank chooses its optimal proportion of investment in∗to maximize its expected return. That is, the bank makes
liquid assets 𝛼𝑖∗ and deposit contract 𝑑0𝑖
lower profit if it chooses any different strategic profile∗∗), ∀(𝛼𝑖 , 𝑑0𝑖 ) ≠ (𝛼𝑖∗ , 𝑑0𝑖);
Π𝑖 (𝛼𝑖 , 𝑑0𝑖 ) < Π𝑖 (𝛼𝑖∗ , 𝑑0𝑖

Zero profit No bank makes positive profit, as a result of Bertrand competition;
Return maximization of depositors At 𝑡 = 0 depositors deposit their endowments at those banks
offering the highest expected return;Market clearing At 𝑡 = 1, if there is no bank run, the liquidity market is cleared by the interest rate 𝑟
which is offered to the early entrepreneurs by banks.
Since the bad state is a low probability event, banks can take more liquidity risk to maximize theirreturn in the good state, ignoring the liquidation cost in the bad state. Banks’ return is maximized
when they can obtain the most liquidity at the lowest interest rate. By market clearing condition, theequilibrium interest rate is determined by the value of banks’ late projects and the volume of liquidity(the rent retained by early entrepreneurs) – both are functions of 𝛼. In the end, banks’ decision
problem boils down to choosing the optimal 𝛼 so that banks can borrow in liquidity market at thelowest cost. The equilibrium can be characterized by the following proposition:
Proposition 2 Market equilibrium in the decentralized real economy is featured by(1) At 𝑡 = 0 all banks set 𝛼 ∗ = 𝛼𝐻 and offer 𝑑0∗ = 𝛾𝛾[𝑅𝐻 ];(2) When 𝑝𝐻 is revealed, banks raise liquidity from liquidity market at 𝑡 = 1 at interest rate𝑟 = 1 and repay depositors; when 𝑝𝐿 is revealed, banks experience runs and have to liquidate
12both safe and risky projects at 𝑡 = ;(3) Depositors’ expected return is 𝜋𝜋𝜋[𝑅𝐻 ] + (1 − 𝜋)𝑐 , which is inferior to that in theconstrained efficient allocation.
Proof See Appendix.

Comparing with the constrained efficient solution as characterized in the last section, it is evident thatin the decentralized economy banks do have the incentive to hold the same share of liquid assets intheir investment portfolio, which maximizes depositors’ return in the good state. However, when the12

bad state is revealed at 𝑡 = and there is an aggregate liquidity shortage in the economy, banks cannot

raise sufficient funding in the liquidity market. Although banks hold enough delayed projects which
may serve as good collateral, depositors will run the bank, resulting in socially costly liquidation of all
assets. In the next section, we show that costly bank run can be eliminated in a monetary economy,
where there is a central bank providing money both as medium of transaction and bank liquidity, so
the constrained efficient solution can be replicated.

3 Constrained efficiency in a decentralized monetary economy
In this section, we show that the constrained efficient allocation can be replicated in a decentralized
monetary economy, improving efficiency comparing with that under market equilibrium.
From now on, following the model outlined in the last section, assume there is a central bank acting as
monetary authority, with all deposit contracts being written in nominal terms. We distinguish two rolesof money:(1) Money as medium of transaction. In our monetary economy, all the transactions arecommitted via exchanges of cash versus goods. Fiat money is issued by the central bank tofacilitate the transactions (“monetizing the economy”), and the quantity of money incirculation is equal to transaction demand for money. For simplicity, we assume that thequantity of money issued in each period is equal to the quantity of real goods in transaction,
thus normalizing the price level in the absence of additional liquidity provision to 1.

(2) Money as liquidity. Banks can borrow additional fiat money from the central bank to meettheir demand for liquidity subject to the central bank’s policy rate, using their illiquid assets as
collateral.
Motivated by these two roles of money, monetary policy of the central bank is conducted through thefollowing two kinds of operations:(1) Money as medium of transaction is issued through intraday loans to banks during the period.
Its quantity is equal to the quantity of real goods in transaction, and banks have to pay backthe loans in the end of the period. This is the working mechanism of Allen, Carletti and Gale(2014);(2) On the other hand, central bank is also a crucial liquidity provider to the banking sector:
Money as liquidity is injected into the economy through interday loans to banks subject to thepolicy rate, using banks’ long assets as collateral. The loans are paid back one period later,
after the return of long assets materializes.

Central bank
1. Money issuance

5. Money
repayment
Gen. 0 safe projects

Banks
2. Loans

Gen. 0 risky projects

3. Capital purchase

4. Deposits
Gen. 0 depositors

Figure 1 Monetizing the economy at 𝑡 = 0. The arrows show the direction of money flows, and thenumbers indicate the sequence of money flows.

With a central bank providing both medium of transaction and liquidity, money is introduced to theeconomy in the following way:
At 𝒕 = 𝟎

Figure 1 shows the flows of money and goods throughout the sequence:
1. Banks get 1 unit of intraday money loans from the central bank, which is equal to theentrepreneurs’ demand for buying the endowments of the depositors;
2. Banks issue loans to the entrepreneurs, a share of 𝛼 for the entrepreneurs running safe projects(short term loans), 1 − 𝛼 for those running risky projects (long term loans);
3. Entrepreneurs use the loans to purchase the depositors’ endowments, as capital input to theirprojects;
4. Depositors deposit the money in banks. Banks offer depositors the nominal demand deposit
contracts that promise 𝑑0 nominal return at 𝑡 = 1 for each unit of deposit;

5. At the end of the period, banks pay back the loans to the central bank.
At 𝒕 = 𝟏 (good state)
12

Suppose at 𝑡 = it is revealed that the economy will in the good state, Figure 2 shows the flows ofmoney throughout the period 𝑡 = 1:

1. Banks get 𝑑0 unit of intraday money loans from the central bank, which is equal to thedepositors’ demand for buying the consumption goods from the entrepreneurs whose projects
return early;
2. Banks repay 𝑑0 to the depositors;
3. The depositors purchase 𝑑0 = 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 consumption goods from the earlyentrepreneurs;
4. Early entrepreneurs repay their loans, 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ], to banks, and
5. Deposit (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] in banks;
6. At the end of the period, banks pay back the loans to the central bank.

Central bank

6. Money
Repayment

1. Money issuanceBanks
2. Repayment
4. Loanrepayment
Depositors

Type 1 entrepreneurs
3. Goods
purchase

Figure 2 Money flows at 𝑡 = 1.

5. Deposits

Type 2 early entrepreneurs

Central bank
5. Money
repayment

1. Money issuanceBanks

4. Loanrepayment

2. Repayment

Type 1 entrepreneurs

Type 2 late entrepreneurs
3. Goods
purchase

Type 2 early entrepreneurs

Figure 3 Sequence of money flows at 𝑡 = 2 following the good state.
At 𝒕 = 𝟐 (after the good state)

After the good state in period 𝑡 = 1, Figure 3 shows the flows of money throughout the subsequent
period 𝑡 = 2:

1. Banks get (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] unit of intraday money loans from the central
bank, which is equal to early entrepreneurs’ demand for buying the consumption goods fromlate entrepreneurs;
2. Banks repay (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] to early entrepreneurs;
3. Early entrepreneurs purchase (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] = 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐻 )𝑅2
consumption goods from the late entrepreneurs;
4. Late entrepreneurs repay their loans, 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐻 )𝑅2 , to banks;
5. At the end of the period, banks pay back all the loans to the central bank.

At 𝒕 = 𝟏 (bad state)

In good times, the market equilibrium is constrained efficient and there is no need for the central bank1
2

in liquidity provision. However, if it is revealed at 𝑡 = that the economy will in the bad state, theliquidity stress of banks can be alleviated by the central bank’s interday facilities. The mechanism isshown as Figure 4:

1. When it comes to the repayment to the depositors, banks can only get a total of 𝛼𝐻 𝑅1 +
(1 − 𝛼𝐻 )𝑝𝐿 𝑅2 < 𝑑0 intraday loans from the central bank. Therefore
1.5 Banks get 𝛾(1 − 𝛼𝐻 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 liquidity through interday loans from the central bankusing 𝛾(1 − 𝛼𝐻 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 illiquid assets as collateral, given the policy rate 𝑟 𝑀 = 1;
2. Banks repay 𝑑0 to the depositors;
3. The depositors purchase 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 consumption goods from the early
𝑑

0
> 1;
entrepreneurs at a price 𝛼 𝑅 +(1−𝛼
)𝑝 𝑅
𝐻 1

𝐻

𝐿 2

4. Early entrepreneurs repay their loans, 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ], to banks, and5. Deposit 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐿 )𝑅2 in banks with gross interest rate 1;

The final step 6 is similar to Figure 2, except that banks only repay 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 intradayloans to the central bank, retaining 𝛾(1 − 𝛼𝐻 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 as liabilities to the central bank backed by
the collateral.
At 𝒕 = 𝟐 (after the bad state)
Central bank

1.5. Liquidity loan

6. Money
repayment

1. Money issuanceBanks
2. Repayment
4. Loanrepayment
Depositors

5. Deposits

Type 1 entrepreneurs
3. Goods
purchase

Type 2 early entrepreneurs

Figure 4 Sequence of money flows at 𝑡 = 1 in the bad state.
Central bank
4.5. Liquidity loan repayment
5. Money repayment

1. Money issuanceBanks

4. Loanrepayment

2. Repayment

Type 1 entrepreneurs
Type 2 early entrepreneurs

Type 2 late entrepreneurs
3. Goods
purchaseFigure 5 Sequence of money flows at 𝑡 = 2 following the bad state.
In the subsequent period 𝑡 = 2, banks need to repay the interday loans when the return of illiquid
collateral is realized. As Figure 5 shows,
1. Banks get 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐿 )𝑅2 − 𝛾(1 − 𝛼𝐻 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 unit of money loans from thecentral bank, which is equal to early entrepreneurs’ demand for buying the consumption goods
from late entrepreneurs less the collateral;
2. Banks repay 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐿 )𝑅2 to early entrepreneurs;
3. Early entrepreneurs purchase 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐿 )𝑅2 consumption goods from lateentrepreneurs;
4. Late entrepreneurs repay their loans, 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐿 )𝑅2 , to banks.

Since at that stage return of the collateral is realized, banks

4.5 Repay 𝛾(1 − 𝛼𝐻 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 inter-date loans to the central bank.

And in the last step 5, the intraday loan is repaid, too.

In the following proposition, we show that the constrained efficient allocation can be replicated in thedecentralized monetary economy:
Proposition 3 In the decentralized monetary economy, with a central bank as monetary authority
providing both interday and intraday loans to the banking sector, the constrained efficient solution
can be replicated such that
1. At 𝑡 = 0 all banks invest a share 𝛼 ∗ = 𝛼𝐻 =

𝛾−𝑝𝐻

𝑅
𝑅2

𝛾−𝑝𝐻 +(1−𝛾) 1

on the safe projects, and promise

the depositors 𝑑0 = 𝛾[𝛼 ∗ 𝑅1 + (1 − 𝛼 ∗ )𝑅2 ] nominal return at 𝑡 = 1 for each unit of deposit;
2. If it is in the good state at 𝑡 = 1 , the depositors receive 𝑑0 = 𝛼 ∗ 𝑅1 + (1 − 𝛼 ∗ )𝑝𝐻 𝑅2
consumption, and the price level is 𝑃𝑔 = 1;
3. If it is in the bad state at 𝑡 = 1, the depositors receive 𝑑0 = 𝛾[𝛼 ∗ 𝑅1 + (1 − 𝛼 ∗ )𝑅2 ] nominal
repayment and purchase consumption good at the price 𝑃𝑏 =𝑑0
> 𝑃𝑔 = 1. The
𝛼 ∗ 𝑅1 +(1−𝛼 ∗ )𝑝𝐿 𝑅2

consumption of depositors is 1 < 𝛼 ∗ 𝑅1 + (1 − 𝛼 ∗ )𝑝𝐿 𝑅2 < 𝑑0 , but there is no bank run;
4. Banks get access to the central bank’s interday loan facility only in the bad state at 𝑡 = 1:
Banks get 𝛾(1 − 𝛼 ∗ )(𝑝𝐻 − 𝑝𝐿 )𝑅2 liquidity through interday loan, using the same amount oflate projects as collateral. The interday loan will be repaid at 𝑡 = 2, when late projects return;(1−𝑝 )𝑅𝐻 2
5. The central bank’s policy rate is above 𝛼∗ (𝑅 −𝑅 )+(1−𝑝> 1 throughout the entire time
)𝑅
1

2

𝐻

2

horizon, but should be lowered to 1 at 𝑡 = 1 when the bad state is revealed.

Proof: See Appendix.

In the good state, banks holding a share 𝛼𝐻 of liquid assets will be able to make nominal repayment 𝑑0
to the depositors, by borrowing from early entrepreneurs against their illiquid assets. In this case,banks need intraday loans from the central bank only for transaction purpose, and depositors’ realconsumption is 𝑑0 , too. In the bad state, by borrowing from early entrepreneurs, banks will not be able
to fulfill 𝑑0 nominal repayment to depositors, since in that state the aggregate early output – hence themoney obtained from intraday loans – is lower than 𝑑0 . However, banks now have more delayed
projects which can be used as collateral for interday loans. Therefore, they can get additional funding

from the central bank’s liquidity line and so meet the depositors’ demand for nominal repayment. In
this case, depositors’ real consumption will now be lower than 𝑑0 with the price level rising above 1,
but depositors have no longer incentive to run on the banks. In other words, the central bank’s interdayloans allow banks to fulfill depositors’ demand for nominal repayment in both states, while the pricelevel adjusts to make depositors’ real consumption contingent on the states. This eliminates thefragility arising from the fixed deposit contracts and improves social welfare.

4 Market equilibrium in the monetary economy and systemic liquidity risk
The key mechanism to replicate the constraint efficient solution hinges on the central bank’s interest
rate policy, which works through the liquidity facility. The policy rate should be so high in the normal
state that banks are induced to implement the first best equilibrium, while it should be low enough inthe crisis state that banks are guaranteed for sufficient liquidity through the central bank’s liquidity
facility.
However, the central bank’s commitment to high policy rate in the normal state, which intends to deterbanks’ incentive of over-investing in illiquid assets, is an incredible threat. Since the social cost ofbank failure is too high, it is ex post optimal for the central bank to bail out the illiquid banks even in
the normal state. Such time-inconsistency problem is characterized by the following proposition:
Proposition 4 In the decentralized monetary economy, with a central bank as monetary authority
providing both interday and intraday loans to the banking sector, the market equilibrium is featuredby
1. At 𝑡 = 0 all banks invest a share 𝛼 = 0 on the safe projects, and promise the depositors
𝑑0 = 𝛾𝑅2 nominal return at 𝑡 = 1 for each unit of deposit;
2. The central bank’s policy rate is 1 at 𝑡 = 1, no matter what state the economy is in;
3. If it is in the good state at 𝑡 = 1, the depositors receive 𝑑0 = 𝑝𝐻 𝑅2 consumption, and theprice level is 𝑃′𝑔 =𝛾> 𝑃𝑔 ;
𝑝𝐻

4. If it is in the bad state at 𝑡 = 1, the depositors receive 𝑑0 = 𝑝𝐿 𝑅2 consumption, and the price𝛾
level is 𝑃′𝑏 = > 𝑃𝑏 ;𝑝𝐿

5. Banks get access to the central bank’s interday loan facility in both states at 𝑡 = 1: Banks get(𝛾 − 𝑝𝑠 )𝑅2 ( 𝑠 = {𝐻, 𝐿} ) liquidity through interday loan, using the same amount of lateprojects as collateral. The interday loan will be repaid at 𝑡 = 2, when late projects return.

Proof: See Appendix.

Proposition 4 says that constrained efficient solution is bound to break down in the monetary economy,
when a central bank conducts active monetary policy, using short-term interest rate as policy
instrument and providing liquidity to the banking sector via interday loans. This is due to the moral
hazard arising from central bank’s time-inconsistent policy: Suppose that one bank 𝑖 deviates from theconstrained efficient solution by setting 𝛼� < 𝛼 ∗ and 𝑑̃0 = 𝛾[𝛼�𝑅1 + (1 − 𝛼�)𝑅2 ] at 𝑡 = 0, then this1
2bank will suffer from liquidity shortage at 𝑡 = even in the normal state. Although in the normal state

the central bank would like to commit to set its policy rate well above 1, it is now ex post optimal forthe central bank to cut its rate to 1 to avoid bank’s failure. However, this means that the total nominal
payoff that bank 𝑖 can get is 𝑑̃0 = 𝛾[𝛼�𝑅1 + (1 − 𝛼�)𝑅2 ] > 𝛾[𝛼 ∗ 𝑅1 + (1 − 𝛼 ∗ )𝑅2 ] = 𝑑0∗ , i.e., thisdeviator can offer higher nominal return to its depositors and outbid all its rivals. Knowing this, nobank will have an incentive to hold liquid assets at 𝑡 = 0 and the central bank will be forced to choose
low interest rates in both states. Although depositors get higher nominal return, their real consumption
is below the constrained efficient solution since too many projects are postponed to 𝑡 = 2. The
economy is trapped in the inferior equilibrium. In the next section, we show that the same result holds
in the long-run equilibrium.

5 Systemic liquidity risk in the long-run equilibrium
Now suppose that the monetary model extends to multiple periods, 𝑡 ∈ {0, … , 𝑇}. The settings remain
almost the same, except that depositors and entrepreneurs come from overlapping generations, whilebanks are infinitely-lived. Assume that
(1) There is a continuum of depositors born in each period 𝑡, call them generation 𝑡 depositors.
Each depositor lives for up to two dates — “young” and “old”: she deposits her endowment
in a bank when she is young, and she consumes when she is old. There is no population
growth;(2) There is a continuum of entrepreneurs born in each period 𝑡 — call them generation 𝑡
entrepreneurs — each running either a safe or a risky project. Each entrepreneur lives for up to
three periods — “young”, “middle age” and “old”: she works on one project (call it project 𝑡)when she is young, then consumes the proceeds later;(3) There is a finite number 𝑁 of active banks engaged in Bertrand competition in the deposit
market by offering fixed nominal deposit contracts 𝑑0,𝑡 to generation 𝑡 depositors. When abank experiences a run, it can be restructured and restart its business in the next period.
1
2

At each intermediate period 𝑡 + , the state of the world 𝑝𝑡 ∈ {𝑝𝐻 , 𝑝𝐿 } for date 𝑡 + 1 is revealed. To

focus on the illiquidity risk, we further assume that the crisis state is a low probability event, that is,
𝜋 → 1. With a central bank providing both medium of transaction and liquidity, the constrained
efficiency can be implemented in the following way:
At period 𝟎

Figure 6 shows the flows of money throughout the period:
1. Banks get 1 unit of intraday loans from the central bank, which is equal to the generation 0
entrepreneurs’ demand for buying the endowments of generation 0 depositors. Central bank
(1−𝑝 )𝑅

𝐻 2> 1 throughout the period;
keeps its interday loan rate 𝑟0𝑀 > 𝑟 = 𝛼 (𝑅 −𝑅 )+(1−𝑝
)𝑅
𝐻

1

2

𝐻

2

2. Banks issue loans to the entrepreneurs, a share of 𝛼0 = 𝛼 ∗ for the safe projects (short termloans), 1 − 𝛼0 for the risky projects (long term loans);
3. Entrepreneurs use the loans to purchase the depositors’ endowments, as capital input to theirprojects;
4. Depositors deposit the money in banks, using the nominal demand deposit contracts;
5. At the end of the period, banks pay back the loans to the central bank.

Central bank
1. Money issuance

5. Money
repayment
Gen. 0 safe projects

Banks
2. Loans

Gen. 0 risky projects

3. Capital purchase

4. Deposits
Gen. 0 depositors

Figure 6 Monetizing the economy at 𝑡 = 0. The arrows show the direction of money flows, and thenumbers indicate the timing of money flows.

Central bank

9. Money
Repayment

1. Money issuance

Gen. 1 safe projects

6. Loans

Gen. 0 depositors

Gen. 1 risky projects

Banks
2. Repayment

3. Goods
purchase

4. Loanrepayment

Gen. 0 type 1 ent.

8. Deposits
5. Deposits

7. Capitalpurchase
Gen. 1 depositors

Gen. 0 type 2 early ent.

Figure 7 Money flows at period 𝑡 = 1.
At period 𝟏

Figure 7 shows the flows of money throughout the period:
1. Banks get 𝑑0,0 unit of intraday loans from the central bank, which is equal to generation 0
depositors’ demand for buying the consumption goods from early generation 0 entrepreneurs.
Central bank keeps its interday loan rate 𝑟0𝑀 > 𝑟 throughout the period;
2. Banks repay 𝑑0,0 to generation 0 depositors;
3. Generation 0 depositors purchase 𝑑0,0 = 𝛼0 𝑅1 + (1 − 𝛼0 )𝑝0 𝑅2 consumption goods from theearly generation 0 entrepreneurs;4. Early generation 0 entrepreneurs repay their loans, 𝛾[𝛼0 𝑅1 + (1 − 𝛼0 )𝑝0 𝑅2 ], to banks, and
5. Deposit (1 − 𝛾)[𝛼0 𝑅1 + (1 − 𝛼0 )𝑝0 𝑅2 ] in banks;
6. Banks issue loans to generation 1 entrepreneurs, a share of 𝛼1 = 𝛼 ∗ for the safe projects (shortterm loans), 1 − 𝛼1 for the risky projects (long term loans);
7. Generation 1 entrepreneurs use the loans to purchase generation 1 depositors’ endowments, as
capital input to their projects;
8. Generation 1 depositors deposit the money in banks, using the nominal demand deposit
contracts;
9. At the end of the period, banks pay back the loans to the central bank.

Central bankGen. 0 type 2 late ent.
12. Money
repayment

1. Money issuance

Gen. 1 depositors

Gen. 0 type 1 ent.

4. Loanrepayment

Banks

Gen. 0 type 2 early ent.

2. Repayment

9. Loans

5. Repayment
6. Goods
purchase

3. Goods
purchase

Gen. 2 safe projects

7. Loanrepayment

Gen. 1 type 1 ent.

Gen. 2 risky projects

8. Deposits

11. Deposits

Gen. 1 type 2 early ent.

10. Capitalpurchase
Gen. 2 depositors

Figure 8 Money flows at period 2.
At period 𝟐

Figure 8 shows the flows of money throughout the period:
1. Banks get (1 − 𝛾)[𝛼0 𝑅1 + (1 − 𝛼0 )𝑝0 𝑅2 ] unit of intraday loans from the central bank, whichis equal to early generation 0 entrepreneurs’ demand for buying the consumption goods fromlate generation 0 entrepreneurs. Central bank keeps its interday loan rate 𝑟0𝑀 > 𝑟 throughout
the period;
2. Banks repay (1 − 𝛾)[𝛼0 𝑅1 + (1 − 𝛼0 )𝑝0 𝑅2 ] to early generation 0 entrepreneurs;
3. Early generation 0 entrepreneurs purchase (1 − 𝛾)[𝛼0 𝑅1 + (1 − 𝛼0 )𝑝0 𝑅2 ] = 𝛾(1 − 𝛼0 )(1 −𝑝0 )𝑅2 consumption goods from late generation 0 entrepreneurs;
4. Late generation 0 entrepreneurs repay their loans, (1 − 𝛾)[𝛼0 𝑅1 + (1 − 𝛼0 )𝑝0 𝑅2 ], to banks;

5. Banks get 𝑑0,1 − 𝛾(1 − 𝛼0 )(1 − 𝑝0 )𝑅2 more unit of money intraday loans from the central
bank and repay 𝑑0,1 to generation 1 depositors;
6. Generation 1 depositors purchase 𝑑0,1 = 𝛼1 𝑅1 + (1 − 𝛼1 )𝑝1 𝑅2 consumption goods from theearly generation 1 entrepreneurs;
7. Early generation 1 entrepreneurs repay their loans, 𝛾[𝛼1 𝑅1 + (1 − 𝛼1 )𝑝1 𝑅2 ], to banks, and
8. Deposit (1 − 𝛾)[𝛼1 𝑅1 + (1 − 𝛼1 )𝑝1 𝑅2 ] in banks with gross interest rate 1;
9. Banks issue loans to generation 2 entrepreneurs, a share of 𝛼2 = 𝛼 ∗ for the safe projects (shortterm loans), 1 − 𝛼2 for the risky projects (long term loans);
10. Generation 2 entrepreneurs use the loans to purchase generation 2 depositors’ endowments, as
capital input to their projects;
11. Generation 2 depositors deposit the money in banks, using the nominal demand deposit
contracts;
12. At the end of the period, banks pay back all the loans to the central bank.
For any period 2 ≤ 𝜏 ≤ 𝑇 after period 2, the intraday money flows are similar as those in period 2, aslong as the economy is in the normal state, with 𝛼𝜏 = 𝛼 ∗. However, it makes difference when theeconomy turns out to be in the crisis state. The difference starts from step 4, as Figure 9 shows:
4. Till this step, the money flows are the same as in Figure 8. However, when it comes to therepayment to generation 𝜏 − 1 depositors, banks can only get a total of 𝛼𝜏−1 𝑅1 + (1 −𝛼𝜏−1 )𝑝𝐿 𝑅2 < 𝑑0,𝜏−1 intraday loans from the central bank. Therefore
4.5 Banks get (1 − 𝛼𝜏−1 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 liquidity through interday loans from the central bankusing (1 − 𝛼𝜏−1 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 long assets as collateral, and central bank reduces its policy rate
(interday loan rate) 𝑟𝜏𝑀 |(𝑝𝜏−1 = 𝑝𝐿 ) = 1;
5. Banks repay 𝑑0,𝜏−1 to generation 𝜏 − 1 depositors;
6. Generation 𝜏 − 1 depositors purchase 𝛼𝜏−1 𝑅1 + (1 − 𝛼𝜏−1 )𝑝𝐿 𝑅2 consumption goods from theearly generation 𝜏 − 1 entrepreneurs at a price𝑑0,𝜏−1𝛼𝜏−1 𝑅1 +(1−𝛼𝜏−1 )𝑝𝐿 𝑅2> 1;

7. Early generation 𝜏 − 1 entrepreneurs repay their loans, 𝛾[𝛼𝜏−1 𝑅1 + (1 − 𝛼𝜏−1 )𝑝𝐿 𝑅2 ], tobanks, and
8. Deposit 𝛾(1 − 𝛼𝜏−1 )(1 − 𝑝𝐿 )𝑅2 in banks with gross interest rate 1;

The rest are the same as in Figure 8, except that banks only repay 𝛼𝜏−1 𝑅1 + (1 − 𝛼𝜏−1 )𝑝𝐿 𝑅2 intradate loans to the central bank at the end of the period, retaining (1 − 𝛼𝜏−1 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 as liabilities
to the central bank backed by the collateral.

Central bank

1. Money issuance

Gen. 𝜏 − 2 type 2 late ent.

12. Money
repayment

4. Loanrepayment

3. Goods
purchase

Gen. 𝜏 − 2 type 1 ent.

Gen. 𝜏 − 2 type 2 early ent.

2. Repayment

4.5. Liquidity injection

Gen. 𝜏 − 1 depositors

6. Goods
purchaseBanks
5. Repayment

9. Loans
Gen. 𝜏 safe projects

7. Loanrepayment

Gen. 𝜏 type 1 ent.

8. Deposits

Gen. 𝜏 type 2 early ent.

Gen. 𝜏 risky projects

11. Deposits

10. Capitalpurchase

Gen. 𝜏 depositors

Figure 9 Money flows at any period 𝜏 in crisis state.
In the following period 𝜏 + 1, suppose that the economy is in the normal state, banks need to repay theinter-date loans when the assets in the collateral return, and central bank raises its interday loan rate to
𝑟0𝑀 > 𝑟 throughout the period. As Figure 10 shows,
5. Banks get 𝛾(1 − 𝛼𝜏−1 )(1 − 𝑝𝐿 )𝑅2 − (1 − 𝛼𝜏−1 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 unit of intraday loans from thecentral bank, which is equal to early generation 𝜏 − 1 entrepreneurs’ demand for buying theconsumption goods from late generation 𝜏 − 1 entrepreneurs less the collateral;
6. Banks repay (1 − 𝛼𝜏−1 )(1 − 𝑝𝐿 )𝑅2 to early generation 𝜏 − 1 entrepreneurs;
7. Early generation 𝜏 − 1 entrepreneurs purchase 𝛾(1 − 𝛼𝜏−1 )(1 − 𝑝𝐿 )𝑅2 consumption goods
from late generation 𝑡 − 1 entrepreneurs;
8. Late generation 𝑡 − 1 entrepreneurs repay their loans, 𝛾(1 − 𝛼𝜏−1 )(1 − 𝑝𝐿 )𝑅2, to banks.

Since the collateral returns, banks

8.5 Repay (1 − 𝛼𝜏−1 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 inter-date loans to the central bank.

The rest are the same as in Figure 8.

Gen. 𝜏 − 2 type 2 late ent.

Central bank

12. Money
repayment

1. Money issuance

4. Loanrepayment

4.5. Liquidity loan
repayment
Gen. 𝜏 − 1 depositors

6. Goods
purchase
Gen. 𝜏 − 1 type 1 ent.

Banks
5. Repayment

3. Goods
purchase

Gen. 𝜏 − 2 type 2 early ent.

2. Repayment

9. Loans
Gen. 𝜏 safe projects

7. Loanrepayment

Gen. 𝜏 − 1 type 2 early ent.

Gen. 𝜏 − 2 type 1 ent.

8. Deposits

Gen. 𝜏 risky projects

11. Deposits

10. Capitalpurchase

Gen. 𝜏 depositors

Figure 10 Money flows at any period 𝜏 + 1 (in normal state) following a crisis state.
Proposition 5 With a central bank providing both medium of transaction and liquidity, the allocation
implemented through Figure 6 to 10 is constrained efficient.
Proof: See Appendix.
In the normal state, central bank sets its policy rate so high that banks rely on the liquidity market
instead of using central bank’s liquidity facilities. As shown in the baseline case, if banks coordinate
on choosing 𝛼𝐻 , the liquidity market in the next period will guarantee the constrained efficiency.
When it is in the crisis state, central bank reduces its policy rate so that banks can obtain interday loansat the lowest cost. In this way, depositors nominal claims can be fulfilled and price in consumption
goods market rises. Although in this case depositors’ real return is lower than that in the normal state,
they are still better off than receiving the poor return if they run on banks. Therefore, by making
depositors’ real return contingent on the state of the world, central banks’ conducting monetary policy
through its liquidity facilities avoids costly bank runs and achieves constrained efficiency.
But monetary policy also affects banks’ behavior: central bank’s commitment to provide cheap accessto liquidity facilities under stress crowds out incentives of private liquidity provision. As a result, theconstrained efficient allocation cannot be implemented as the market equilibrium.
1
2

To see this, suppose that at period 𝑡 the economy is in the normal state, while at 𝑡 + it is revealed

that the economy will be in the crisis state at period 𝑡 + 1 so that central bank needs to reduce its
𝑀
policy rate to 𝑟𝑡+1
= 1. At period 𝑡 + 1, banks make decision on their investment portfolios: (1) theycan either invest 𝛼 ∗ = 𝛼𝐻 as is suggested by the constrained efficient allocation, (2) or nothing inliquid assets, 𝛼 ∗ = 0. Then at period 𝑡 + 2 (most likely, the economy is in the normal state), when𝑀
central bank raises its policy rate to 𝑟𝑡+2
> 𝑟, if banks opted for (1), they will (be most likely to)survive since they can get obtain sufficient funding from liquidity market, while if banks opt for (2),they will become insolvent and experience runs since they cannot raise enough liquidity to fulfill𝑀depositors’ claims. However, if central bank remains its policy rate as 𝑟𝑡+2
= 1, banks will survive
under both (1) and (2) since they can access to cheap liquidity from central bank.
𝑀
𝑀
Ex post, central bank will be indifferent between 𝑟𝑡+2
> 𝑟 and 𝑟𝑡+2
= 1 under (1) since banks receive𝑀𝑀
the same payoff, 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑅2 ]; under (2), it will prefer 𝑟𝑡+2 = 1 to 𝑟𝑡+2
> 𝑟 to avoid costly∗∗bank runs. However, knowing this, banks will strictly prefer 𝛼 = 0 to 𝛼 = 𝛼𝐻 at period 𝑡 + 1, since
their payoff is strictly higher, 𝛾𝑅2 > 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑅2 ]. Therefore, in the dynamic consistentmarket equilibrium, banks choose 𝛼 ∗ = 0 and central bank remains 𝑟𝑡𝑀 = 1. By backward induction,
this applies for all 𝑡 ∈ {0, … , 𝑇}.

Notice that in such equilibrium, although depositors receive higher nominal return 𝛾𝑅2 >𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑅2 ], their real consumption 𝛾𝑝𝑡 𝑅2 is lower, 𝛾𝑝𝑡 𝑅2 < 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝑡 𝑅2 . Thisis due to the fact that all resources are invested in the illiquid assets so that the early return fordepositors is too low. That is, depositors are worse off in the dynamic consistent market equilibrium.
In summary, the dynamic consistent market equilibrium can be characterized in the following
proposition:
Proposition 6 With a central bank providing both medium of transaction and liquidity, the uniquedynamic consistent market equilibrium is characterized by(1) Banks choose 𝛼𝑡 = 0 and offer nominal deposit contracts 𝑑0𝑡 = 𝛾𝑅2 to depositors, ∀𝑡 ∈{0, … , 𝑇};
12(2) Central bank sets 𝑟𝑡𝑀 = 1, irrespective to the signals observed at 𝑡 − ;(3) Depositors become worse off than they are under the constrained efficient allocation.
Proof: See Appendix.
It can be also easily seen that the same conclusion holds even if the time horizon is infinite:
Corollary The dynamic consistent market equilibrium remains the same when 𝑇 → +∞.

6 Discussion
In this section, we discuss two extensions of the model. First, instead of aggregate liquidity shocks, we
assume that liquidity shocks are idiosyncratic to the banks. We show in this case, the market will reachconstrained efficient allocation even in the real economy, while efficiency again breaks down in themonetary economy. Then we discuss adequte mechanism design to implement constrained efficiency
in the monetary economy. We show that monetary policy rule fails to work due to time inconsistency
problem, and ex ante liquidity requirement is needed to implement the constrained efficient outcome.
6.1 Idiosyncratic liquidity shocks
Instead of aggregate liquidity shocks to all banks, suppose that each bank experiences an i.i.d. shock 𝑝𝑖(𝑖 ∈ {1, … , 𝑁}) which is randomly drawn from a probabilistic distribution 𝑓(𝑝𝑖 ) with support [𝑝𝐿 , 𝑝𝐻 ]

(0 < 𝑝𝐿 < 𝑝𝐻 < 𝛾 < 1). Further, we assume that the number of banks 𝑁 is large enough so that the
Law of Large Numbers holds.
When liquidity shocks are idiosyncratic, the market equilibrium will be constrained efficient in thereal economy. At 𝑡 = 0 banks invest 𝛼𝐸[𝑝] in liquid assets, based on the expected value of 𝑝; as shock
to each bank is revealed, banks’ liquidity problem will be solved by liquidity market: the demand for
liquidity (from those banks with liquidity shortages 𝑝𝑖 < 𝐸[𝑝]) will be met by the supply of liquidity(from those banks with liquidity surplus 𝑝𝑖 > 𝐸[𝑝]). Such equilibrium is summarized by the following
proposition:
Proposition 7 The market equilibrium in the real economy is constrained efficient, featured by(1) At 𝑡 = 0 each bank invests a share 𝛼𝐸[𝑝] =𝛾−𝐸[𝑝]𝑅𝑅2

𝛾−𝐸[𝑝]+(1−𝛾) 1

in the safe assets, and thedepositors’ expected return at 𝑡 = 1 is 𝑑0 = 𝛾�𝛼𝐸[𝑝] 𝑅1 + �1 − 𝛼𝐸[𝑝] �𝑅2 �;(2) Liquidity market at 𝑡 = 1 will redistribute liquidity among banks with liquidity surplus andbanks with liquidity shortages, and the market is cleared at interest rate 𝑟 = 1. All banks areable to fulfill depositors’ claims, and there is no bank run.
Proof See Appendix.
Proposition 7 says that as long as the ex ante symmetric banks hold a share 𝛼 of liquid assets, – 𝛼
being chosen based on their expectation on liquidity shock 𝑝 – after 𝑝 gets revealed for all the banks,
there will be banks with liquidity shortage (more delayed risky projects than they expected) and bankswith liquidity surplus (less delayed risky projects than they expected). The market mechanism will
ensure that the former can borrow from the early entrepreneurs of the latter, so that depositors of all
the banks can get fully repaid at 𝑡 = 1.

However, this conclusion will no longer hold in the nominal economy, with a central bank injecting
money into the economy via intraday and interday loans. Since there is only illiquidity risk, at 𝑡 = 1
the banks can always use the risky assets as good collateral to get liquidity through the central bank’s
interday loans. On the other hand, banks with liquidity shortage have to borrow from the central bankat 𝑡 = 1. It is always ex post optimal for the central bank to lower the interest rate of interday loans to
reduce the likelihood of bank failure. Knowing this, the banks will minimize their liquid assets holding
at 𝑡 = 0, aiming to maximize liquidity borrowed through interday loans at 𝑡 = 1, – hence the nominal
return at 𝑡 = 1 – and the central bank will be forced to cut the interest rate down to 𝑟 = 1 to avoid
bank runs. So the central bank’s commitment to providing standing liquidity facilities crowds out the
incentive of private liquidity provision. Although in nominal terms depositors get 𝑑0 = 𝛾𝑅2irrespective to the banks’ idiosyncratic shocks, higher than that in the constrained efficient solution,
their real consumption is 𝐸[𝑝]𝑅2 which is below the constrained efficient solution. Depositors areworse off.
6.2 Liquidity requirement and monetary policy
In the monetary economy, constrained efficiency breaks down because the central bank’s commitment
to providing interday loans against good collateral encourages banks to take excessive illiquidity risks.
To restore the constrained efficiency, one may suggest that the central bank can use monetary policy tocorrect banks’ incentives (such as Freixas, Martin and Skeie, 2011), i.e., when the good state isobserved, the policy rate should be prohibitively high so that banks need to borrow from liquidity
market; while when the bad state is observed, the policy rate should be low enough that banks cansurvive through the central bank’s liquidity line. Knowing this, banks will have the incentive to holdliquid assets so that market mechanism works to allocate liquidity in the good state, and costly bankruns will be eliminated in the bad state since banks can access central bank’s liquidity facilities at low
cost.
Unfortunately, such monetary policy rule fails to work in our dynamic setup since the rule itself is notcredible: If banks do not hold enough liquid assets at 𝑡 = 0, ex post it is always optimal for the central
bank to cut down policy rate at 𝑡 = 1 to avoid costly bank run. Knowing this, banks will not invest on
liquid assets at 𝑡 = 0 and the central bank will be forced to set policy rate to 1 despite the state of theeconomy. The time inconsistency problem makes the proposed monetary policy fail to save theeconomy out of the inferior equilibrium.
To avoid supporting banks with excessive investment on illiquid assets, one might suggest the central
bank should commit to provide interday loans only to those banks who hold sufficient liquid assets,i.e., 𝛼 ≥ 𝛼 ∗ in our model. However, such policy suffers from the same time inconsistency problem: If
one bank doesn’t hold enough liquid assets, 𝛼 < 𝛼 ∗, at 𝑡 = 0, it is still ex post optimal for the central
bank to allow the bank to access interday loans and avoid costly liquidation from any bank runs.
Therefore, there will be no incentive for banks to hold liquidity in the first place.
Therefore, rather than relying on implausible commitment mechanisms, the solution to fix the time
inconstancy problem and restore constrained efficiency in the monetary economy is to combine ex
ante liquidity requirement and ex post liquidity facilities. In the first place, banks should be obliged tomeet a certain requirement of holding liquid assets (such as Liquidity Coverage Ratio defined in Basel
III) in the normal time, and when there is a systemic liquidity shortage, central bank commits toproviding liquidity through interday loans and eliminates bank runs. This deters the moral hazard thatbanks are engaged in investing on excessive illiquid assets, raises despositors’ expected real return andrestores constrained efficiency.

7 Conclusion
Our paper developed a framework for analyzing the roles of money in banking, both as medium fortransaction and as bank liquidity. In the model, banks provide maturity transformation service todepositors. With fixed deposit contracts and aggregate liquidity risk, the fragile structure of banking
triggers bank run in the bad state, leading to socially costly liquidation.
Ideally, in a monetary economy with nominal deposit contracts, a central bank conducting active
monetary policy can eliminate such costly liquidation and replicate the first best solution. Money isissued by the central bank through (1) intraday loans to the bank, as a medium facilitating transaction,and (2) interday loans to banks, or standing liquidity facilities, to accommodate their demand for
liquidity, using illiquid long assets as collateral. In good times, banks can borrow from the liquidity
market to meet depositors’ demand; while in bad times when banks suffer from liquidity shortage,
central bank will inject liquidity to the market through liquidity facilities, making sure banks can still
fulfill their nominal deposit contracts. Central bank’s policy rate should be high in good times toencourage the efficient market outcome, while it should be low in bad times to avoid runs.
Unfortunately, such a scheme cannot be equilibrium market outcome. Absent liquidity regulation,banks always have incentives to invest excessively on the illiquid assets and obtain liquidity from thecentral bank to maximize depositors’ nominal return. Banks are more likely to have liquidity shortage
even in good times. So the central bank will be forced to cut down its policy rate to prevent bankfailure. As a result, the economy will end up in an inferior equilibrium: banks overinvest in illiquid
assets, the central bank has to keep policy rate low, and depositors are worse off from lower realconsumption.
We show that using interest rate rules to deter banks’ excessive risk taking is not credible for
implementing the constrained efficient allocation, because of a time inconsistency problem: Oncebanks engage in excessive liquidity risks ex ante, it is always ex post optimal for the central bank to
cut down interest rate. An additional instrument, such as imposing liquidity coverage requirement ex
ante, is needed to restore efficiency in the monetary equilibrium.

Appendix

A. Proof of Proposition 1
The proposition is proved by construction. First, suppose that there is no aggregate risk and 𝑝𝑠(𝑠 ∈ {𝐻, 𝐿}) is known at 𝑡 = 0, the social planner’s problem is to maximize depositors’ return bychoosing 𝛼𝑠 :max 𝛾 �𝛼𝑠 𝑅1 + (1 − 𝛼𝑠 ) �𝑝𝑠 𝑅2 +𝛼𝑠 ∈[0,1]

with market clearing condition(1 − 𝑝𝑠 )𝑅2��,𝑟𝑠𝑟𝑠 (1 − 𝛾)[𝛼𝑠 𝑅1 + (1 − 𝛼𝑠 )𝑝𝑠 𝑅2 ] = 𝛾(1 − 𝛼𝑠 )(1 − 𝑝𝑠 )𝑅2

and incentive compatibility constraint for entrepreneurs 𝑟𝑠 ≥ 1.

The solution is 𝛼𝑠 =

𝛾−𝑝𝑠

𝑅
𝑅2

𝛾−𝑝𝑠 +(1−𝛾) 1.

With aggregate risk, the incentive compatibility constraint for entrepreneurs still holds, implying that
depositors’ expected return at 𝑡 = 0 is
𝐸[𝑅(𝛼)] = 𝜋 min{𝛼𝑅1 + (1 − 𝛼)𝑝𝐻 𝑅2 , 𝛾[𝛼𝑅1 + (1 − 𝛼)𝑅2 ]}+ (1 − 𝜋) min{𝛼𝑅1 + (1 − 𝛼)𝑝𝐿 𝑅2 , 𝛾[𝛼𝑅1 + (1 − 𝛼)𝑅2 ]} , ∀𝛼 ∈ [0,1].

𝛾𝛾[𝑅𝐿 ]−𝑐, 1�), depositors’ expected return under different 𝛼 is𝛾𝛾[𝑅𝐻 ]−𝑐

When 𝜋 is sufficiently large (𝜋 ∈ �(1) If 𝛼 ∈ [𝛼𝐿 , 1] , depositors’ expected return is 𝐸[𝑅(𝛼)] = 𝛾[𝛼𝑅1 + (1 − 𝛼)𝑅2 ] <𝜋𝜋[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑅2 ] + (1 − 𝜋)𝛾[𝛼𝐿 𝑅1 + (1 − 𝛼𝐿 )𝑅2 ] = 𝐸[𝑅(𝛼𝐻 )];(2) If 𝛼 ∈ [0, 𝛼𝐻 ) , depositors’ expected return is 𝐸[𝑅(𝛼)] = 𝜋[𝛼𝑅1 + (1 − 𝛼)𝑝𝐻 𝑅2 ] +(1 − 𝜋)[𝛼𝑅1 + (1 − 𝛼)𝑝𝐿 𝑅2 ] < 𝜋[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] + (1 − 𝜋)[𝛼𝐻 𝑅1 + (1 −𝛼𝐻 )𝑝𝐿 𝑅2 ] = 𝐸[𝑅(𝛼𝐻 )];(3) If 𝛼 ∈ (𝛼𝐻 , 𝛼𝐿 ) , depositors’ expected return is 𝐸[𝑅(𝛼)] = 𝜋𝜋[𝛼𝑅1 + (1 − 𝛼)𝑅2 ] +(1 − 𝜋)[𝛼𝑅1 + (1 − 𝛼)𝑝𝐿 𝑅2 ] . When 𝜋 is sufficiently large, 𝐸[𝑅(𝛼)] < 𝜋[𝛼𝐻 𝑅1 +
(1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] + (1 − 𝜋)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] = 𝐸[𝑅(𝛼𝐻 )].

Therefore, when 𝜋 is sufficiently large, choosing 𝛼𝐻 maximizes depositors’ expected return.

B. Proof of Proposition 2
To show that the allocation is the market equilibrium in real economy, we need to show that theallocation is feasible and it is not profitable for any bank to deviate unilaterally.
If all banks choose 𝛼 ∗ = 𝛼𝐻 , when 𝑝𝐻 is revealed, depositors will get maximized return, 𝑑0∗ = 𝛾𝛾[𝑅𝐻 ],1
2

and this is exactly the solution to the planner’s problem. However, when 𝑝𝐿 is revealed at 𝑡 = , banks

can at most collect 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 < 𝑑0∗ from early entrepreneurs and liquidity market at12

𝑡 = 1. This triggers bank run at 𝑡 = , forcing banks to liquidate both safe and risky projects for aninferior return 𝑐 . Therefore, depositors’ expected return at 𝑡 = 0 is 𝛾𝛾𝛾[𝑅𝐻 ] + (1 − 𝜋)𝑐 <𝛾𝛾𝛾[𝑅𝐻 ] + (1 − 𝜋)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] = 𝐸[𝑅(𝛼𝐻 )], inferior to the planner’s solution.
Suppose one bank 𝑖 chooses 𝛼𝑖 ≠ 𝛼𝐻 :(1) If 𝛼𝑖 < 𝛼𝐻 , when 𝑝𝐻 is revealed, the interest rate in liquidity market at 𝑡 = 1 is determined by𝑟′{(1 − 𝛾)[𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 )𝑝𝐻 𝑅2 ] + (𝑁 − 1)(1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ]}= 𝛾(1 − 𝛼𝑖 )(1 − 𝑝𝐻 )𝑅2 + (𝑁 − 1)𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐻 )𝑅2 .

Remember that in the planner’s problem, equilibrium interest rate is determined by𝑟𝐻 (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] = 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐻 )𝑅2 ,and 𝑟𝐻 = 1. Therefore, 𝑟 ′ > 1. For the non-deviators, their return at 𝑡 = 1 becomes𝛾 �𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 ) �𝑝𝐻 𝑅2 +(1 − 𝑝𝐻 )𝑅2
�� < 𝑑0∗𝑟′

When 𝑝𝐿 is revealed, both deviator and non-deviators will experience runs.

Knowing that non-deviators will not even be able to meet the contracted 𝑑0∗ at 𝑡 = 1, thedepositors will only deposit at bank 𝑖 at 𝑡 = 0. If so, the deposit return that bank 𝑖 can offer is
𝑑0𝑖 = 𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 )𝑝𝐻 𝑅2 < 𝑑0∗ ,implying that the deviator becomes worse off;(2) If 𝛼𝑖 > 𝛼𝐻 , when 𝑝𝐻 is revealed, the aggregate liquidity supply at 𝑡 = 1 exceeds aggregateliquidity demand because(1 − 𝛾)[𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 )𝑝𝐻 𝑅2 ] + (𝑁 − 1)(1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ]> 𝑁(1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] = 𝑁𝑁(1 − 𝛼𝐻 )(1 − 𝑝𝐻 )𝑅2 ,Implying that the liquidity market interest rate remains at 𝑟 = 1 and non-deviators are able tomeet 𝑑0∗ at 𝑡 = 1. However, the deviator’s return at 𝑡 = 1 is
𝑑0𝑖 = 𝛾[𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 )𝑅2 ] < 𝑑0∗ .
When 𝑝𝐿 is revealed, non-deviator banks experience liquidity shortages and bid up theliquidity market interest rate. As a result, even if the deviator invests more on liquid assets, itcannot meet the deposit contract either. Therefore, both deviator and non-deviators will
experience runs.
Since the expected return from non-deviator banks is higher than that from the deviator,
implying that the deviator will not get any deposit at 𝑡 = 0 and is hence worse off.

Therefore the suggested allocation is indeed the market equilibrium in real economy.

C. Proof of Proposition 3
By investing a share 𝛼 ∗ = 𝛼𝐻 =

𝛾−𝑝𝐻

𝑅
𝑅2

𝛾−𝑝𝐻 +(1−𝛾) 1

at 𝑡 = 0 on the safe projects, at 𝑡 = 1(1) If it is in the good state, banks collect 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] real return from earlyentrepreneurs. Using the late projects (1 − 𝛼𝐻 )(1 − 𝑝𝐻 )𝑅2 = (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 −𝛼𝐻 )𝑝𝐻 𝑅2 ] as collateral, the banks are able to borrow (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] fromthe early entrepreneurs at the interest rate 𝑟𝑔 = 1. The central bank’s aggregate supply ofmoney to the banks through intraday loans is thus 𝑀𝑔 = 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] +(1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 ] = 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 = 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑅2 ] = 𝑑0 ,the same as the depositors’ nominal return. On the other hand, the aggregate output at 𝑡 = 1 is𝑌𝑔 = 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2, the aggregate demand for money in goods transaction 𝑃𝑔 𝑌𝑔 =𝑀𝑔 implies that the price level is 𝑃𝑔 = 1;
(2) If it is in the bad state, banks collect 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] real return from earlyentrepreneurs. Using (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] late projects as collateral, the banks areable to borrow (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] from the early entrepreneurs at the interest
rate 𝑟𝑏 = 1. The central bank’s aggregate supply of money to the banks through intraday loansisthus
𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] + (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] = 𝛼𝐻 𝑅1 +
(1 − 𝛼𝐻 )𝑝𝐿 𝑅2 . The banks can further borrow from the central bank at 𝑟 𝑀 = 1 throughinterday loans, using the rest of late projects 𝛾(1 − 𝛼𝐻 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 as collateral. In the end,
the depositors’ nominal return, or aggregate money supply in the economy, is 𝑀𝑏 =𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] + (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] + 𝛾(1 − 𝛼𝐻 )(𝑝𝐻 − 𝑝𝐿 )𝑅2 =𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐻 𝑅2 = 𝛾[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑅2 ] = 𝑑0 , the same as that is promised in thedeposit contracts. On the other hand, the aggregate output at 𝑡 = 1 is now 𝑌𝑏 = 𝛼𝐻 𝑅1 +
(1 − 𝛼𝐻 )𝑝𝐿 𝑅2 , the aggregate demand for money in goods transaction 𝑃𝑏 𝑌𝑏 = 𝑀𝑏 implies that𝑑𝛼 𝑅 +(1−𝛼 )𝑝 𝑅

0
𝐻 𝐻 2
= 𝐻 1
> 1.
the price level is 𝑃𝑏 = 𝛼 𝑅 +(1−𝛼)𝑝 𝑅
𝛼 𝑅 +(1−𝛼 )𝑝 𝑅
𝐻 1

𝐻

𝐿 2

𝐻 1

𝐻

𝐿 2

Furthermore, there will be no bank run in such decentralized monetary economy. In the good state, thedepositors get 𝑑0 nominal repayment and the same amount of real consumption, so there will be no
incentive to run on the bank. In the bad state, the depositors get at most 𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 < 𝑑0
1
2

consumption; however, if they run on the bank at 𝑡 = , they can only get 𝑐 < 1 < 𝛼𝐻 𝑅1 +

(1 − 𝛼𝐻 )𝑝𝐿 𝑅2 real consumption. Therefore, they will prefer not to run on the bank and wait instead toget 𝑑0 nominal repayment at 𝑡 = 1.
The banks’ nominal repayment to depositors at 𝑡 = 1 is 𝑑𝑠 = 𝛾 �𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 ) �𝑝𝑠 𝑅2 +(1−𝑝𝑠 )𝑅2
𝑟𝑠��,in which 𝑟𝑠 is the rate of the banks’ borrowing from entrepreneurs and / or the central bank in the state𝑠 ∈ {𝐻, 𝐿}:(1) In the bad state, if the banks only borrow from early entrepreneurs, market clearing condition𝑟𝐿 (1 − 𝛾)[𝛼𝐻 𝑅1 + (1 − 𝛼𝐻 )𝑝𝐿 𝑅2 ] = 𝛾(1 − 𝛼𝐻 )(1 − 𝑝𝐿 )𝑅2
implies that 𝑟𝐿 > 1, hence 𝑑𝐿 < 𝑑0 . However, if the central bank commits to setting 𝑟 𝑀 = 1
for its interday loans so that banks can borrow from both early entrepreneurs and the central
bank, depositors’ nominal repayment will be 𝑑𝐿 = 𝑑0 since 𝑟𝐿 = 𝑟 𝑀 = 1. The constrained
efficiency is achieved;(1−𝑝 )𝑅𝐻 2(2) In the good state, if the central bank’s policy rate is 𝑟𝑔𝑀 = 𝛼∗ (𝑅 −𝑅 )+(1−𝑝> 1, the banks
)𝑅
1

2

𝐻

2

𝛾(1−𝛼 )(1−𝑝 )𝑅will only borrow from the early entrepreneurs at the rate 𝑟𝐻 = (1−𝛾)[𝛼 𝑅𝐻+(1−𝛼𝐻 )𝑝2 𝑅 ] = 1.
𝐻 1
𝐻 𝐻 2

The depositors’ nominal repayment is therefore 𝑑𝐻 = 𝑑0 and the constrained efficiency isachieved.
D. Proof of Proposition 4
The proposition is proved by backward induction. The game consists of two subgames:(1) At 𝑡 = 1, for the banks with strategic profile (𝛼, 𝑑0 ), the central bank needs to set its policy
rate 𝑟𝑆𝑀 that maximizes the banks’ return in the state 𝑠 ∈ {𝐻, 𝐿}, i.e.
(1 − 𝑝𝑠 )𝑅2(1 − 𝛼) �𝑝𝑠 𝑅2 +
max
𝛾
+
��
�𝛼𝑅
1
𝑟𝑠
𝑟𝑆𝑀
in which 𝑟𝑠 = min{𝑟𝑠0 , 𝑟𝑆𝑀 }. The banks may borrow from the entrepreneurs at the rate 𝑟𝑠0 ,𝛾(1−𝛼)(1−𝑝 )𝑅𝑠 2, and 𝑟𝑠0 ≥ 1 to allow entrepreneurs towhich is determined by 𝑟𝑠0 = (1−𝛾)[𝛼𝑅 +(1−𝛼)𝑝
𝑅 ]
1

𝑠 2

participate. Obviously, the central bank’s optimal policy rate is 𝑟𝑆𝑀 = 1 irrespective with thestate;(2) Known that 𝑟𝑠 = 𝑟𝑆𝑀 = 1 at 𝑡 = 1, the banks’ problem at 𝑡 = 0 is to choose the strategic
profile (𝛼, 𝑑0 ) that maximizes depositors’ nominal return, i.e.
(1 − 𝑝𝑠 )𝑅2
max 𝑑0 = 𝛾 �𝛼𝑅1 + (1 − 𝛼) �𝑝𝑠 𝑅2 +�� = 𝛾{𝛼𝑅1 + (1 − 𝛼)𝑅2 }.
(𝛼,𝑑0 )
𝑟𝑠
Since 𝑅2 > 𝑅1 , it is optimal to set 𝛼 = 0 and 𝑑0 = 𝛾𝑅2 .

As for the aggregate real output and price level,(1) In the good state, banks collect 𝛾𝑝𝐻 𝑅2 real return from early entrepreneurs. Using part of thelate projects (1 − 𝛾)𝑝𝐻 𝑅2 as collateral, the banks are able to borrow (1 − 𝛾)𝑝𝐻 𝑅2 from theearly entrepreneurs at the interest rate 𝑟′𝑔 = 1. The central bank’s aggregate supply of money
to the banks through intraday loans is thus 𝛾𝑝𝐻 𝑅2 + (1 − 𝛾)𝑝𝐻 𝑅2 = 𝑝𝐻 𝑅2 , The banks canfurther borrow from the central bank at 𝑟 𝑀 = 1 through interday loans, using the rest of lateprojects 𝛾(1 − 𝑝𝐻 )𝑅2 − (1 − 𝛾)𝑝𝐻 𝑅2 = (𝛾 − 𝑝𝐻 )𝑅2 as collateral. In the end, the depositors’
nominal return, or aggregate money supply in the economy, is 𝑀′𝑔 = 𝛾𝑝𝐻 𝑅2 + 𝛾(1 −𝑝𝐻 )𝑅2 = 𝛾𝑅2 = 𝑑0 , the same as that is promised in the deposit contracts. On the other hand,the aggregate output at 𝑡 = 1 is now 𝑌′𝑔 = 𝑝𝐻 𝑅2 , the aggregate demand for money in goods
transaction 𝑃′𝑔 𝑌′𝑔 = 𝑀′𝑔 implies that the price level is 𝑃′𝑔 =𝛾> 𝑃𝑔 = 1;𝑝𝐻(2) In the bad state, banks collect 𝛾𝑝𝐿 𝑅2 real return from early entrepreneurs. Using part of thelate projects (1 − 𝛾)𝑝𝐿 𝑅2 as collateral, the banks are able to borrow (1 − 𝛾)𝑝𝐿 𝑅2 from theearly entrepreneurs at the interest rate 𝑟′𝑏 = 1. The central bank’s aggregate supply of money
to the banks through intraday loans is thus 𝛾𝑝𝐿 𝑅2 + (1 − 𝛾)𝑝𝐿 𝑅2 = 𝑝𝐿 𝑅2 , The banks canfurther borrow from the central bank at 𝑟 𝑀 = 1 through interday loans, using the rest of lateprojects 𝛾(1 − 𝑝𝐿 )𝑅2 − (1 − 𝛾)𝑝𝐿 𝑅2 = (𝛾 − 𝑝𝐿 )𝑅2 as collateral. In the end, the depositors’
nominal return, or aggregate money supply in the economy, is 𝑀′𝑏 = 𝛾𝑝𝐿 𝑅2 + 𝛾(1 −𝑝𝐿 )𝑅2 = 𝛾𝑅2 = 𝑑0 , the same as that is promised in the deposit contracts. On the other hand,the aggregate output at 𝑡 = 1 is now 𝑌′𝑏 = 𝑝𝐿 𝑅2 , the aggregate demand for money in goods
transaction 𝑃′𝑏 𝑌′𝑏 = 𝑀′𝑏 implies that the price level is 𝑃′𝑏 =𝛼𝐻 𝑅1 +(1−𝛼𝐻 )𝑝𝐻 𝑅2
𝛼 𝑅 +(1−𝛼𝐻 )𝑝𝐻 𝑅2> 𝐻 1= 𝑃𝑏 .
[𝛼)𝑅]
𝑝𝐿 𝐻 𝑅1 +(1−𝛼𝐻 2
𝛼𝐻 𝑅1 +(1−𝛼𝐻 )𝑝𝐿 𝑅2

E. Proof of Proposition 5𝛾𝛾[𝛼 𝑅 +(1−𝛼 )𝑅 ]= 𝑝 [𝛼𝐻 𝑅1 +(1−𝛼𝐻 )𝑅2 ] =
𝑝𝐿
𝐿 𝐻 1
𝐻 2

Since the planner’s problem is symmetric at any 𝑡, it is the same as the one in the static modelcharacterized in section 2. The allocation implemented through Figure 6 to 10 is the same as in
Proposition 1, therefore, it replicates the constrained efficient solution.
F. Proof of Proposition 6
The proposition is proved by backward induction. At any period 𝑡(1) If the banks choose (𝛼 ∗ , 𝑑0∗ )𝑀
a. If the central bank chooses 𝑟𝑡+1
= 1, banks’ nominal return is 𝑑0∗ despite the state at
𝑡 + 1;
𝑀
b. If the central bank chooses 𝑟𝑡+1
> 1, banks’ nominal return is 𝑑0∗ in the good state,while in the bad state the maximal nominal return is 𝛾 �𝛼 ∗ 𝑅1 + (1 − 𝛼 ∗ )𝑝𝐿 𝑅1 +(1−𝛼∗ )(1−𝑝𝐿 )𝑅1
𝑀
𝑟𝑡+1� < 𝑑0∗ so that banks will experience runs. Banks’ expected nominal

return at 𝑡 is therefore 𝜋𝑑0∗ + (1 − 𝜋)𝑐 < 𝑑0∗ ;(2) If the banks choose �𝛼�, 𝑑̃0 � ≠ (𝛼 ∗ , 𝑑0∗ )𝑀= 1 , banks’ nominal return is 𝑑̃0 = 𝛾[𝛼�𝑅1 +a. If the central bank chooses 𝑟𝑡+1
(1 − 𝛼�)𝑅2 ] which is maximized at 𝛼� = 0. In this case, 𝑑̃0 = 𝛾𝑅2 > 𝑑0∗ ;𝑀
b. If the central bank chooses 𝑟𝑡+1
> 1, banks’ nominal return is , 𝑑̃0 = max �𝛼�𝑅1 +(1 − 𝛼�)𝑝𝑅2 , 𝛾 �𝛼�𝑅1 + (1 − 𝛼�)𝑝𝑅1 +(1−𝛼� )(1−𝑝)𝑅1
𝑀
𝑟𝑡+1�� < 𝑑0∗ .

The unique subgame perfect equilibrium is that bankers choose 𝛼 = 0, 𝑑0 = 𝛾𝑅2 at 𝑡, and the central
𝑀bank stays with 𝑟𝑡+1
= 1, irrespective of the signal observed at 𝑡 + 1.5. However, the real return theinvestors receive is 𝑝𝑅2 < 𝑑0∗ .

G. Proof of Proposition 7

Under idiosyncratic liquidity shocks, the social planner’s problem is to maximize depositors’ expected
return by choosing 𝛼𝑖 for bank 𝑖 at 𝑡 = 0:
max 𝑑0 = 𝐸0 �𝛾 �𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 ) �𝑝𝑖 𝑅2 +𝛼𝑖 ∈[0,1]

with market clearing condition
𝑁

𝑁

𝑖=1

𝑖=1

(1 − 𝑝𝑖 )𝑅2���,𝑟𝑟 �(1 − 𝛾)[𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 )𝑝𝑖 𝑅2 ] = � 𝛾(1 − 𝛼𝑖 )(1 − 𝑝𝑖 )𝑅2

and incentive compatibility constraint for entrepreneurs 𝑟 ≥ 1.

The solution is 𝛼𝑖 =𝛾−𝐸[𝑝]𝑅𝑅2

𝛾−𝐸[𝑝]+(1−𝛾) 1, 𝑑0 = 𝛾�𝛼𝐸[𝑝] 𝑅1 + �1 − 𝛼𝐸[𝑝] �𝑅2 � and 𝑟 = 1.

In the market equilibrium, bank 𝑖’s problem is to maximize its depositors’ expected return by choosing
𝛼𝑖 at 𝑡 = 0:max 𝑑0 = 𝐸0 �𝛾 �𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 ) �𝑝𝑖 𝑅2 +𝛼𝑖 ∈[0,1]

taking into account the market clearing condition
𝑁

𝑁

𝑖=1

𝑖=1

(1 − 𝑝𝑖 )𝑅2���,𝑟𝑟 �(1 − 𝛾)[𝛼𝑖 𝑅1 + (1 − 𝛼𝑖 )𝑝𝑖 𝑅2 ] = � 𝛾(1 − 𝛼𝑖 )(1 − 𝑝𝑖 )𝑅2

and incentive compatibility constraint for entrepreneurs 𝑟 ≥ 1.

Since all the banks are ex ante symmetric, bank 𝑖’s problem in the market equilibrium is the same as
the social planner’s problem, the solution should be the same and constrained efficient. In addition,
depositors’ return 𝑑0 doesn’t depend on the bank’s realized liquidity shock 𝑝𝑖 , therefore, there is nobank run.

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