IMES DISCUSSION PAPER SERIES

Federal Reserve Tools for Managing Rates and Reserves

Antoine Martin, James McAndrews, Ali Palida,and David Skeie
Discussion Paper No. 2015-E-8

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
JAPAN

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IMES Discussion Paper Series 2015-E-8
July 2015

Federal Reserve Tools for Managing Rates and Reserves

Antoine Martin*, James McAndrews**, Ali Palida***and David Skeie****Abstract
Monetary policy measures taken by the Federal Reserve as a response to the
2007-09 financial crisis and subsequent economic downturn led to a large increasein the level of outstanding reserves. The Federal Open Market Committee (FOMC)
has a range of tools to control short-term market interest rates in this situation. We
study several of these tools, namely interest on excess reserves (IOER), reverse
repurchase agreements (RRPs), and the term deposit facility (TDF). We find thatovernight RRPs (ON RRPs) provide a better floor on rates than term RRPs because
they are available to absorb daily liquidity shocks. Whether the TDF or RRPs best
support equilibrium rates depends on the relative intensity of the frictions thatbanks face, which are bank balance sheet costs and interbank monitoring costs in
our model. We show that when both costs are large, using the RRP and TDF
concurrently most effectively raises short-term rates. While public money suppliedby the Federal Reserve in the form of reserves can alleviate bank liquidity shocks
by reducing interbank lending costs, large levels of reserve increase banks’ balance
sheet size and can induce greater bank moral hazard. RRPs can reduce levels of
costly bank equity that banks are endogenously required to hold as a commitment
device against risk-shifting returns on assets.
Keywords: monetary policy; fixed-rate full allocation overnight reverse
repurchases; term deposit facility; interest on excess reserves; FOMC;
banking
JEL classification: E42, E43, G12, G20*Federal Reserve Bank of New York (E-mail: antoine.martin@ny.frb.org)**Federal Reserve Bank of New York (E-mail: jamie.mcandrews@ny.frb.org)***Federal Reserve Bank of New York (E-mail: ali.palida@ny.frb.org)****Federal Reserve Bank of New York (E-mail: david.skeie@ny.frb.org)
We thank Alex Bloedel and Sean Myers for research assistance and seminar participants atseveral seminars and conferences for helpful comments. The views expressed in this paper are
those of the authors and do not necessarily reflect the official views of the Bank of Japan, theFederal Reserve Banks of New York or the Federal Reserve System.

1

Introduction

This paper studies new monetary policy tools for managing short-term market rates.
The tools we consider are interest on excess reserves (IOER), reverse repurchaseagreements (RRPs) with a wide range of market participants, and the term deposit
facility (TDF).
The Federal Reserve responded to the 2007-09 …nancial crisis and its aftermath
with a variety of monetary policy measures that dramatically increased the supplyof reserves. In part, this has led the federal funds rate, and other money market
interest rates, to be somewhat more variable than before. In October 2008, theFederal Reserve began paying IOER to depository institutions (DIs); despite this,money market rates have consistently remained below the IOER rate. In Juneof 2011, the Federal Open Market Committee (FOMC) announced new tools, term
RRPs and the TDF, aimed at managing short-term market interest rates and keeping
them close to the IOER.2 In August 2013, the FOMC also announced a …xed-rate,
full-allotment overnight RRP (ON RRP) as another potential tool.3
The sets of institutions that have access to each tool vary. IOER is paid to DIs
holding reserve balances at the Federal Reserve. 4 Similarly, the TDF is a facilityo¤ered to DIs, who are eligible to earn interest on balances held in accounts at theFederal Reserve, that allows them to hold deposits for longer term for an interest
rate generally exceeding the IOER. In contrast, term and ON RRPs are available
to a range of bank and non-bank counterparties, giving them the opportunity tomake the economic equivalent of collateralized loans to the Federal Reserve.5 An
institutional background and explanation of these tools is provided in Section 2.
We develop a general equilibrium model to study how the Federal Reserve canuse its tools to manage short-term interest rates and the large level of reserves
on its balance sheet. Our model extends Martin, McAndrews, and Skeie (2013)(henceforth referred to as MMS) to include two separate banking sectors, liquidity
shocks occurring in an interim period, interbank lending frictions, and bank moral
hazard. The model provides a framework within which to study the e¤ectiveness ofthe Federal Reserve tools in supporting interest rates and delivers insight into theeconomic mechanisms that determine equilibrium rates and quantities.
2

Source: http://www.federalreserve.gov/monetarypolicy/…les/fomcminutes20110622.pdf
Source: http://www.federalreserve.gov/newsevents/press/monetary/20130821a.htm
4
IOER di¤ers from interest on reserves (IOR) in that IOER is paid to reserve holdings in excessof the reserve requirement.
5
http://www.newyorkfed.org/markets/expandedcounterparties.html provides a list of the counterparties.
3

1

The framework allows for addressing several questions about the optimal provision of public money, such as reserves, and private money, such as bank deposits.
In particular, how do reserves di¤er from government bonds as a source of publicmoney and other types of public assets created by the Federal Reserve? How does
public money di¤er from private money provided by bank deposits? What is thewelfare bene…ts and costs of public and private money-like assets? What is the optimal quantity of public versus private money? Speci…cally, what is the optimal levelof central bank reserves held by banks? Finally, what is the optimal compositionof di¤erent types of public versus private money, and liquid money versus illiquid
assets?
Banks in our model face two main frictions. First, they have the ability to riskshift returns on assets. So that banks do not risk-shift, regulators must impose aleverage ratio on banks, so that they hold an appropriate amount of equity on theirbalance sheet. Equity can be costly for many reasons, in particular if investors havea natural preference for liquidity. The cost of having to hold additional equity is
captured in a lower deposit rate. Second, banks face interbank lending frictionsin the form of monitoring costs when they borrow in the interbank market. Theinterbank lending frictions lead to higher rates on larger interbank loans.
We assume that there are two banks.6 At date 1, one of the banks may face a“liquidity shock”, which we model as a withdrawal by depositors to purchase government bonds. This shock reduces the amount of deposits at the bank experiencing
the shock and increases deposits at the other bank. The bank receiving additional
deposits faces an unplanned balance sheet expansion and requires more capital, due
to the risk-shifting incentive. This will lead the bank to lower its deposit rate. In
equilibrium, the liquidity shock leads to downward pressure on both deposit rates
and government bond yields.
Liquidity shocks a¤ect banks’asset returns as well. When a bank faces stochastic
withdrawals by its depositors, liquid reserves serve as a bu¤er, allowing the bankto fund these withdrawals with accumulated reserves. If the bank does not haveenough reserves, it can borrow in the interbank market, which is costly because of
the interbank lending friction. Hence, the liquidity shock gives banks an incentive
to hold their assets as liquid reserves, rather than tie them up in illiquid assets suchas loans to …rms. For illiquid assets to be held in positive amounts, they must earna premium over reserve holdings in equilibrium.
The central bank has the ability to issue public money in the form of reserves
or RRPs. If banks hold su¢ ciently many reserves, they do not need to borrow in
6

For simplicity, we will refer to DIs as “banks” in our framework.

2the interbank market in case of a liquidity shock, reducing the interbank market
friction. However, reserves can only be held by banks, and a large supply of reserves
can lead to increased bank balance sheet size. Balance sheet expansion is costly
when incentives to risk-shift are increasing in balance sheet size, as in our model. In
contrast, RRPs can be held by non-banks as well as banks. As more non-banks hold
RRPs, the balance sheet of the banking sector decreases, reducing the balance sheetcost. Because overnight interest rates re‡ect daily liquidity shocks, the …xed-rate,
full-allotment ON RRP is the most e¤ective facility for setting a …xed reservation
rate for those intermediaries; term or …xed-quantity RRPs cannot achieve the samelevel and stability of interest rates.
In comparison to the RRP, the TDF absorbs liquid reserves without reducing thesize of bank liabilities and increases bank asset returns more directly. If RRPs or the
TDF are used in su¢ ciently large size, they can increase interbank market activity
by reducing the size of liquid reserves used to ward o¤ liquidity shocks and can raise
equilibrium bank asset returns. We …nd that utilizing both the TDF and the RRP
together may support rates most e¤ectively if both bank risk-shifting incentives andinterbank lending frictions are large enough and quickly increasing.
We also …nd that limited competition may be an additional reason for spreads
between IOER and money market rates. In particular, limited competition gives
banks more oppertunities to extract surplus from households. Monopolistic banks
will o¤er a contract to households that makes them completely indi¤erent between
storing wealth and holding deposits. Depending on model parameters, this can o¤erhouseholds expected returns that are signi…cantly below the competitive contractand further below IOER. When banks are monopolistic within their sector but
engage in Bertrand competition across sectors, many contracts are possible withthe potential of some of them having equilibrium money market returns also below
IOER and the competitive outcome. In either case, the tools can be used to raise
rates and limit the amount of surplus extracted from the household.
Our paper …ts broadly into the existing literature on the public supply of liquidity and money, monetary policy implementation, IOER, and reserves. Holmstrom
and Tirole (1998) highlight the failure of private economies to e¤ectively supply
necessary quantities or quality of private liquidity to investors in the case of aggregate liquidity shocks. Our paper shows that public money supply is critical even
without aggregate shocks. Poole (1970) shows that the e¤ectiveness of policy basedon targeting interest rates versus money stocks is not well determined and depends
on parameter values, but policy using both is always weakly superior to either of
the two used alone. Ennis and Keister (2008) provide a general framework for un3

derstanding monetary policy implementation with IOER. They show that IOER
can help implement a ‡oor on market rates and allows the Federal Reserve to manage interest rates. Bech and Klee (2011) analyze the federal funds market in thepresence excess reserves. They argue that since government-sponsored enterprises(GSEs) do not have access to IOER, they have lower bargaining power and trade at
rates lower than IOER, thus resulting in the observed IOER-federal funds e¤ective
rate spread.
Kashyap and Stein (2012) show that, with both IOER and reserve quantity
control, the central bank can simultaneously maintain price stability and address
externalities resulting from excessive bank short-term debt issuance. MMS focuseson the e¤ects of excess reserves on in‡ation, interest rates, and bank credit. They
…nd that these parameters are largely independent of bank reserve holdings unlessexternal frictions are present. In particular, very large reserves can be contractionary
in bank lending and be de‡ationary at the zero lower bound of interest rates. The
current paper is the …rst to analyze the additional Federal Reserve tools and theire¤ectiveness in controlling short-term money market rates and managing Federal
Reserve liabilities. We provide positive-result predictions on the e¤ects of the tools
on a dispersion of interest rates and normative results that indicate the optimalquantity and composition of the provision of public and private money.
The paper is organized as follows: Section 2 explains institutional details on theFederal Reserve’s monetary policy before, during, and after the 2007-09 …nancial
crisis and provides descriptions of IOER, RRPs and the TDF. Section 3 presents
and solves the benchmark model. Section 4 incorporates the RRP and the TDF
into the benchmark model and analyzes their equilibrium results and e¤ectiveness.
Section 5 o¤ers some brief extensions to monopolistic and oligopolistic banks. Section
6 concludes. Proofs of most propositions and …gures are in the Appendix.

2

Institutional Background

Prior to the …nancial crisis of 2007-2009, the Federal Reserve closely controlled
the supply of reserves in the banking system through its open market operations(OMOs). In an OMO, the Federal Reserve buys or sells assets, either on a temporary basis (using repurchase agreements) or on a permanent basis (using outrighttransactions), to alter the amount of reserves held in the banking system.7 For
7Assets eligible for OMOs are Treasuries, agency debt, and agency mortgage-backed securities(MBS).

4example, purchasing Treasuries will increase the amount of reserves in the system.
By adjusting the supply of reserves in the system, the open market trading desk(the Desk) at the Federal Reserve Bank of New York (NY Fed) could in‡uence thelevel of the federal funds rate, the rate at which DIs lend reserves to each other. DIs
in the US are required to maintain a certain level of reserves, proportional to speci…eddeposit holdings, which, in addition to precautionary demand for reserve balances,
creates a demand curve for reserves. The interest rate at which this demand andthe supply curves intersect increases when the Desk reduces the supply of reserves,for example.8 Through arbitrage, the level of the federal funds rate in‡uences othershort-term money markets rates.
In response to the 2007-09 …nancial crisis and subsequent economic downturn,monetary policy measures included large-scale lending to provide liquidity to …nancial institutions, and large-scale asset purchases through the large-scale asset purchase program (LSAP) to stimulate the economy by lowering longer term interest
rates.9 This facilitated a very large increase in the supply of reserves.10 Moreover,in December 2008, the FOMC lowered the target federal funds rate to a range of 0
to 25 basis points, its e¤ective zero bound, to help stimulate the economy.11
The e¤ective federal funds rate, a weighted average of federal funds trades
arranged by brokers, remained below 25 basis points, as shown in …gure 1.12 Figure 1
highlights that the federal funds rate ‡uctuated closely with other short-term money
market rates, including the overnight Eurodollar rate and the overnight Treasury
repo rate. These rates are seen to be typically decreasing in the level of reserves.
See …gure 1
Following the LSAPs and the large expansion of the balance sheet, the Federal
Reserve has been preparing a variety of tools to ensure that short-term rates canbe lifted when needed. IOER has been used as one of these tools since October
2008; however two of these tools, RRPs with an extended range of counterparties,
and the TDF, have not been implemented in large-value facilities as of yet. In
a 2009 speech, NY Fed President William Dudley, referred to the RRP and the
TDF as the “suspenders” that will support IOER, i.e. the “belt,” in allowing the8
See Ennis and Keister (2008) and Keister, and Martin, and McAndrews (2008) for a moredetailed introduction to traditional Federal Reserve monetary policy and OMOs
9
The LSAPs are sometimes referred to as "quantitative easing" (QE)10
See Gagon, Raskin, Remanche, and Sack (2010) for more information on the LSAPs.
11
Source: http://www.federalreserve.gov/monetarypolicy/…les/fomcminutes20081216.pdf
12
One common explanation for this is the currently large presence of Government SponsoredEnterprises (GSE) lending in the fed funds market. GSEs are not eligible for IOER and therefore
tend to lend at rates below 25 basis points (see Beck and Klee (2011)).

5

Federal Reserve “retain control of monetary policy.”13 In August 2013, the FOMC
announced potential use of an additional tool, the ON …xed-rate, full-allotment
RRP.

2.1

Interest on Excess Reserves

To manage short-term rates in the face of large excess reserves, the Federal Reservebegan to pay DIs IOER in October 2008. IOER di¤ers from interest on reserves(IOR) in that IOER is paid to reserve holdings in excess of the reserve requirement.
The Financial Services Regulatory Relief Act of 2006 originally granted the Federal
Reserve the ability o¤er IOER. However, the original authorization was only applicable to balances held by DIs starting October 2011. The Emergency Economic
Stabilization Act of 2008 accelerated the start date to October 2008.14
The interest owed to a balance holder is computed over a maintenance period,
typically lasting one to two weeks depending on the size of the DI. Interest payments are typically credited to the holder’s account about 15 days after the closeof a maintenance period.15 IOER was …rst o¤ered in October of 2008 at 75 basis
points, but is currently at 25 basis points where it has been since December 2008.
Institutions that are not DIs are not eligible to earn IOER.16

2.2

Reverse Repurchase Agreements

An RRP is economically equivalent to a collateralized loan made to the Federal Reserve by a …nancial institution. RRPs have historically been used, though somewhat
infrequently, by the Federal Reserve in the conduct of monetary policy, arranged
with a set of counterparties called “primary dealers.”17
In October 2009, the Federal Reserve announced that it was considering o¤ering
RRPs on a larger scale to an expanded set of counterparties.18 The expanded set ofcounterparties include DIs as well as non-DIs, such as MMFs, GSEs, and dealers,
increasing both the number and the type of Federal Reserve counterparties.19 In
13

Source:http://www.newyorkfed.org/newsevents/speeches/2009/dud090729.html
Source:http://www.federalreserve.gov/newsevents/press/monetary/20081006a.htm
15
Source:http://www.federalreserve.gov/newsevents/press/monetary/monetary20081006a2.pdf
16
Source:http://www.federalreserve.gov/newsevents/press/monetary/20081006a.htm
17
See http://www.newyorkfed.org/markets/primarydealers.html.
18
Source:http://www.newyorkfed.org/newsevents/news/markets/2009/an091019.html
19
A
full
list
of
current
eligiblecounterparties
isavailableat
http://www.newyorkfed.org/markets/expanded_counterparties.html
146

addition, in August 2013 the Federal Reserve announced it would further study thepotential for adopting a …xed-rate, full-allotment ON RRP facility.20 In his September 2013 speech, President Dudly discussed this new facility as a way to support
money market rates by allowing counterparties a ‡exible amount of investment at
a …xed rate when needed.21
RRPs do not change the size of the Federal Reserve’s balance sheet, but modify
the composition of its liabilities. Indeed, each dollar of RRPs held by counterparties
reduces one-for-one reserves held by DIs.22
The Federal Reserve Bank of New York has held numerous small-scale temporary operational exercises of RRPs for eligible counterparties starting in the fall of2009.23 Small-scale operational exercises held in April, June and August of 2013
were limited in terms of their overall size (less than $5 billion) and were focusedon ensuring operational readiness on the part of the Federal Reserve, the tri-party
clearing banks, and the counterparties.24 While the Desk has the authority to conduct RRPs at maturities ranging from 1 business day (overnight) to 65 businessdays, the operational exercises thus far have typically ranged from overnight to 5
business days, with several of the August 2013 operational exercises consisting ofovernight RRPs. Overnight RRPs were originally settled the day after auction;
however overnight RRPs with same day settlement were o¤ered starting in August
2013.
At the September FOMC meeting, the committee authorized the Desk to implement …xed-rate RRP exercises with per-counterparty bid caps to limit the aggregate
size of the facility. In comparison to previous exercises, these exercises should better
simulate the …xed-rate, full-allotment facility, which was discussed in the July 2013
FOMC minutes.25
20

Source:http://www.federalreserve.gov/monetarypolicy/…les/fomcminutes20130731.pdf
Source:http://www.newyorkfed.org/newsevents/speeches/2013/dud130923.html
22
See
the
New
York
Fed
page
on
RRPs
for
more
information:
http://data.newyorkfed.org/aboutthefed/fedpoint/fed04.html
23
See the New York Fed page on temporary operations for a listing of recent RRP excercises:
http://www.newyorkfed.org/markets/omo/dmm/temp.cfm
24
These
excercises
wereapproved
by
the
FOMC
in
November2009.
See:http://www.federalreserve.gov/monetarypolicy/…les/fomcminutes20091216.pdf
25
Source:http://www.newyorkfed.org/markets/rrp_faq.html
21

7

2.3

Term Deposit Facility

The TDF is another policy tool that can reduce reserves but it is available only to
DIs.26 The TDF was approved April 2010, following the approval of amendments
to Regulation D (Reserve Requirements of Depository Institutions), allowing Federal Reserve Banks to o¤er term deposits to institutions eligible to earn intereston reserves.27 Small value temporary operational exercises of term deposits haveoccurred since June 2010, and recent small value operational exercises have been
held in March, May, and July, and September of 2013.28
As was the case for RRPs, the TDF does not change the size of the Federal
Reserve’s balance sheet, but alters the composition of its liabilities. Reserves used
to …nance purchases of term deposits are unavailable to DIs until the term deposit
matures. The TDF therefore directly absorbs reserves when banks substitute reserve
holdings for TDF holdings.

3

Benchmark Model

3.1

Agents

The economy lasts three periods t = 0; 1; 2 and consists of two sectors, i = 1; 2; which
are partially segmented. Each sector contains three agents: a bank, a …rm, and ahousehold. In addition, a …nancial intermediary that we associate with an MMF
operates across both sectors. The banks, the …rms, and the MMF act competitively
and are risk-neutral. There is also a central bank and the government that bothissue liabilities across sectors but do not behave strategically. We consider an ex-ante
symmetric case where the sectors, agents, and initial asset holdings are identical.
By symmetry, returns for assets issued at date t = 0 will be equal across sectors inequilibrium.
At date 0, households in each sector receive an endowment (W) that can be heldin the form of deposits, D0 , or equity, E 0 , in the bank of their sector or in MMF
shares (F 0 ). No other agent has an endowment.
The supply of reserves and government bonds are set exogenously and denoted by
M and B, respectively. The interest paid on excess reserves, or IOER, is set exogenously and denoted by RM paid each period, while the interest paid on government
26

Source:http://www.federalreserve.gov/newsevents/press/monetary/20100430a.htm
Source:http://www.federalreserve.gov/newsevents/press/monetary/20100430a.htm
28
Source:http://www.frbservices.org/centralbank/term_deposit_facility_archive.html
27

8

bonds is determined in equilibrium and denoted RB .
Banks take deposits from households and can invest them in loans to the …rmfrom the same sector (L) or hold them as reserves at the central bank (M 0 ). Notethat only banks can hold reserves. Reserves are injected into the economy by thecentral bank purchasing bonds, so the quantity of bonds held by the central bank,
B CB , is equal to the supply of reserves M . Firms borrow from banks and …nanceprojects with a concave and strictly increasing production function, with marginal
real return given by r(L). The …rms’output is sold as consumption goods to households at date t = 2.29 The MMF can sell shares to households and invests ingovernment bonds. We denote the MMF’s bond holdings as B H .
Banks, …rms, and the MMF are pro…t maximizers, while households seek to maximize consumption. There are centralized markets for goods, bonds, and reserves,which imply they have common prices and returns across sectors. In contrast, deposits and bank equity have separate markets in each sector and their returns canvary across sectors. We abstract from credit risk for simplicity, as the focus of thepaper is the use of monetary policy tools in a stable, non-crisis environment.

3.2

Timeline

At t = 0; the household of sector i deposits D0 and holds E 0 of equity in thelocal bank and invests F 0 in MMF shares. Banks accept deposits, issue equity, holdreserves, and lend to …rms at a rate RL . The MMF sells shares and purchases bonds.
At t = 1; a liquidity shock hits one of the two sectors. The probability that sector
i is hit is 21 for i = 1; 2: The nature of the liquidity shock is that the household in theshocked sector demands an additional quantity of MMF shares equal to a fractionof its bank assets (D0 +E 0 ). The household receives an interest rate RW on deposits
withdrawn. 30 The household in the non-shocked sector can redeem MMF shares at
an equilibrium price of P B : The quantity redeemed is denoted by B 1 . The revenue
from MMF redemptions can be deposited in the bank of the same sector or invested
in additional bank equity. New deposits are denoted D1 and new equity investment
is denoted E 1 : For simplicity, we assume that households cannot deposit or holdequity in a bank if they have just withdrawn from the same bank in t = 1.
Banks can use reserves to meet withdrawals. Reserves receive a return of RM ,corresponding to the IOER, at each date t = 1; 2 per unit held in the previous
29

Note that uppercase variables denote nominal values while lowercase variables denote realvalues. Also subscripts always represent the sector.
30
We assume that equity cannot be withdrawn or sold but thatis small enough so that
(Di0 + Ei0 ) < Di0 :9period: If a bank does not have enough reserves, it can borrow I from the otherbank in the interbank market at an interest rate of RI . A key friction in the modelis interbank lending costs in the form of a strictly increasing and convex real cost,f(I), for the lending bank, which represents interbank monitoring costs.
At t = 2, returns on remaining assets are paid, …rms sell their output to households at a price of P per unit, and households consume the goods they purchase.
Deposits made at t = 0 and not withdrawn at t = 1 yield a return of RD0 at date
2, while deposits that are made at t = 1 yield a return of RD1 at date 2. Similarly,
equity issued at date t yields a return of REt at date 2, t = 1; 2. MMF sharespurchased at date t o¤er a competitive return of RF t at date 2, t = 0; 1. Thus,
we consider shares of the MMF o¤ered in di¤erent periods as investments in di¤erent funds. Finally, households pay a lump-sum tax ( ) such that the government
maintains a net balanced budget.
Households obtain a nonpecuniary real liquidity bene…t > 0 for holding a liquid
asset. In our model, all assets held by households are liquid except equity. Thus,
the total nominal utility bene…t to holding a liquid asset with pecuniary nominal
return R is R+ , where = P : This implies that equity is socially costly, a second
key friction in the model.
The …nal key friction is bank moral hazard in the form of ine¢ cient risk-shifting.
Banks have the ability to shift risk at the end of dates t = 0 and t = 1: If a bankrisk-shifts at date t = 0; it obtains an additional (A0 )A0 in pro…ts at the end oft = 2 with probability 21 ; where A0 L + M 0 is de…ned as the banks total assets,
and where (:) 0 is a weakly increasing and weakly convex function that allows
the bank to risk-shift their assets in an amount at the margin that increases with thebank’s balance sheet size. Since bank assets equal bank liabilities, L+M 0 = D0 +E 0 :Alternatively, with probability 12 ; the bank loses (A0 )A0 in pro…ts, where (:) is aweakly increasing and weakly convex function. Similarly, the bank can risk-shift at
the end of t = 1 on new assets acquired at t = 1 to obtain an additional (A0 +A1 )A1or lose (A0 + A1 )A1 ; each with probability 12 ; where A1 are new bank assets (hence
equal to new bank liabilities). Note that only the bank in the non-shocked sector
can risk-shift at t = 1; since it is the only bank with new assets at that date.
Risk-shifting that results in gains is observable but not veri…able nor contractible.
The loss function (:) is large enough such that, for any asset size, all bank pro…ts
and equity returns are zero, and with limited bank liability, the bank incurs a partial
default on depositors. If the bank chooses large enough equity, it will not risk-shift,
as the bank operates to ensure suitable returns for the equity shareholders. We
assume that the government cannot commit not to bailout depositors when there10

is a default. Risk-shifting would impose the cost of bailouts on the government.
Assuming (:) su¢ ciently larger than (:); bank risk-shifting is socially ine¢ cient.
A bank regulator would require an external leverage requirement in place such thatbanks hold enough equity that risk-shifting is not incentive compatible for banks.
This leverage ratio acts as a “balance-sheet cost”, since it is costly for banks toincrease the size of their balance sheet.31 Indeed, the leverage ratio is, in itself,a source of economic ine¢ ciency since households derive a non-pecuniary bene…tfrom holding debt rather than equity. Thus, regulators will require that banks hold
only the minimal amount of equity so that risk-shifting will be suboptimal in both
periods for both banks.
Denote B as the bank’s pro…ts in the absence of risk shifting and B;RS
as thet
bank’s pro…ts under risk shifting at time t (given risk shifting is sucessful): We will
have
B;RS
0
B;RS
1

B

+ (A0 )A0E 0 RE0

B+ (A0 + A1 )A1(1)

E 1 RE1

and where the additional equity terms, E 0 RE0 and E 1 RE1 ; are paid by thebank to equityholders, in the outcome that the a(:) is received, so that equity holders
are fairly compensated under bank risk-shifting and are indi¤erent.
The constraint for banks to not risk-shift in t = 0 and t = 1 is
B;RS
t

B

for t = 0; 1

Note that we could add a third condition stating that banks should not want torisk-shift in both periods togeather, but that would be redundant as ful…llment of
the two conditions in 1 guarantees ful…llment of the third. Two su¢ cient conditions
for the constraint to hold is that
E0 =
E1 =

(A0 )A0RE0(A0 + A1 )A1;
RE1While these need not be necessary conditions, as will be shown later, these are
the equity levels eliminate risk-shifting with minimal welfare costs. Thus, we assume
the regulator imposes these leverage requirements on banks. The regulator can alter
31

Bank balance sheet costs were introduced in MMS, in which each bank bears an exogenous
cost that is increasing in the size of their balance sheet.

11requirements in the face of changes of equilibrium parameters (e.g. RE0 and RE1 ).
However, banks take this requirement function as exogeneous and unchanging.
Frictions related to bank balance sheet size are motivated in part by the analysis
of market observers. For example, interbank broker Wrightson ICAP (2008) voiced
concerns that large reserves could “clog up bank balance sheets.”In July 2013, theFederal Reserve and the FDIC proposed a new rule to strengthen leverage ratios
for the largest, most systemically important banks. Under the proposed rule, bankholding companies with more than $700 billion in consolidated total assets would
be required to maintain a tier 1 capital leverage of 5 percent, 2 percent above theminimum supplementary leverage ratio of 3 percent. Such proposals suggest thatexogenous regulatory-based balance sheet costs may be relevant in the near futurein addition to the market-discipline based endogenous balance sheet costs that wederive.32 An additional explanation for balance sheets costs in addition to capitalrequirements and leverage ratios as in the model is the FDIC deposit insurance
assessment that is applied to all non-equity liabilities.
Furthermore, as MMS explains, banks tended to reduce the size of their balance
sheets during the recent crisis, in line with the presence of balance sheet costs.
Evidence for this cost is also suggested by …gure 1. We observe that the quantity ofreserves is clearly negatively correlated with all of the deposit and related short-termmoney market rates plotted. The table below lists these correlation coe¢ cients.33
Rate

Correlation

Federal Funds E¤ective -.59
O/N Eurodollar-.57

4 Week T-Bill-.53

In MMS, this negative correlation is explained by balance sheet frictions bearing
exogenous costs on banks, which in equilibrium are pushed onto depositors. Thus,
when reserves, and consequently bank balance sheets, are large, the resulting frictions are imposed on depositors through a lower deposit rate. 34
32

Source:http://www.federalreserve.gov/newsevents/press/bcreg/20130709a.htm
Source: http://www.federalreserve.gov/releases/h15/update/ provides the Federal Reserve
Board H.15 report and http://www.federalreserve.gov/Releases/h41/ provides the Federal Reserve
Board H.4.1 report.
34
Federal Reserve Bank of New York President Dudley states that “to the extent thatthe banks worry about their overall leverage ratios, it is possible that a large increasein excess reserves could conceivably diminish the willingness of banks to lend.” Source:
http://www.newyorkfed.org/newsevents/speeches/2009/dud090729.html
33

12

3.3

Optimizations

In this section we describe each agent’s optimization. A bank’s optimization is given
by:max

L;M 0 D0 ;D1 ;E 0 ;E 1 ;I

B

=

1
2
f2RL L 2RE0 E 0 + RM M 0 RD0 D0
2+RM (D1 + E 1 ) RD1 D1 RE1 E 1 + (RI RM )I
Z I^ I^ RI maxf0; (D0 + E 0 )RW RM M 0 g
f (I)dP
0

M+R maxf0; RM M 0RD0 [(1s:t:)(D0 + E 0 )(D0 + E 0 )RW g
E 0 ]g

L + M 0 = D0 + E 0
RM M 0 + D 1 I 0(A0 )A0
E0 =; E1 =RE0(A0 + A1 )A1
RE1The bank receives a certain return of RL L on its loans and pays RE0 E 0 in
2
all states. If the bank is not in the shocked sector they obtain RM (M 0 L)
on their t = 0 reserve holdings and pay out RD0 D0 on all t = 0 deposits. It
also receives new deposits and equity holdings which are invested in reserves at
t = 1 and paid back at t = 2, which is captured by the quantity RM (D1 + E 1 )RD1 D1 RE1 E 1 : The bank has the opportunity to make interbank loans which
RI^ I:^ If the bank is in the shocked sector,
have a return of (RI RM )I P 0 f (I)dit will pay RD0 [RW (1)(D0 + E 0 ) E 0 ] for non-withdrawn deposits, and either
earn RM maxf0; RM M 0 RW (D0 + E 0 )g or pay RI maxf0; (D0 + E 0 ) RM M 0 g;
depending on whether M 0 is large enough to cover withdrawals. The …rst two
constraints are simple budget balance constraints for t = 1 and t = 2: The last
constraints are the no risk-shifting constraints.
Firms seek to maximize pro…ts obtained from sales of real goods in t = 2: The…rm in sector i solves:
Z L
F
^ L
^ RL L
max
=P
r(L)d
L

0

The MMF maximizes pro…ts and solves:
max RB B H

B H ;F 0

s:t: B H = F 0
13

RF 0 F 0

The MMF simply arbitrages, in the bond market, the funds obtained from selling
their shares to households. They maximize the spread between the total bond returnand the claims that they pay out to shares in t = 2:
The household of sector i solves:
max
0
0
1

D0 ;E ;F ;B ;D1 ;E 1

1f2(RF 0 + )(F 0 ) + (RD0 + )D0 + 2RE0 E 0 + RE1 E 1
2P+(RD1 + )(B 1 P B E 1 ) + (RD0 + )[(D0(D0 + E 0 )](D0 + E 0 )RW+(RF 0 + )(RF 0 + )B 1 2
PB+ B + Fg
D0 + E 0 + F 0 < W

s:t:

B1 < F 0
Households value real consumption, thus nominal returns are divided by theprice level. MMF shares issue a riskless return of RF 0 + ; which captures the …rst
term in the objective function. With 21 probability, the household is not shocked
and earns (RD0 + )D0 + 2RE0 E 0 on t = 0 assets. They also can sell B 1 of their
MMF shares and invest them in equity or deposits to obtain (RD1 + )(B 1 P B
E 1 ) + RE1 E 1 (RF 0 + )B 1 : With 21 probability, the household is hit with the0
shock and hearns (RD0 +(D0 + E 0 ] on non-withdrawn deposits. They earn
i )[D
0
0
W
)R
(RF 0 + ) (D +E
on deposits withdrawn to purchase MMF shares. Finally,
PB

they pay in all cases and receive residual claims on the banks and …rms, B + F .
The …rst constraint assures household budget balance at t = 0; while the second
mandates that MMF share sales at t = 1 do not exceed the household’s holding of
MMF shares.

3.4

Equilibrium Analysis

We use general equilibrium as our solution concept. In particular, an equilibriumin this economy is a set returns, RD0 ; RD1 ; RE0 ; RE1 ; RB ; RF 0 ; P B ; RL ; and RI ; anda t = 2 price level P; such that all markets clear at the agents’optimizing levels ofinvestment and consumption.
We assume standard regularity conditions:r(L) > 0; r0 (L) < 0; r(0) = 1; r(1) = 1

f (I) > 0; f 0 (I) > 0; f (0) = 0; f (1) = 1
(D)

0;

0

(D)

0;

14

(0) = 0;(1) = 1:

Since asset holdings are ex-ante identical, M 0 = M2 and F 0 = B 2M : This implies
that
A0 = W
L = W

B

M
2

; A1 = A0 RW

B
:
2

(2)
(3)

We can now turn to the determination of equilibrium rates and quantities. We…rst discuss equity. By (2), A0 and A1 must both be positive in equilibrium. Thus,
we must have both E 0 and E 1 positive in equilibrium to achieve no risk-shifting for0(:) > 0. Because deposits have a liquidity premium for households over equity, we
have that RE0 = RD0 + and RE1 = RD1 + : Since deposits are a cheaper form ofcapital, it is clear the the risk-shi…ting constraint binds and no excess equity will beheld, i.e.:
E0 =
E1 =

(A0 )A0RE0(A0 + A1 )A1
RE1Before we state the main proposition of this section, our model necessitates
several regularity conditions. First, we make the following assumption:
2

RM > RM + max( (2
2

);

0

(W )W + (W ))(4)2This assumes that RM is su¢ ciently larger than RM and will be necessary toallow equilibria without unnecessary withdrawals at t = 0: Second, we will assume
that RM > RW ; so that increases in reserves weakly decrease interbank lending.
Third, we will assume that

RM
RM

> 0

( 0 (A0 + A1 )A1 + (A0 + A1 )) > 0

This will ensure that equilibrium solutions for RD1 exist and remain positive.
Third, we de…ne15

2(x; y; z)

[2R

M2

+ (R

1
(RM
2
W M

M

W

M

R )f (y

R z)

2r(E
0
(x + y)y

(2

RM z)
1

) + 12 f(RM + )2 + 4 [RM

R R

2RM
B
) RM f (y2( 0 (x + y)y + (x + y))]g 2 +

) ]

2

f2R

'(x; y; z)

M2

+ (R

1
(RM
2
W M

R R

M

W

R )f (y

M

R z)

2r(E
0
(x + y)y

2RM
B) RM f (y
2

RM z)
1

) + 12 f(RM + )2 + 4 [RM( 0 (x + y)y + (x + y))]g 2 +2[ 0 (x)x + (x)]gWWand assume that both (W; RW W; RRMW ); '(W; RW W; RRMW ) > 0: This will
ensure that equilibrium solutions for RD0 exist and remain positive.
The following proposition establishes the remainder of the equilibrium.
Proposition 1 A unique competitive equilibrium is given by:
2

1. RL = RM + P2 RM f (I)p 0 1 02
(A0 ;A1 ;M 0 )+(A ;A ;M ) 4(2D0
2. R =
2(2 )3. RD1 = 21 (RM)'(A0 ;A1 ;M 0 )+ [(RM + )2 + 4 [RM

1( 0 (A0 + A1 )A1 + (A0 + A1 ))] 2 )B

4. P B = RRD1++
5. RF 0 = RB = RD0 (1
6. RW

2

) + 2 [RW RD1

(RW

1) ]

RD0 +
RD1 +

7. I = maxf0; A0 RW

RM M 0 g

8. RI = RM + P f (I)M2

9. P = 2r(L)2RRM f (I) :2

Item 1 states that when I > 0; loan rates are at a spread above RM increasing
in I: Items 2 and 3 describe the equilibrium deposit rates. Item 4 shows that theequilibrium spot price of t = 0 MMF shares in t = 1 is given by the ratio of
the total nominal utility value of bonds to t = 1 deposits. Item 5 describes the16equilibrium bond rate, while item 6 places an upper bound on RW : Item 7 showsthat interbank loans are zero when they are not needed to fund shocks and areequal to the de…ciency otherwise. The interbank loan rate, given in item 8, is above
RM at a spread also increasing in the amount of interbank loans, re‡ecting the realinterbank monitoring cost f (I).
Note at a bank’s balance sheet size will increase with increases in reserves. We
de…ne bank balance sheet costs as
C0 (M 0 )

RM

0

M

C1 (M )

R

2

RD0

(5)

D1

(6)

R

:

The following corollary to the proposition states that balance sheet costs increase,re‡ected in deposit rates RD0 and RD1 decreasing, with increases in reserves.
@C0@C1
Corollary 2 @M
0 < 0; @M 0 < 0Equity requirements act as a form of balance sheet costs for banks. Taking
on more liabilities forces banks to hold more equity which is an expensive form of…nance. Thus, there is friction in balance sheet expansion. Figure 2 provides agraphical illustration of the t = 0 bond, bank deposit, and bank loan markets.
See …gure 2
A useful case is when RW A0 < RM M 0 ; so that withdrawals can be funded
without interbank trading. The level of reserves required for this is given by:
M
When M

3.5

2 RW W B
RM 2 RW(7)2

M ; we have I = 0 and RL = RM :Economic Welfare and Optimal Policy

Our model highlights two drivers of welfare: interbank monitoring costs and balance
sheet costs. There is potentially a trade-o¤ between these two costs. Increasing the
supply of reserves reduces the need for interbank transactions and, thus, the costassociated with interbank monitoring. If reserves are large enough, however, theincrease in the size of a bank’s balance sheet raises equity requirements, which iscostly.
In this section we consider a social planner, who chooses the optimal level ofreserves, M ; to maximize social welfare. Welfare is de…ned as the real utility of17households, which is equivalent to the total sum of the real consumption and liquidity
bene…t enjoyed by households, minus interbank lending costs. More formally, M
can be written as the solution to:

max
M

Z L

r(l)dl + (B

0

0

1

(2E + E )M ) + [2 + (RW
Z I

f (i)di1)] (W +

M
2

B
)
2

(8)
(9)

0

The …rst term represents total consumption in the economy determined by real
production. This term is independent of the level of reserves as in MMS. The second
term (B M ) is simply the liquidity bene…t households enjoy resulting from holding
MMF shares that were invested in government bonds. The term (W + M2 B2 ) (2 +(RW 1)) captures the liquidity bene…t derived from households investing in bankassets (deposits and equity): Households would gain 2 (W + M2 B2 ) from investingin bank assets at t = 0 if no one withdrew at t = 1: Because of withdrawals andB
consequent new deposits at t = 1; a net bene…t of (RW 1)(W + M2
) is added2W
to the total bank bene…t. If R > 1; this net bene…t is positive and withdrawals
multiply the bank asset bene…t. If RW < 1, the withdrawals decrease the totalbene…t from holding bank assets. The …nal term is the interbank lending cost.
The term (2E 0 + E 1 ) represents the fact that a portion of bank assets mustbe held in equity, which does not yield a liquidity bene…t and therefore must be
subtracted o¤ the prior term. Notice that equity issued at time zero is sociallymore costly than equity issued at time one. This is because both banks must issue
equity at t = 0 and only the non-shocked bank issues equity at t = 1: Thus, equity
costs are minimized when banks issue only the exact amount of equity needed toeliminate risk-shifting at t = 0: In other words, it is socially ine¢ cient for banks
to issue extra equity issued at t = 0 and reduce the amount of equity they must
hold at t = 1. With this result, the leverage requirement we imposed is the socially
optimal mechanism for eliminating risk-shifting.
It should …rst be noted that if the social planner has the ability to alter RM
as well as M a solution to his problem does not exist. To see this, note that for
RI
any chosen level of M; If we increase RM we will weakly decrease 0 f (i)di while
simultaneously strictly decrease 2E 0 + E 1 : Thus, for any selection of M; RM we can…nd a Pareto superior policy simply by increasing RM : However, increases in RM will
be extremely in‡ationary. It is widely accepted that high in‡ation yields economicine¢ ciencies, these however are not captured in our model. For this reason, we
18

do not consider RM as a choice variable for the planner and instead take this as
exogenous to his problem.
The …rst order condition for the planner’s optimization can be given as:2(RW1) + (RM

2)f (I) = (2@E 0 @E 1
+
)
@M
@M

(10)

If the balance sheet cost is positive for M = M ; then M < M ; since the monitoring
cost goes to zero when loans go to zero. If the balance sheet cost is zero at M = M ,then M > M . In particular, for RW
1; it is clear that M < M for 0 (0) > 0;10
+ @E
> 0: Thus, some interbank lending is desirable. It is also clear that,since 2 @E
@M
@M
as long as equity requirements are not too fast increasing when M = M ; M > M ;and some reserves are also desirable to mitigate monitoring costs. For RW > 1 and0
M depending on how high RW is. But given item 6 in proposition(0) > 0; M
1, we shouldn’t expect RW to be excessively high.
This result suggests that public money in the form of reserves is desirable for
economic welfare. Reserves are liquid and reliable claims to future wealth which
can be used to …nance liquidity shocks, and thus are valuable to banks as well ashouseholds. In an optimal solution, the central bank may decide to provide banks
public money in the form of reserves, instead of allocating it all to households inthe form of government bonds. At the same time, outstanding reserves increasebanks’balance sheet size, which exacerbates banks’incentives to risk-shift. Thus,
quantities of reserves need to be moderated. At the end of the next section, we
discuss how a social planner can mitigate the latter cost of a large size of the central
bank’s balance sheet when additional central bank public money tools are available
to provide for di¤erent compositions of the central bank’s liabilities.

4

Central Bank Tools

We now use this framework to analyze the RRP and TDF. We assume that the MMF
can invest in RRPs in addition to government bonds.35 In contrast, only banks caninvest in the TDF. The TDF serves as a substitute to reserves for banks, but does
not bear the liquidity bene…t to banks that reserves do. Both RRPs and the TDF
substitute for reserves on the balance sheet of the central bank one for one. Thus,
these new central bank liability facilities essentially “absorb” traditional central
bank reserve liabilities. In this section we assume that we are in the no interbank
35

Banks would not invest in the RRP at equilibrium rates below IOER, as we consider.

19

lending case, i.e. M M , so we need not consider whether or not reserves are large
enough to support necessary interbank lending.

4.1

Overnight RRPs: Fixed-Quantity vs. Fixed-Rate

The purpose of an overnight RRP is to o¤er a short-term investment that is available
whenever needed. The ON RRP is o¤ered at t = 1 and allows the MMF to purchaseadditional assets. This, in turn, allows households selling bonds to purchase MMF
shares as an alternative to redepositing in their bank. We do not consider ON
RRPs at date 0 to focus instead on the role of ON RRP in mitigating the liquidity
shock. In this section, we also assume that we are in the no interbank lending case,M , so we need not consider whether or not reserves are large enough toi.e. M
support necessary interbank lending.
Proposition 4 considers the case of a …xed-quantity operation (ON FQ RRP).
The central bank o¤ers a perfectly inelastic supply of RRPs (RP F Q ) at a market
equilibrium competitive rate (RF Q ).
Proposition 3 For RP F Q(D0 + E 0 )RW ; we have in equilibrium that RF Q =RD1 and D1 = (D0 + E 0 )RW RP F Q . Furthermore, in this equilibrium RD0 , RD1 ,and RB are all higher than the corresponding rates in proposition 1.
Proposition 5 shows that a …xed-rate, full-allotment ON RRP (ON FRFA RRP)
can achieve the same allocation as an ON FQ RRP. ON FRFA RRP o¤ers an interest
rate RF R at t = 1 for any quantity demanded. The rate RF R is set exogenously by
the central bank.
Proposition 4 If the central banks sets RF R = RF Q from proposition 4, we havethat RP F R = RP F Q , RF R = RF Q = RD1 , and D1 = (D0 + E 0 )RW RP F Q =(D0 + E 0 )RW RP F R :Figure 5 illustrates the relationship between the two overnight RRP policies.
See …gure 3
We can see from the previous two propositions that the RRP creates a transferfrom the non-shocked bank to the shocked household by absorbing some of theliquidity shock. As a result, the equilibrium t = 1 deposit rate is increased, up to
the facility rate, and the quantity decreased, by the size of the facility. The overall
result of this is a decrease in bank pro…ts (because a lower amount of t = 1 deposits
20

are issued at a higher rate) and an increase in direct returns to households throughhigher returns on MMF shares.
Note that the presence of the ON RRP indirectly exerts upward pressure on
bond rates in t = 0. The increase in the t = 1 deposit rate for a shocked householdincreases the overall expected return of investing in deposits. Arbitrage then require
the bond and t = 0 MMF return to increase as well.
4.1.1

Uncertainty in Shock Size

While proposition 5 shows that a ON FQ RRP can implement the same allocation
as an ON FRFA RRP, we consider how the two tools di¤er in a richer setting.
In particular, the two facilities would have di¤erent implications if the fraction ofhousehold that are relocated, , is uncertain. In such a case, an ON FQ RRP would
result in ‡uctuations in the RRP rate, while an ON FRFA RRP would result in‡uctuations in the quantity of RRPs. Hence, a policymaker who dislikes ‡uctuationsin the interest rate more than ‡uctuations in the quantity of reserves would prefer
the ON FRFA RRP.
To formalize this, we assume in this section that is random and can take two
realizations, L and H ; with H > L . We assume that the central bank knows
the two possible values of but does not know which one will occur when they
implement their ON RRP policy.36 We also assume that the central bank would
like to target a speci…c t = 1 investment rate of RD1 and can choose either an ON
FRFA RRP or an ON FQ RRP to do so. We will show that, in general, the ON
FRFA RRP can implement a t = 1 investment rate equal to RD1 with less interest
rate volatility than the ON FQ RRP.
To make the problem interesting, we analyze the case where:
1 M
(R
2
> RD1
1 M
>
(R
2

1
) + f(RM + )2
2

4[ 0 (A0 )A1 + (A0 )]g 2

1) + f(RM + )2
2

4[ 0 (A0 +1L

A0 RW )A1 + (A0 +

(11)
L

1

A0 RW )]g 2

so that the target t = 1 investment rate is higher than the outcome that would
occur in either state without central bank intervention, but not so high that t = 1
deposit markets become completely inactive in all situations. We …rst show that an
ON FRFA RRP policy can implement RD1 in either state.
36

For the analysis we conduct here, it is actually not necessary for us to de…ne probabilities of
the two states occuring.

21

Proposition 5 If the central bank sets RF R = RD1 ; and (11) holds, then RD1 =RF R = RD1 when either L or H occurs:
The proposition establishes that an ON FRFA RRP can impose a target RD1
for any state under (11) and thus completely eliminate t = 1 interest rate volatility.
It is rather clear that this cannot be achieved with the ON FQ RRP. This is because
when L is realized, a smaller quantity facility will be needed to impose RD1 thanwhen H occurs. A central bank cannot achieve RD1 for all realizations of if itcannot condition its policy on the state of the world. We do however show thatthe central bank can implement a ‡oor on rates at RD1 : First, we de…ne RP F Q to
be the quantity of the facility that is needed to impose RD1 when H occurs; i.e.
RP F Q will solve:
1RD1 = (RM
2
1
M
+ f(R + )2
2)
4[ 0 (A0 +H

A0 R W

RP F Q )A1 + (A0 +H

A0 R W

Such an RP F Q will exist by continuity as well as (11).
Proposition 6 Under (11), if the central banks sets RP F Q = RP F Q ; then RD1 =RD1 when H occurs, and RD1 > RD1 when L occurs.
This proposition shows that a ON FQ RRP can e¤ectively provide a ‡oor on
rates at the target rate RD1 : However, t = 1 investment rates may be very volatile,especially if the di¤erence between the two shock sizes are large and both occur withhigh probabilities. A central bank with the intention of implementing a target rate
while minimizing interest rate volatility may thus prefer a ON FRFA RRP over an
ON FQ RRP.
4.1.2

Discussion

Other advantages of a FRFA RRP may also exist that are outside the scope of thismodel. For example, …xed-rate RRPs provide MMFs with certainty regarding …xedrates, and certainty regarding (unlimited) quantities, both of which would haveadditional bene…ts to MMFs in the face of uncertainty on demands and supplies
in short-term money markets. In practice, MMFs have e¤ective risk aversion, inpart caused by the requirement for stable NAVs. The certainty provided by ON
FRFA RRPs creates a bene…t for a more stable transmission of monetary policy.
FQ RRPs, in contrast, will not eliminate the uncertainty regarding equilibrium rates
and quantities that MMFs can receive.
22

1

RP F Q )]g 2

Furthermore, …xed-rate RRPs tend to better facilitate an overnight RRP facility.
As shown above, such daily availability provides MMFs with greater certainty tosupport their elastic demand, at or above the RRP rate, for other assets. Fixedrate RRPs also allow for one-day maturity RRPs, which can be rolled over. A…xed-quantity, operation-based RRP is not as amenable to daily operations. FQ
RRPs would therefore tend to require longer term RRPs for operational cost reasons. Shorter-term RRPs provide MMFs the ability to substitute with shorter-term
alternative assets. Such RRPs will support money market and bank deposit ratesof all maturities, even as short as overnight rates. Typically, money market rates
increase with the tenor of the instrument. While ON FRFA RRPs can be rolledover, and therefore provide a better ‡oor to overnight rates as well as to longer-term
rates, a longer-tenor RRP does not provide such support to rates of shorter tenors.
As a result of this, longer-term RRPs may possibly have little e¤ect on shorterterm rates. For example, a one-month RRP may increase one-month money market
rates, but this may not be well transmitted down to provide support for overnightrates, which could be best supported with ON RRPs. One-month RRPs may rather
simply increase the steepness of the one-month yield curve. In the case of ON FRFARRPs, MMFs may even take up quantities of RRPs at quite low rates.

4.2

Term vs. Overnight RRPs

We model long-term RRPs as an alternative source of investment for MMFs in
t = 0. Term RRPs (denoted RP T M ) are supplied inelastically by the central bankat a …xed-rate RT M ; paid o¤ in t = 2. Term RRPs are available only to the MMF
and reduce the quantity of reserves. We restrict our analysis to the case where
the interbank market is inactive at date 1, as in proposition 1. If RP T M RRPs
are issued in t = 0, the total supply of reserves decreases from M to M 0 whereM 0 = M RP T M ; so we consider M 0 M :Item 5 of proposition 1 must hold for bonds to be held in equilibrium. We de…nethe equilibrium bond rate in the benchmark case of proposition 1 to be RB . If
term RRPs are introduced at rate below RB , then no RRPs are held by the MMF.
Therefore we have the following lemma:
Lemma 7 If RT M is set less than RB , we will have RT M < RF 0 = RB = RB =RB = (1 2 )RD0 + 2 (RW RD1(RW 1)) and RP T M = 0; M = M 0 :If RT M is set above RB , then there will be a positive demand for term RRPs.
The following proposition shows that the central bank has the freedom to set the
RRP rate within an appropriate range without inducing interbank lending.
23

T M such that for RB < RT M < R
T M , in equilib[[
Proposition 8 There exists a R(RW 1)),
rium we must have RB < RB = RT M = RB = (1 2 )RD0 + 2 (RW RD1
RP T M > 0; M 0 < M , and M 0 M . Furthermore, both RD0 and RD1 will increase,
while their respective equilibrium quantities decrease.

RRP holdings increase when the term RRP rate is set at a higher level. This leads
to higher MMF investment and lower bank liabilities, reducing equity requirements.
Hence, both RD1 and RD0 increase until RT M = (1 2 )RD0 + 2 [RW RD1 (RW 1)].
The term RRP rate, when it is set su¢ ciently high, creates a ‡oor for the bond anddeposit rate. Figure 6 graphically illustrates the e¤ect of the term RRP policy on
RD0 , and the a¤ect on RD1 is similar.
See …gure 4
We have modeled term RRPs as having a …xed-price and full-allotment. As
in the previous section, one can also consider an operation for a …xed quantity,
RP T M , of the RRP asset, where the rate is market determined. The equilibriumin the …xed-price, full-allotment case where RB = RB = RF 0 = RT M is identicalto any equilibrium in which a quantity RP T M of RRPs are set that yield RT M
in equilibrium. The main di¤erence is that in the …xed-rate setting we could haveequilibria where RT M < RB . In this situation, MMFs only hold bonds and no
RRPs are held. Such an equilibrium (where the RRP rate is strictly below theother rates) is impossible in the operation style setting of the …xed-quantity RRP
market.37 Arbitrage forces the RRP rate to be equal to the bond rate, since it isalso a safe asset. For similar reasons, the bank deposit rates increase with the RRP
quantity supplied. The mechanism is that an increase in the quantity supplied of
RRPs decreases the quantity of deposits which reduces balance sheet costs.
Both the term and the ON RRP increase t = 1 and t = 2 deposit rates. However,they do so through di¤erent mechanisms. The ON RRP (that is o¤ered in t = 1
only) sets a reservation t = 1 deposit rate, which directly raises RD1 up to that level.
RD0 increases because the ON RRP absorbs shocked withdrawals and lowers t = 1
expected balance sheet costs for the bank. The term RRP (o¤ered only in t = 0)
directly lowers balance sheet costs in t = 0 and partially raises t = 0 deposit rates.
The decrease in deposits at t = 0 reduces the size of the shock in t = 1; which thenindirectly increases t = 1 deposit rates by reducing the balance sheet cost burden.
This, then, feeds back into t = 0 deposit rates, as banks now expect lower t = 1
deposits. Thus, an additional increase in RD0 occurs.
37
One could argue that this case corresponds to a zero quantity auctioned, where the equilibrium
rate i is indeterminate within a range.

24

An ON RRP (of either …xed-rate or …xed-quantity) o¤ered at t = 0 would also
reduce D0 and thus reduce balance sheet costs and raise t = 0 deposit rates. Term
RRPs may be preferable, however, if o¤ering RRPs of open-ended size on a daily
basis is operationally di¢ cult or expensive compared to a term RRP that attracts
large, long-term deposits.

4.3

TDFs vs. RRPs

As shown in the previous sections, RRPs provide a tool for the central bank to bothmanage reserves and provide ‡oors for rates. Another such tool is the Term Deposit
Facility (T ). The TDF is comparable to the term RRP with the exception that they
are only o¤ered to banks. As we will show in this section, the two tools vary in theire¤ect on equilibrium rates depending on various parameters in the model. For thissection, we assume that the central bank chooses a su¢ ciently large quantity of the
TDF or RRP such that reserves are reduced to the point where M < M at the endof t = 0: That is, the central bank is using its policy tools in large enough size tools
so as to rekindle the interbank market and promote interbank lending in t = 1:
We model the TDF as a …xed-quantity operation (T ) o¤ered by the central bankin t = 0 maturing in t = 2 with a competitive equilibrium return RT : Important tonote is that TDF holdings cannot be used to ward o¤ liquidity shocks. In this sense,
they are a perfect substitute for real sector bank lending and in equilibrium wemust have that RT = RL : The key di¤erence between the RRP and the TDF is that
the TDF forces substitution of liquid reserves to illiquid TDF holdings completely
within the banking sector. RRPs, on the other hand, divert reserve holdings to theassets held by the universal MMF which is not prone to shocks. Thus, utilizing
RRPs as opposed to TDFs reduces liquid assets while bearing less of an increase
on interbank borrowing costs and liquidity premia. For simplicity, and for ease ofcomparison, we will assume that the RRP is a term RRP in this case.
To see this, …rst suppose the central bank increases the TDF supply by 2 units.
Each bank would decrease its liquid reserve holdings by 1 unit. In doing so, theylose RM units of reserves that could have been used to ward o¤ a potential liquidity
shock. This means that interbank loans will increase by RM ; which implies that theequilibrium interbank rate will increase by f (I + RM ) f 0 (I): On the other hand,
if the central bank were to supply 2 units more RRPs instead, each bank would
still decrease its liquid reserve holdings by 1 unit (assuming the MMF borrows
equally from both sectors), sacri…cing RM in liquid assets in t = 1. However,each household is also converting one deposit to an MMF share. Thus, interbank
25

loans will only increase by RM< RM and the interbank rate will increase byf (I + RM
RW ) f 0 (I) < f (I + RM ) f 0 (I): This implies that the liquidity
premium will be smaller in this case than the former. This is shown in …gure 7.
See …gure 5This di¤erence also factors into the e¤ect on deposit rates. From part 2 ofproposition 1, we can see that the t = 0 deposit rate is increasing in interbank
lending and decreasing in equilibrium t = 0 and t = 1 liabilities due to equity
requirements. This leads to the following proposition for interbank costs and equity
requirements that are largely convex.
Proposition 9 When marginal equity requirements are large relative to marginal
D0
@RD0
> @RP
interbank lending costs and both are convex, we will have @R
T M : This@T
occurs when liabilities and liquid reserves are relatively large and interbank lending
is relatively small: Reversing these relationships will yield the opposite inequality.
This result is immidiate due to the convexity of the two costs. Nevertheless,there are relatively interesting implications of the above proposition. The sizesof f (:) and c(:) will increase when interbank loans and deposits, respectively, arelarger. Therefore, when both of the cost functions are very convex, we suspect that apolicymaker who has a primary goal of increasing deposit rates may want to mediate
between usage of both tools. Speci…cally, he may want to …rst use term RRPs toreduce deposit size, then divert to implementation of the TDF after decreases inmarginal balance sheet costs diminish. On the other hand, a policymaker who seeks
to absorb reserves with a smaller e¤ect on increasing deposit rates may want tofocus on one facility, namely, the TDF if marginal balance sheet costs are very fast
increasing, and the RRP if interbank lending frictions are instead more prominent.

4.4

Optimal Monetary Policy with An Extended Set of Tools

In this section, we extend the analysis performed at the end of section 3, but allow
the planner access to RRPs and the TDF in addition to reserves. The same tradeo¤ remains at the heart of the planner’s problem. Liquidity helps banks reduce
monitoring costs but comes at a cost of larger equity requirements.
The …rst thing that is immediately clear is that the TDF provides no welfare
bene…ts. The TDF increases interbank lending without reducing equity requirements, as it keeps bank’s asset size constant. In any economic welfare maximizing
monetary policy, the size of the TDF should be zero.
26

The ON RRP, however, should be used to its maximum potential. Reserves
at t = 0 are necessary bu¤ers against liquidity shocks. However, adding reservesto the banking system beyond that point can be wasteful in terms of extra equity
requirements. Investment is better made into the ON RRP. Thus the ON RRP
should clearly be set to completely eliminate the shock. In particular, for whateverbank balance sheet size, A0 ; the planner chooses as optimal, the size of the ON RRP
should be set to completely cover the size A0 RW : If is uncertain, the size of the
ON RRP should be set to eliminate the shock in the worst state (highest ).
The planner’s problem then becomes choosing the M that maximizes:

max
M

Z L
0

M
r(l)dl+ (B M )+(W +2B) [2+ (RW 1)]
2

0

(2E )

Z I

f (i)di (12)0Note that (12) is identical to (8) except that now unnecessary t = 1 equity
requirements are eliminated as withdrawn funds are entirely invested in overnight
RRPs instead of bank assets. Thus, we will have that the M that maximizes (12)will be weakly greater than the M that maximizes (8). When equity requirements
diminish, public money becomes more valuable to the economy. In particular, M <M when RW is su¢ ciently small and (:) > 0, but still greater than zero as long as
t = 0 equity requirements are not too fast increasing when M = 0:
Obtaining the M that maximizes (12) can either be done through standard
bond sales or by utilizing term RRPs at t = 0: Our model does not distinguishbetween the two. However, if the central bank prefers to keep longer term bonds ontheir balance sheet, term RRPs may be the more desirable approach.

5

Extensions

In this section we consider two extensions introducing limited banking competitionin the deposit market to the model. In the …rst extension, we assume banks aremonopolistic suppliers of deposits within their sectors. In the second, the two banks
across sectors engage in Bertrand competition in their deposit contracts.
The purpose of this section is twofold. First, it serves as an opportunity for us todiscuss our tools in a slightly more elaborate framework. Second, and perhaps more
importantly, it provides additional reasons as to why we observe money market rates
signi…cantly below IOER. In the competitive case, the spread was entirely due topecuniary costs of balance sheet expansion that the bank had to be compensatedfor. When there is limited competition, banks gain strong opportunities to extract
27

surplus from households by o¤ering lower rates. Indeed, in the following two sections,banks may o¤er deposit contracts with even lower returns then in the competitivecase.
In the monopolistic case, we …nd that banks naturally o¤er the lowest rate at
which households would be willing to hold deposits. This indi¤erence rate can beraised by usage of the tools. In the Bertrand case, many contracts are possible
in equilibrium. Some of these contracts can be well below the competitive level
depending on parameters. The tools can also be used in this situation to eliminate
those potential contracts in equilibrium.
A key assumption of these two sections that was not used in previous sections
is the ability for households to store wealth. This assumption is not necessary forthese sections either, but are more realistic and make the problem signi…cantly moreinteresting by introducing a strong individual rationality constraint for households
to be willing to hold …nancial assets. In particular, in the following two sections,
we assume households can store their endowment at t = 0 until t = 2 for a real
return of 1 + (nominal return of P + ). At t = 1; we assume that households
can store any cash from …nancial investments at t = 0 for a nominal return of 1 + :Should investment opportunities prove less pro…table than storage, households willstore their wealth instead of investing it.

5.1

Monopolistic banks

In this section we consider the bank of each sector being the monopolistic supplierof deposits to households. A key insight is that all quantities are pinned down inequilibrium by the economic constraints. Banks will therefore take these constraints
as given and choose their deposit rates (RD0 and RD1 ) and withdrawal rates (RW )optimally. We also assume that banks have full knowledge of how other equilibrium
returns and prices will evolve as a result of their actions.
First, if households have the ability to store cash, then it is clear that the optimal
D1
R will be 1: Banks will give households exactly their outside option on deposit
holdings at t = 1: Note that households of the non-shocked sector will still bewilling to sell their bonds to the shocked sector as the price of bonds will rise inB
equilibrium: P B = R1++ : This occurs because shocked depositors have a perfectly
inelastic demand for bonds at t = 1:
Moving backwards, banks will choose their optimal RD0 and RW : Their optimization is essentially the same as in the benchmark model, however given thatbank asset and liability quantities are pinned down in equilibrium, we can simplify
28the banks optimization to:max

RD0 ;RW

1
f RD0 D0 (RM f ( (D0 + E 0 )RW RM M 0 )) maxf0; (D0 + E 0 )RW
2+ RM maxf0; RM M 0
A0 RW g RD0 [(1)A0 E 0 ]P+
[RW (1
)+ ]
D0
2
s:t:
R =
1 2

The constraint is the household’s individual rationality constraint that mandates
that they are indi¤erent between storing their endowment or depositing in the bank.
First consider that the bank sets RW low enough that interbank lending is notnecessary in the event of the shock. Substituting the constraint in for RW in theobjective function we have that the derivative of the objective function with respect
to RW is given by:
(1

)(D0
2

E 0)

(1

+

)(1
2

)A0

(1

A0 ( R M

)(1
2

)

)

(14)

With some algebra we can show that this derivative will be less than zero when:0

A >Note that A0 > (1(2
)RM (1)(1
followning proposition

(2

(1
)(D0
)RM (1

E 0)
)(1

)

(15)

)(D0 E 0 ) always and since (1) < 1 and RM > 1;
) > 1: Thus, (15) will always hold. This leads to the

Proposition 10 A monopolist will set RW = 0; no interbank lending will occur,
P + (1)and RD0 = 1 2
2

The monopolist faces a trade o¤ in that o¤ering a lower deposit rate meansthat he must o¤er a higher withdrawal rate so that the expected return on a bankdeposit is high enough to satisfy the households individual rationality constraint.
The proposition states that the monopolistic bank would rather o¤er a higher t = 0
deposit rate than face the potential of any withdrawals at t = 1: Note that RD0 =P + (1 2 )is strictly greater than P + which is the households nominal return on
1
2

a unit of storage. Also note that for smaller ; RD0 will be smaller and closer
to the households indi¤erence rate P + : If P + is signi…cantly lower than thecompetitive deposit rate given in Proposition 1, then the monopolistic equilibrium
can have money market returns well below IOER.
29

0
RM M
(13)
g

Tools can have interesting e¤ects in this environment. First consider the ON
RRP. The ON RRP will raise RD1 above 1 to the ON RRP rate. Thus, condition(15) will become
A0 >

(2

(RD1
)RM

)(D0 E 0 )
(1
)(RD1

)

(16)

For an ON RRP rate high enough, RD1 can be raised so high that condition (16)will no longer hold and the derivative of the bank’s objective function with respect
to the withdrawal rate will become positive. In such a situation, we will have thatthe bank may optimally choose a withdrawal rate high enough so that interbank
lending will occur. Therefore, in a monopolistic setting, an aggressive ON RRP
policy may in fact incentivize interbank lending. The intuition behind this is that
increases in the ON RRP rate exogeneously increases the households return on adeposit. Thus the bank can o¤er much lower t = 0 deposit rates with small increases
in the withdrawal rate. Similarly, a term RRP o¤ered at t = 0 will reduce A0 andalso incentivize interbank lending at large enough sizes. Fixed rate RRP policies
can clearly dramatically raise rates by increasing the pro…tability of the household’s
outside options. In fact, o¤ering these facilities at a high enough rate can inducein the monopolist to again o¤er competitive rates, and no take up is necessary to
induce these rate raises. If we are in a situation where (16) holds, then the TDF
will have no e¤ect. However, if the RRP has been used so that (16) is violated, theTDF can make interbank lending more costly and thus have a countervailing e¤ect
by decreasing RW :5.2

Bertrand competition

We now consider the two banks of the two sectors engaging in Bertrand competition,that is, households have the ability to choose between holding their deposits in eitherof the two sectors at t = 0.38 One may think that this induces more competition
than the competitive case. However, it is important to realize that while households
are competitive across sectors, they hold full market power with their sector. In thissituation, as we show below, there will be many equilibrium outcomes. As before
we will have that RD1 = 1; as the non-shocked bank is the monopolistic supplier oft = 1 deposits. We aim to classify potential equilibrium RD0 and RW :Denote B (RD0 ; RW ; A0 ) as the bank’s expected pro…t from o¤ering households
RD0 and RW on a deposit contract and holding A0 in assets at t = 0. We focus on
38

For simplicity we maintain the assumption that only the non-shocked bank o¤ers t = 1 deposits.

30

pure symmetric strategies and assume that the market is split evenly across sectors
when households are indi¤erent between the contracts o¤ered in the two sectors.
That is, banks face the choice of either o¤ering an equally attractive contract astheir competitor and holding half the total equilibrium deposit amount across sectors(E B2 + M2 ) or o¤ering a more attractive contract and attracting all deposits fromboth sectors (E B + M ). The latter option may be attractive because of increased
revenue from investment, however banks also face increasing leverage requirements
for maintaining larger balance sheets, so there is a trade o¤ and balance sheet
expansion may therefore not be optimal.
Let us de…ne the set C as follows:
C = f(RD0 ; RW ) :B(RD0 ; RW ; E

B M
+ )
2
2

B

(RD0 ; RW ; E

B M0+ ); 8(RD0 ; RW 0 )
2
2

0

0

0

0

0

) + [RW (RW 1) ] = RD0 (1) + [RW
(RW
1) ]g
2
2
2
2
Elements of C are optimal in the sense that they o¤er the the bank the highest
expected pro…t out of all contracts that give households the same expected payo¤
when the deposit quantity is equal to half the market (E B2 + M2 ). If banks arecomitted to o¤ering households a certain expected payo¤ for E B2 + M2 deposits,
the optimal contract will necessarily be an element of C:
Similarly de…ne the set C as:s.t. RD0 (1C = f(RD0 ; RW ) :s.t. RD0 (1B(RD0 ; RW ; E) + [RW
22

B + M)

(RW

B

0

0

(RD0 ; RW ; E
01) ] = RD0 (10B + M ); 8(RD0 ; RW 0 )) + [RW
2
2

0

(RW

0

1) ]g

C is the equivalent of C with the deposit quantity being the whole market (EB + M ) as opposed to only half the market in C: If a bank wants to service thewhole market while maintaining a certain level of household expected payo¤, theoptimal contract will be a member of C:
Finally de…ne C as:
C= f(RD0 ; RW ) : (RD0 ; RW ) 2 C and
B M
0
0
B
B
(RD0 ; RW ; E+ )(RD0 ; RW ; E
220

B + M ); 8(RD0 ; RW 0 ) 2 C
0s.t. RD0 (1

0

0

(RW
1) ]g
) + [RW (RW 1) ] = RD0 (1) + [RW2
2
2
2
The below proposition characterizes the space of potential equilibrium deposit
contracts.
31

Proposition 11 A necessary and su¢ cient condition for a pair (RD0 ; RW ) to bean equilibrium contract is that (RD0 ; RW ) 2 C and RD0
D0

W

P+

2

[RW (11)+ ]

2(R ; R ) 2 C ensures that neither bank has the incentive to deviate to acontract that is either more pro…table but equally preferred by the household, or toa contract that is slightly more preferred by the household and will yield the entire
market. The second condition ensures households want to make positive deposits.
Once again, general equilibrium and even market split necessitates that each bankwill hold exactly E B2 + M2 in assets at t = 0 in any equilibrium:
It is clear that many contracts can satisfy the above proposition and it is di¢ cult to characterize the space of admissable equlibrium contracts further than this.
However, some additional intuition is perhaps necessary. Consider a potential equilibrium contract (RD0 ; RW ): Since (RD0 ; RW ) 2 C it is clear that no bank would
want to deviate to a contract that yields only half the market. Consider a bank de0
ciding whether to deviate to a contract (RD0 ; RW 0 ) that will yield them the entire
market. In making this decision the bank faces a tradeo¤. On the one hand, more
assets allows the bank more investments and more raw revenue. However, a larger
balance-sheet size also yeilds higher equity requirements. A contract (RD0 ; RW ) canonly be an equilibrium if the costs from balance-sheet expansion exeed the bene…ts.
When RD0 is too low, the equity requirement costs will be small and the added
revenue bene…t of balance-sheet expansion will be large in comparison. However,a higher RW is more costly for a large balance-sheet than a small one, so a low
deposit rate can be accomodated with a higher withdrawal rate. So increasing thewithdrawal rate in equilibrium relaxes both of the Bertrand oligopoly constraints.
However, one thing that is clear in the Bertrand case is that the monopolistic equilibrium may not be feasible. The monopolist only needed to accomodate the second
constraint of the above proposition, but a Bertrand equilibrium is also subject tothe bank’s incentive compatibility constraint that they should not want to capture
the whole market. When the incentive compatibility constraint binds more strongly
over the individual rationality constraint depends on parameters. In particular,when balance sheet costs are low or slowly increasing, there is more incentive toexpand the balance sheet at low rates. Thus, if banks are subject to less pro…tablerisk-shifting abilities, the incentive compatibility constraint may be tighter than theindividual rationality constraint and rule out the monopolistic equilibrium in the
Bertrand case.
While many equibria are possible, many equilibria can exist where large spreads
exist between IOER and money market rates. We can also consider our tools in thissituation. The ON RRP will work only through the households individual rationality
32

constraint and will thus have the same e¤ect of mandating higher withdrawal rates
for a given deposit rate in equilibrium. The e¤ect of the term RRP is somewhat
ambiguous. It can expand the set of possible equilibria as it decreases the absolute
size of the banking sector and thus the extra return to deviating and doubling asset
holdings relaxing the incentive compatibility constraint (as well as the individual
rationality constraint. However, it will also reduce the costs of doubling balance
sheet size, thus increasing the incentives to deviate on any contract. In eithercase however, a …xed rate RRP will tighten the household’s individual rationality
constraint and raise the household’s returns on permissible contracts. In the nointerbank lending case, the TDF will have no e¤ects on the contract space. However,in the positive interbank lending case, a large TDF increases the the costs interbank
lending for bigger balance sheet sizes and can make deviation less attractive.

6

Conclusion

In response to the 2007-09 …nancial crisis and subsequent economic conditions, theFederal Reserve engaged in large-scale lending to …nancial institutions to provide
liquidity and large-scale asset purchases to stimulate the economy by lowering interest rates. As a result of these policies, the amount of reserves in the banking systemincreased dramatically to over $2 trillion during 2013.
We introduce a simple general equilibrium model to analyze the tools available to
the Federal Reserve to manage interest rates and the supply of reserves. Our model
highlights numerous important results regarding IOER, RRPs, and the TDF. As
in MMS, IOER represents the riskless return for holding marginal reserve on abank’s balance sheet and, thus, in‡uences the deposit rate o¤ered by banks. The
IOER also sets a short-term reservation rate in the interbank market below which
DIs will not lend. RRPs help reduce the balance sheet costs faced by banks andreduce the supply of reserves. RRPs also set a reservation interest rate for shortterm deposits that occur as a result of liquidity shocks. While the model showsthat …xed-rate and …xed-quantity ON RRPs would be identical in the absence of
informational frictions, …xed-rate RRPs may be advantageous for a central banker
trying to set a ‡oor on short-term rates without perfect knowledge on the intensityof shocks or exogenous costs agents face. Term RRPs in this model o¤er little
bene…t over ON RRPs if the ON RRP can be o¤ered on a consistent basis at
low cost in implementation. However, they can further reduce balance sheet costs
and raise deposit rates. TDFs, in contrast to RRPs, do not reduce bank equity

33

requirements as they do not reduce bank balance sheet size. Therefore, when thebanking system incurs costly monitoring on interbank markets, RRPs increase thesize of this interbank market by less than the TDF. This motivates usage of the
RRP and the TDF concurrently. Namely when marginal risk-shifting incentives
and interbank lending costs are very convex, policy makers with the intention ofraising deposit rates to high levels may want to use both facilities together.
In addition to our analysis of how interest rates and quantities are determined inequilibrium, we also analyze how the Federal Reserve can provide optimal levels of
public money. In our model, public money refers to special assets that the Federal
Reserve o¤ers that are not available in the private economy alone. These include
all of our tools, as well as reserves. Public money has bene…ts that show positive
amounts of public money should be provided in an optimal equilibrium. Reserves
serve as a liquid asset for banks that can be used to pay o¤ withdrawals from depositors. Substantial reserves can prevent banks from having to excessively incur
monitoring costs in interbank markets. However, too many reserves in the banking system exacerbates the size of banks’ balance sheets with an increased moral
hazard problem, which further increases equity that banks will hold as a commitment device not to risk-shift on assets. These equity requirements are socially costly
when households place extra value on liquidity. Thus, RRPs are of use to absorb
reserves from the banking system in excess of those used for mitigating liquidity
shocks, which reduces the need for equity requirements. The Federal Reserve is able
to provide these public assets as it is a reliable counterparty backed by government
taxes not subject to equity requirements as commercial banks are.
We conclude our analysis with an extension to limited banking competition in
the deposit market. We …nd that this may be an additional reason for spreads
between IOER and money market rates. In this case, banks are able to extract
more surplus from households by exerting market power. Monopolistic banks willo¤er a contract to households that makes them completely indi¤erent between storing wealth and holding deposits, which may yield household expected returns thatare signi…cantly below the competitive contract and IOER. When banks engage in
Bertrand competition across sectors, many contracts are possible, but some of themcan have equilibrium money market returns also below IOER and the competitive
outcome. In both cases, the tools can be used to raise rates by providing households
a stronger alternative option to the deposit market.
Our model provides a broad framework in which many additional questions about
central bank policy for managing large levels of reserves can be analyzed. As in
MMS, we can ask how balance sheet frictions, like more pro…table risk-shifting34

on larger balance sheets, e¤ect in‡ation and consumption. Additionally, we canconsider heterogeneity across and within sectors. Other extensions to the model can
include incorporating additional …nancial intermediaries, such as securities dealersand GSEs. The central bank can manage its liabilities facilities with regard to these…nancial institutions to further absorb reserves, in‡uence the broad range of money
market rates, and impact the level of lending, output, in‡ation, and consumption.
The framework developed is also versatile to analyze a variety of further institutional
details for …nancial intermediaries and money markets.

35

Appendix: Proofs
Proof of Proposition 1. Item 1 comes from the …rst order condition of the banks
optimization with respect to loans. Item 4 comes from the …rst order condition of thebanks optimization with respect to B 1 : Item 5 comes from the …rst order conditionof the household’s optimization with respect to D0 : Item 6 is necessary to prevent
unshocked deposits from being withdrawn and invested in bonds. If this were tooccur no one would want to hold t = 0 deposits at t = 1: Everyone would withdraw
their deposits at t = 1 to invest in bonds (since we do not allow redeposits in thesame bank). However, the total bond supply at t = 1 would not allow this, so wemust have item 6 hold in equilibrium. Item 8 comes from the …rst order conditionof the banks optimization with respect to I: Given item 8, if I > 0 then RI > RM .
Since RM is the bank’s best investment option, there is no reason to demand I unless
it is necessary to fund withdrawals. Thus, I = A0 RW RM M 0 if this is positiveand 0 otherwise. I is feasible as RW A0 is deposited into the lending bank. Item
9 comes from the …rst order condition of the …rm’s optimization with respect to L:
Items 2 and 3 come from the …rst order conditions banks optimization with respect
to D0 and D1 respectively, taking into consideration the equity requirements. For
item 3, we will obtain
1
1[(RM + )2 + 4 [RM ( 0 (A0 + A1 )A1 + (A0 + A1 ))] 2
RD1 = (RM2
The negative root actually corresponds to a negative quantity of required equity,
so we rule this out. Similarly, we will get(A0 ; A1 ; M 0 ) +p(A0 ; A1 ; M 0 )2 4(2)'(A0 ; A1 ; M 0 )2(2)
Once again the negative root can be ruled out as it implies a negative quantityof equity. By our regularity assumptions, RD0 and RD1 will exist and be positive.
It remains to show however that an RW exists that satis…es item 6. Note that when
RW = 0; and A0 = W we will have RD0 > RD1 by (4). Since by items 2 and 3 RD0
and RD1 are both continuous in RW there exists and interval (0; RW ) such that if0
RW 2 (0; RW ) we will still have RD0 > RD1 :Consider a RW from this interval. We
0will have RD0 > RD1 for any possible A0 < W: Then choosing an RW
min(1; RW )will satisfy item 6 for any A0 < W: Since items 1-9 were necessary conditions for an
equilibrium, it is clear that this equilibrium speci…cation is unique.
D1< 0:We have that
Proof of Corollary 2. We will start with showing that @R@M 0
the …rst order condition of the banks optimization with respect to A1 = D0 + E 0 is
given by:
RD0 =

36

2

RD1

(RM

)RD1

RM

( 0 (A0 + A1 )A1 + (A0 + A1 ) = 0(17)

Implicitly di¤erentiating this with respect to A1 gives us
@RD1
=
@A1

[ 00 (A0 + A1 )A1 + 2 0 (A0 + A1 )]2RD1 (RM)

When (:) is convex, the numerator of the right hand side is negative. Also when
R takes the value given in proposition 1, the dominator will be given by
D1[(RM + )2 + 4 [RM

1( 0 (A0 + A1 )A1 + (A0 + A1 ))] 2
D1

which must be greater than zero. Thus @R< 0: Similarly, di¤erentiating (17)@A10with respect to A yields
@RD1
=
@A0

[ 00 (A0 + A1 )A1 +2RD1 (RM0(A0 + A1 )]
)
0

1

@A
@A
Which is negative for the same reason as above. Since @M
0 ; @M 0 > 0 we will have
D0
D1
@R
; @R < 0@M 0 @M 0
The …rst order condition of the bank’s optimization with respect to A0 = D0 +E 0
can be written as:(2

2)RD0 + RD0 + ' = 0

Di¤erentiating this with respect to A0 gives us:
@
@'
D0
( @A
+ @A
@RD0
0R
0)
=
@A0
2(2
)RD0 +
D1

D1

@
@'
D0
Since @R
; @R
< 0 and RD0 > 0, we will have that @A+ @A
0R
0 > 0: Substi@A0
@A1
p
D0
2
tuting in R from proposition 1 makes the denominator equal to
4(2
)'
@RD0
@RD0
which must be positive. So we have that @A0 < 0 which implies that @M 0 < 0:
Since RM is exogenously given, the corollary is then proved.
Proof of Proposition 3. If RF Q > R1 , demand for RRPs exceed supply. If
RF Q < R1 , demand for RRPs is zero and the market does not clear. Neither of theseare possible equilibria, hence RF Q = R1 in any equilibrium. This directly implies
that households are indi¤erent between re-investing in the unshocked bank andinvesting in the MMF at t = 1, and therefore market clearing implies the expression
for A1 . Then, A1 < (D0 + E 0 )RW implies that (D0 + D1 ) < ( (D0 + E 0 )RW ).
Item 2 and 3 in proposition 1 implies that the long- and short-term deposit rates

37both increase relative to the equilibrium without RRPs due to the decrease in A1 ,and Item 5 implies that the bond rate increases accordingly..
Proof of Proposition 4. Equality of the rates is immediate from proposition 4.
Suppose that RP F R < RP F Q , and denote by A1;F R and A1;F Q the corresponding
re-deposit volumes. Date t = 1 market clearing implies that A1;F R > A1;F Q , which
in turn implies R1;F R < R1;F Q and hence R1;F R < RF R = RF Q = R1;F Q , a contradiction. An analogous argument for RP F R > RP F Q implies that RP F R = RP F Q in
any equilibrium. Equality of the t = 1 deposit volumes follows from market clearing.
Proof of Proposition 5. First assume L occurs. Suppose RD1 > RF R : Then
there would be no demand for the ON FRFA RRP, and A1 = L (D0 + E 0 )RW :However, since we have that
RD1
> RF R
1 M
>
(R
2

1
) + f(RM + )2
2

4[ 0 (A0 +L

A0 RW )A1 + (A0 +L

1

A0 RW )]g 2

by assumption, the unshocked bank will not be willing to supply L A0 RW in t = 1
deposits, which is inconsistent with market clearing. Thus, we cannot have RD1> RF R : Now suppose that RD1 < RF R : Then there will be no demand for t = 1
bank liabilities (D1 ; E 1 = 0). However since
1
RF R = (RM2

1) + f(RM + )2
2

1

4[ 0 (A0 )A1 + (A0 )]g 2 ;banks will want to supply positive deposits, which is also inconsistent with market
clearing. Therefore, we cannot have RD1 < RF R . Thus, we have established that
RD1 = RF R = RD1 when L occurs. Now suppose that H occurs and that RD1> RF R : Since H > L ; we will have that
RD1
> RF R
1 M
>
(R
2

1
) + f(RM + )2
2

4[ 0 (A0 +H

A0 RW )A1 + (A0 +H

1

A0 RW )]g 2

by (11). By the same argument as in the previous part, we have that RD1 > RF R isa contradiction. The case where RD1 < RF R is eliminated by an identical argument
as in the previous part. Thus we have that RD1 = RF R = RD1 in either state.
Proof of Proposition 6. The fact that RD1 = RD1 when H occurs is true by thede…nition of RP F Q : When L occurs and t = 1 deposits are demanded in positive38

amounts, we must have
RD1 = RF Q
1 M
=
(R
)+
2
1
f(RM + )2
2
1 M
>
(R
)+
2
1
f(RM + )2
2= RD1 :

4[ 0 (A0 +L

4[ 0 (A0 +H

A0 R W

A0 R W

1

RP F Q )A1 + (A0 +L

A0 R W

RP F Q )]g 2

RP F Q )A1 + (A0 +H

RP F Q )]g 2

A0 R W

1

Thus, we will have that RD1 = RF Q RD1
Proof of Lemma 7. This is immediate as there will be no demand for RRPs
when bonds are the more attractive option for the MMF.
Proof of Proposition 8. We de…ne RD0 (RP T M ) to be the equilibrium RD0 given
in proposition 1 when the central bank o¤ers RP T M term RRPs. Similarly we de…ne
TM = M\
RD1 (RP T M ): We de…ne RP
M : We let
T M = RD0 (RPT M )(1
[
\
R

2

TM)
\
) + [RW RD1 (RP
2

(RW

1) ]

(18)

T M : The[
Consider a the central bank setting RT M such that RB < RT M < R
MMF will demand only RRPs. Since there is excess bond supply the bond rate
must rise up to RB = RT M so that MMFs are willing to hold positive quantities
of RRPs and bonds. We also must have households willing to hold deposits formarket clearing. Thus D0 ; and consequently D1 ; will fall, being replaced with MMF
share holdings, so that (18) is satis…ed by increasing RD0 and RD1 . This can beaccomplished as RD0 (RP T M ) and RD1 (RP T M ) are continuous functions and we canthus apply intermediate value theorem to …nd the appropriate deposit levels. Since
T M = M no interbank lending will occur.
[
M RT M > M R

39

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40

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