Payments, Credit and Asset Prices∗
Monika Piazzesi
Stanford & NBER

Martin Schneider
Stanford & NBER

June 2016

Abstract
This paper studies a monetary economy with two layers of transactions. In endusertransactions, households and institutional investors pay for goods and securities with payment instruments provided by banks. Endusers’ payment instructions generate interbank
transactions that banks handle with reserves or interbank credit. The model links thepayments system and securities markets so that beliefs about asset payoﬀs matter for the
price level, and monetary policy matters for real asset values.

∗

Email addresses: piazzesi@stanford.edu, schneidr@stanford.edu. We thank Fernando Alvarez, Saki Bigio,
Gadi Barlevy, Gila Bronshtein, V.V. Chari, Veronica Guerrieri, Moritz Lenel, Guido Lorenzoni, Robert Lucas,
Vincenzo Quadrini, Nancy Stokey, Harald Uhlig, Alonso Villacorta, Randy Wright and seminar participants
at Chicago, the Chicago Fed, Columbia, ECB, Minnesota, MIT Sloan, NYU, Princeton, UCL, Wisconsin andvarious conferences for helpful comments and suggestions.

11Introduction

This paper studies the joint determination of payments, credit, and asset prices. The starting
point is that, in modern economies, transactions occur in two layers. In the enduser layer, nonbanks — for example, households, firms and institutional investors — trade goods and securities
and pay for them using payment instruments supplied by banks. Payment instruments include
not only short-term demandable assets such as deposits and money-market fund shares, butalso credit lines that can be drawn on demand such as credit cards. Credit lines also pay a keyrole in payment for securities. For example, institutional investors have sweep arrangementswith their custodian banks. Participants in the triparty repo market obtain intraday credit
from clearing banks.
A common denominator of diﬀerent payment instruments is that banks commit to accept
payment instructions from their clients. As a result of those payment instructions, transactionsin the enduser layer generate interbank transactions in the bank layer. Perhaps the most obvious
example are direct payments out of bank deposit accounts by check or wire transfer: payments
between customers of diﬀerent banks generate interbank transfers of funds. In many securities
markets, transactions are cleared by specialized financial market utilities such as clearinghouses
that provide some netting of transactions. Institutional investors then settle netted positions
with those utilities through payment instructions to their banks.
Interbank payments are often made with reserves, but may also be handled through various
forms of short-term credit. For example, in the United States, utilities like NSS and CHIPS
allow for intraday netting of a share of interbank transactions, so only net positions are periodically settled with reserves. The central bank may also provide intraday overdraft credit tobanks. Nevertheless, the bulk of interbank payments goes through gross settlement systems
provided by central banks, such as Fedwire in the United States or Target in the Euro Area.
In the aftermath of recent financial crises, central banks have made unprecedented changes
to the quantity as well as the price of reserves. Several central banks have dramatically increased
the quantity of reserves relative to the value of transactions. These policy shifts have reduced therelative importance of interbank credit. For example, in the United States, the use of intraday
overdraft as well as interbank overnight Fed funds borrowing have essentially disappeared.
Moreover, a number of central banks have begun charging negative nominal interest rates forthe use of reserves.
This paper proposes a stylized model of an economy with two layers of transactions. Endusers are households and institutional investors who must pay for some goods and securities
with payment instruments — deposits or credit lines — supplied by a competitive banking sector.
Banks handle endusers’ payment instructions and must make some interbank payments withreserves supplied by the government. Both banks and the government incur costs of leveragethat decline with the quantity and quality of available collateral, in particular securities andclaims to future taxes.
The model determines asset prices, the nominal price level and agents’ portfolios as a function of government policy and investor beliefs about asset payoﬀs. It also determines the shareof resources used up as costs of leverage. An eﬃcient payment system allocates collateral so
as to minimize that share of resources and hence maximize consumption. Asset prices reflect2

not only cash flow expectations and uncertainty premia, but also the collateral and liquidity
benefits that assets provide to endusers and banks.
We use the model think about links between securities markets and the payment system.
The key properties that generate such links are illustrated by the quantity equation  = ̄( + )
Here the total volume of transactions  includes institutional investors’ securities purchases, notsimply the value of goods traded. Moreover, the only medium of exchange in the enduser layer
is “inside money” — deposits  and credit lines  — provided by banks who rely on securities
as collateral. Outside money — that is, reserves — is only one input to the production of inside
money, albeit a special one since it not only serves as collateral, but also provides liquidity formaking interbank payments.
Consider an increase in uncertainty about asset payoﬀs that lowers asset values. As the valueof collateral that banks can purchase declines, supplying payment instruments becomes morecostly. A decline in inside money  +  then puts downward pressure on the price level. At thesame time, however, an increase in uncertainty also lowers institutional investors’ demand for
payment instruments, which has the opposite eﬀect. The details of financial structure, including
the use of payment instruments by institutional investors and the scope of netting arrangements— are thus important in order to assess the eﬀects of asset market shocks on inflation.
In our model, some assets are priced by exclusively by intermediaries. Segmentation of asset
markets arises endogenously because banks receive collateral benefits from assets but households
do not. Banks invest in assets to back payment instruments and thus bid up the prices of thoseassets. In particular, the real interest rate on short-term credit is so low in equilibrium thathouseholds choose not to lend short term. It therefore does not satisfy a consumption-based
pricing equation. Instead, banks are the only marginal investors, and bank Euler equations saythat interest rates are lower when bank leverage is higher and collateral is scarcer.
We also use the model to think about recent policy shifts, with a focus on two policy tools.
First, the government can trade in securities markets to change the mix of collateral available
to banks. Second, the government controls the real return on reserves. The central banksets the nominal rate on reserves. Moreover, the inflation rate is given by the growth rate of
nominal government liabilities. This result follows from the quantity equation and the fact thatprices are flexible. Importantly, what matters is not the growth rate of reserves, but instead
the growth rate of nominal payment instruments  + , which in turn depends on nominal
collateral available to banks.
The government can select one of two policy regimes. Reserves are scarce if banks do
not always have suﬃcient reserves to handle all interbank payments but instead turn to theshort-term credit market for liquidity. The liquidity benefit of reserves then generates a spread
between the short-term interest rate and the interest rate on reserves. Reserves are scarce if
the real return on reserves is suﬃciently low, that is, the opportunity cost of holding reserves
is high. Banks then choose higher leverage to maintain a high return on equity in spite of ahigher eﬀective tax on reserves.
As long as reserves are scarce, open market purchases of short-term debt for reserves change
the collateral mix towards more liquid bank assets. In our model, open market purchases
3permanently lower the real short-term interest rate. Indeed, when more liquid reserves areavailable, competition drives banks to produce more payment instruments, pushing the pricelevel up. As a result, the real value of nominal collateral falls — banks become more levered andbid up the prices of all collateral including short bonds, a permanent “liquidity eﬀect” on thereal interest rate.
In the second policy regime, reserves are abundant: the quantity of reserves is suﬃciently
large relative to the volume of transactions that overnight borrowing is never needed. Once
reserves lose their liquidity benefit, short-term loans and reserves become perfect substitutes
and earn the same interest rate — the economy enters a “liquidity trap” where conventional
open market policy becomes ineﬀective. If the interest rate on reserves is zero, then reserves
become abundant at the zero lower bound. More generally, however, reserves are abundant
whenever the real return on reserves is suﬃciently high, which can also happen with positiveor negative interest on reserves.
The fact that payments occur in two layers has important implications for what it means to
be in a “liquidity trap”. The textbook view is that equality of interest rates on outside money
and short bonds implies that the medium of exchange and a safe store of value become perfect
substitutes for all agents. In our model, this is true only for banks who are the only investors
in both reserves and short bonds. In contrast, payment instruments for endusers require costlybank leverage and never become abundant. In particular, they retain their liquidity benefit evenwhen reserves are abundant. At the same time, collateral remains scarce in the liquidity trap,
so unconventional policy that exchanges reserves for lower quality collateral can still matter bychanging the collateral mix.
Which regime is better depends on the relative leverage costs of banks versus the government.
If the government can borrow more cheaply than banks, then it makes sense to move to abundantreserves, as several central banks have done recently. An extreme version would be narrowbanking. In contrast, if the government has trouble to credibly commit to a path for nominal
debt, then it is beneficial to have banks rely more on collateral other than government debt orreserves. Since the optimal system depends on the quality of collateral, it may make sense toswitch between regimes over time in response to asset market events.
The availability of two separate policy tools implies that the stance of policy cannot be easily
summarized by a single variable, such as the short-term nominal interest rate. For example,
when reserves are scarce, the government can lower the nominal interest rate either throughopen market purchases or by lowering the real return on reserves. However, the eﬀect on realinterest rates and inflation is generally diﬀerent. The reason is that asset values reflect not only
liquidity benefits — as in many monetary models — but also collateral benefits. Policy aﬀects
interest rates by altering both benefits.
Our model assumes that markets are competitive and all prices are perfectly flexible. Banksand other financial firms maximize shareholder value and operate under constant returns toscale. Moreover, they do not face adjustment costs to equity. The eﬀects we highlight thus
do not follow from a scarcity of bank capital. We think of our model as one of large banks
that provide payment services in a world where credit markets are highly securitized. This
perspective also motivates our leading example for a shock: a change in uncertainty that moves
asset premia.

4Financial frictions are formally introduced as follows. First, nominal payment instruments
and reserves relax liquidity constraints in the enduser and bank layer, respectively. In thissense, those assets are more liquid than other assets. By assuming generalized cash-in-advanceconstraints for households and institutional investors, we abstract from eﬀects of interest rates
on the volume of transactions in units of goods and securities, respectively. While adding such
eﬀects is conceptually straightforward, our goal here is to provide a tractable setup that zeros
in on novel eﬀects for the demand and supply for money.
Second, banks and the government face an upward-sloping marginal cost of making commitments. This cost is smaller the more collateral the institution has available, that is, the larger
and safer is a bank’s asset portfolio or the larger the tax base, respectively. It can have either an
ex post or an ex ante interpretation. For example, if more levered banks and governments are
more likely to renege on certain promises, more labor may be required to ex post renegotiate
those promises so that less labor is available for producing goods. Alternatively, more leveredbanks and governments may have to exert more eﬀort ex ante to produce costly signals of their
credibility.
Our analysis below starts with a baseline model in which the quantities of both real payment
instruments provided by banks and securities on their balance sheets are fixed. In two extensions, we then introduce institutional investors whose demand for loans or payment instruments
responds to changes in interest rates. We first consider carry traders who hold real assets andborrow against those assets using short-term credit supplied by banks. Carry traders haveno demand for payment instruments, but supply collateral to banks in the form of short-termloans. An example are asset-management firms who finance securities holdings with repurchaseagreements.
The new feature in an economy with carry traders is that the price level now depends on
carry traders’ demand for loans. For example, lower uncertainty increases the demand for loans
and hence the quantity of collateral for banks, the supply of payment instruments and the pricelevel. An asset-price boom can thus be accompanied by inflation even if the supply of reserves
as well as the amount of goods transacted remains constant and banks hold no uncertain
securities themselves. Moreover, monetary policy that lowers the real short rate lowers carrytraders’ borrowing costs and boosts the aggregate market by allowing more leverage.
The second extension introduces active traders who hold not only securities but also payment
instruments, since they must occasionally rebalance their portfolio using cash payments. An
example are asset management firms who sometimes want to exploit opportunities quickly
before they can sell their current portfolio. Active traders’ portfolio choice responds to thedeposit interest rate oﬀered by banks and the fee for credit lines they charge: if payment
instruments are cheaper, active traders hold more of them, and the value of their transactions
is higher. The strength of their response depends importantly on how much netting takes place
among active traders though intraday credit systems.
The new feature in an economy with active traders is that the price level now depends on
active traders’ demand for payment instruments. For example, lower uncertainty increases theirdemand for deposits and credit lines. As more of payment instruments provided by banks areused in asset market transactions, fewer instruments are used in goods market transactions andthe price level declines. During an asset price boom, we may thus see low inflation even if the5supply of reserves increases. Moreover, monetary policy that lowers the real short term interest
rate lowers active traders’ trading costs and further boosts the aggregate market.
Our model can be interpreted as describing the subset of worldwide transactions in a currency, rather than the closed economy of a country. The former interpretation is appropriate
for economies like the United States that have banking systems and financial markets tightly
integrated with those of other countries. We thus think of households in our models as agents
who pay for goods out of dollar deposit accounts, while institutional investors may include foreign firms who obtain credit or payments from banks in terms of dollars. With this perspectivein mind, the model can be used to think about how events in worldwide asset markets may
aﬀect nominal prices in the US.
The broad questions we are interested in are the subject of a large literature. The main newfeatures of our model are that () transactions occur in layers, with payment instruments (inside
money) used exclusively in the enduser layer and reserves (outside money) used exclusively
in the bank layer, () endusers include institutional investors, and () both banks and thegovernment face leverage costs. Relative to earlier work, these properties change answers topolicy questions as well as asset pricing results, as explained in more detail in Section 6.
The paper is structured as follows. Section 2 presents a few facts about payments. Section
3 describes the model. Section 4 looks at the baseline model that features only households andbanks. It shows how steady state equilibria can be studied graphically and considers diﬀerent
monetary policy tools. Section 5 introduces uncertainty and studies the link between thepayment system and securities markets. It also extends the model to accommodate institutional
investors as a second group of endusers. Finally, Section 6 discusses the related literature.

2

Facts on payments

This section presents a number of facts that motivate our model. We combine data from the
BIS Payments Statistics, the Payments Risk Committee sponsored by the Federal Reserve Bankof New York, the Federal Reserve Board’s Flow of Funds Accounts and Call Reports, as wellas publications of individual clearinghouse companies.
Enduser transactions
Figure 1 gives an impression of enduser and interbank payments in US dollars. The left
hand panel considers enduser payments, that is, payment instructions to various types of intermediaries. The blue area labeled “nonfinancial” adds up payments by cheque as well as various
electronic means, notably Automated Clearinghouse (ACH) transfers as well as payments bycredit card. While the area appears small in the figure, it does amount to several multiple ofGDP. For example, in 2011, nonfinancial transactions were $71trn whereas GDP was $15trn.
This is what one would expect given that there are multiple stages of production and commerce
before goods reach the consumer. Moreover a share of trade in physical capital including realestate also is contained in this category.
Payment for securities in U.S. markets is organized by specialized financial market utilities
who clear transactions and see them through to final settlement. A major player is the Depository Trust & Clearing Corporation (DTCC). One of its subsidiaries, the National Securities
6Interbank w/ reserves

1800

1800

1600

1600

1400

1400

1200

1200

nonfinancial
NSCC/DTCC
FICC + FedSec

1000
800

$ Trillions$ Trillions

Enduser

1000
800

600

600

400

400

200

200

0

NSS
FedSecFedFunds
other settlement
residual

0
05

07

09

11

05

07

09

11

Figure 1: Selected US dollar transactions, quarterly at annual rates. Left panel : "nonfinancial" = chequeand electronic payments reported by banks, "NSCC/DTCC" = securities transactions cleared by NSCC,"FICC+FedSec" = securities transactions cleared by FICC and Fedwire Securities Service. Right panel : "NSS"= NSS settlement, "FedSec" = Fedwire Securities settlement, "FedFunds" = estimate of payments for Fed
Funds borrowing by US banks, "other settlement" = estimate of payments to financial market utilities.

Clearing Corporation (NSCC) clears transactions on stock exchanges as well as over-the-counter
trades in stocks, mutual fund shares and municipal and corporate bonds. NSCC cleared $221trn
worth of such trades in 2011. In the left hand panel of Figure 1, transactions cleared by NSCC
are shaded in brown.
NSCC has a customer base (“membership”) of large financial institutions, in particular
brokers and dealers. When a buyer and a seller member agree on a trade — either in an exchange
or in an over-the-counter market — the trade is reported to NSCC which then inserts itself asa counterparty to both buyer and seller. In the short run, members thus eﬀectively pay forsecurities with credit from NSCC. To alleviate counterparty risk, members post collateral that
limits their position relative to NSCC. Over time, NSCC nets opposite trades by the same
member. Periodically, members settle net positions via payment instructions to members’ bankwhich then make (receive) interbank payments to (from) DTCC. Netting implies that settlement
payments amount to only a fraction of the dollar value of cleared transactions.
Another DTCC subsidiary, the Fixed Income Clearing Corporation (FICC) oﬀers clearing
7

for Treasury and agency securities. FICC payments are settled on the books of two “clearing
banks”, JP Morgan and Bank of New York Mellon. Interbank trades of Treasury and agency
bonds can alternatively be made via the Fedwire Securities system oﬀered by the Federal
Reserve System to its member banks. The left hand panel of Figure 1 shows the sum of FICC
and Fedwire Securities trades in red. This number is high partially because every repurchaseagreement involves two separate security transactions (that is, the lender wires payment for apurchase to the borrower and the borrower wires payments back to the lender at maturity).
Figure 1 does not provide an exhaustive list of US dollar transactions. First, it leaves out
financial market utilities handling derivatives and foreign exchange transactions. For example,
the Continuous Linked Settlement (CLS) group is a clearinghouse for foreign exchange spot
and swap transactions that handled trades worth $1,440 trn in 2011. Netting in these markets
is very eﬃcient so that CLS payments after netting were only $3Trn. Second, even for goods
and securities covered, Figure 1 omits purchases made against credit from the seller that involves no payment instruction to a third party. This type of transaction includes trade credit
arrangements. In securities markets, a share of bilateral repo trades between broker dealers
and their clients is settled on the books of the broker dealers. Finally, the figure also leaves outtransaction made with currency.
Even given these omissions, the message from the left panel of Figure is clear: transaction
volume is large, and especially so in securities markets. The volume in securities markets also
exhibits pronounced fluctuations in the recent boom-bust episode. We also emphasize that not
all of these payment instructions are directly submitted to traditional banks. Financial market
utilities that provide netting are also important. Moreover, customers of money market mutual
funds may also pay by cheque or arrange ACH transfers. The payment instruction is thenfurther relayed by the money market fund to its custodian bank.
Transactions in the bank layer
The right hand panel of Figure 1 shows transactions over two settlement systems provided
by the Federal Reserve Banks. The blue area represents interbank payments via the National
Settlement Service, which allows for multilateral netting of payments by cheque and ACH. To
a first approximation, one can think of it as the counterpart of the blue area in the left handpanel, that is, non-securities payments after netting. All other areas in the right panel representinterbank payments over Fedwire, the real time gross settlement system of the Federal Reserve.
Fedwire is accessed by participating banks who send reserves to each other.
The coloring of areas is designed to indicate roughly how the interbank payments weregenerated. The red area represents payments for Treasury and agency securities over Fedwire
Securities. Since there is no netting involved, large securities transfers correspond to largetransfers of reserves. For the years after 2008, the brown area is an estimate of payments madeover Fedwire to settle positions with financial market utilities. The estimate includes not only
NSCC and FICC, but also CHIPS, a private large value transfer system used by about 50 largebanks. CHIPS uses a netting algorithm to simplify payments among its member banks; in 2011,it handled $440trn worth of transactions.
The green area in the figure represents payments for interbank credit in the Fed Funds market, also sent over Fedwire. As for repo transactions, a relatively small amount of outstanding
overnight credit can generate a large number for annual Fedwire transfers. The transition from8

a regime of scarce reserves to one with abundant reserves after the financial crisis is apparent
by the drop in Fed Funds transactions. The presence of government sponsored enterprises andFederal Home Loan banks implies that the Fed Funds market has not dried up completely.
The red and brown areas suggest that payment instructions generated by securities trading
are responsible for a large share of interbank payments. This is true even though netting byfinancial market utilities reduced the cleared transactions from the left panel to much smaller
numbers. At the same time, during times of scarce reserves, bank liquidity management via the
Fed Funds market also generates a large chunk of payments. The figure also contains a grayarea which we cannot assign to one of the payment types.
Payments by custodian banks
Figure 2 provides a closer look at the activities of banks who make a lot of payments. We
consider 27 bank holding companies that are "systematically important" and hence report individual data on payments to regulators. Along the horizontal axis, we measure 2014 payments
via large value transfer systems like Fedwire and CHIPS, normalized by bank assets. The 27
banks’ joint payments are large: they account for over 75% of total payments over CHIPS and
Fedwire.
Along the vertical axis, we measure securities held in custody, again normalized by assets.
There is a strong positive relationship: banks who have more assets in custody also tend to makemore payments. The fact that the relationship holds after normalization by assets suggests saysthat is it not merely a scale eﬀect. Moreover, the color of a dot indicates the size of the bankin terms of total assets. It is not obviously related to the two ratios measured along the axes.
In fact, the largest banks — JP Morgan, Citi, Wells Fargo and Bank of America all appear as
bright pink dots — do not have the largest payments relative to assets. Instead, it is somewhat
smaller banks specializing in the custodian business — State Street, Northern Trust and BoNY
Mellon appear in the top right corner of the figure — who have the largest payments/assets
ratios.
There are two plausible reasons why banks who are large custodians might be expected tomake lots of payments. One is already apparent in Figure 1: there is simply a lot of "churn"in securities markets, in part due to frequent short term changes in positions. This possibility
motivates the inclusion of liquidity constraints for securities traders together with a netting
system, in our model below.
A second reason is that custodian banks hold the portfolios of money market funds, who inturn oﬀer payment instruments to endusers. While the front oﬃce of the money market fundreceives payment instruments, those instruction are still executed by banks who access large
value transfer systems, in particular Fedwire. For the purposes of our model, our perspective
on money market funds is thus to consolidate them with the banking system.

3

Model

Time is discrete, there is one good and there are no aggregate shocks. Households consume
an endowment of goods as well as fruit from trees. The total amount of goods available for

9

50
20

1

10
5
0.5

2
1
0.5

Assets, $trn

Securities in Custody / Assets, 2014

2

0.2

0.2
0.1
0.05

0.1

0.02
1

2

5

10

20

50

100 200

500

Payments / Assets, 2014
Figure 2: Payments / Assets vs Securities in Custody / Assets for systemically important bank holding
companies. Flows for the year 2014, stocks for 2014 Q4. Color indicates assets in $trn.

consumption is constant. Households also own competitive financial firms. For now, the only
financial firms are banks who issue payment instruments. Below we introduce diﬀerent types ofasset management companies. All financial firms issue equity and participate in tree and credit
markets along with households.
Layers and frictions
The model describes transactions and asset positions in both the enduser layer and thebank layer. In the enduser layer, households and nonbank financial firms trade goods andassets. In the bank layer, banks trade securities and also borrow from and lend to each other.
A key connection between layers is that enduser transactions generate payment instructions tobanks.
The model incorporates two frictions. First, to capture the costs of supplying payment
instruments and debt as well as the benefits of safe collateral, we assume that there is a costof leverage for banks as well as other financial firms. The government similarly faces a cost ofleverage, with the tax base serving as collateral. Second, to capture why some assets provide
liquidity benefits or costs, we impose liquidity constraints in both the enduser and bank layer.
In the enduser layer, liquidity constraints require payment instruments supplied by banks.
We allow for both deposits held for one period and for credit lines arranged a period in advance. Those instruments are equivalent in equilibrium — both provide liquidity to endusers and10both require costly leverage by banks. A binding enduser liquidity constraint thus generates aliquidity benefit of inside money.
Liquidity constraints in the bank layer require that interbank payments generated by customer withdrawals are paid with borrowed or unborrowed reserves. A binding bank liquidity
constraint generates liquidity benefits for outside money (reserves) as well as interbank credit.
It also generates a liquidity cost of providing payment instruments.
We assume that financial firms can be costlessly recapitalized every period and that theirobjective function exhibits constant returns to scale. As a result, a firm’s history does notconstrain its future portfolio and capital structure decisions. As in Lagos and Wright (2005),
the distribution of heterogeneous agents’ (here firms’) histories thus plays essentially no role inthe model.
AssetsThere are four diﬀerent asset classes. Reserves serve as numeraire; they are issued by thegovernment, pay a nominal interest rate  and are held only by banks. Overnight credit pays
an interest rate ; it is less liquid than reserves as funds lent out at date  cannot be used tohandle customer withdrawals at date  + 1.
Trees are infinitely lived assets that each pay an exogenous quantity of goods  in period
. Trees diﬀer only in who is allowed to invest in them, as detailed below. For now, we index
trees by types  ∈ [0 1] The nominal price of a type  tree is denoted  , and the nominal
value of its fruit is   , where  is the nominal price level.

3.1

Households

Households have linear utility, discount the future at rate  = − log  and receive an endowmentΩ every period. Households enter the period with deposits  and outstanding credit lines  ;they buy consumption  at the nominal price  measured in units of reserves. Their liquidity
constraint is  ≤ ̄( +  )
where  is a fixed parameter that determines the velocity of money available to households. A
cash-in-advance approach helps zero in on the role of endogenous inside money.
The liquidity constraint allows for two types of nominal payment instruments. Deposits
require investment one period in advance and earn the nominal interest rate 
−1 . Credit
lines must be arranged one period in advance with a bank, but require no investment — they
represent intraday credit extended by banks on demand. In exchange for the commitment toaccept payment instructions, banks charge a fee −1  proportional to the (nominal) amountof credit.1
1

We assume that an interest rate  is earned on deposits regardless of whether they are spent, and thatthe fee  is paid on credit lines regardless of whether they are drawn. These assumptions help simplify thealgebra. More detailed modeling of the fee structure of diﬀerent payment instruments is possibly interesting
but not likely to be first order for the questions we address in this paper.

11

The household budget constraint is  + −1 = Ω + ¡¢
1 + 
−1 − +1 +

Z 1
0

¡¡ 
¢
¢
 +   −1 −   (1)+ dividends + fees + government transfers.

Expenditure on goods as well as fees paid for credit lines must be financed through either ()the sale of endowment, () changes in household asset positions in trees and deposits, or ()exogenous income from dividends, fees or government transfers, described in more detail below.
Households are allowed to invest in all trees  ∈ [0 1], but they cannot sell trees short,that is, we impose  ≥ 0. We interpret the endowment as labor income (payoﬀs from humancapital), while trees represent other long lived assets such as equity in nonfinancial firms or
claims to housing services. Both trees and human capital are less illiquid in the sense that theycannot be used to pay for consumption. The diﬀerence between them is that human capitalmust be held by households, whereas trees can also be held by financial firms or the government,
and their ownership aﬀects the production of payment instruments. A key equilibrium outcome
is where in the economy trees are held.
We think of our model period as a short period such as a day, and we do not study hyperinflation periods, so that nominal and real rates of return are always small decimal numbers.
We thus simplify formulas throughout by using the approximation  = 1 +  for any small rateof return , and setting any products of rates of return to zero. For example, with an inflation
rate   = log  −1 , the real rate of return on deposits is 
−1 −   Household choices

Let   denote households’ marginal utility of wealth and let ̂  = − log( +1  ) denoteminus the logarithm of the MRS of wealth next period for wealth this period.2 The household
first-order condition for type  trees is¡¢  ≥ −̂ +1 + +1 +1+1The condition holds with equality if the household has a positive position in type  trees. The
rate of return on trees held by the household is ̂  ; such trees always exist in equilibrium sincebanks can hold only a subset of trees.
We focus on equilibria in which −  +1 = ̂  −   ̂  (2)The equality implies that households are indiﬀerent between the two payment instruments.
Indeed, households who invest in deposits must provide funds a period in advance on which
they receive real return. Households who arrange a credit line can instead invest the funds intrees that yield ̂, but must then pay the fee  . The inequality says that payment instruments
are costly. Households’ optimal choice is then to hold as few payment instruments as necessary,that is, the liquidity constraint binds.
2

Here   is the Lagrange multiplier on the date  budget constraint (1). In a steady state,   is constant overtime and ̂ = . More generally, the MRS of wealth may diﬀer from  if the liquidity cost of buying consumption
goods changes over time.

12

3.2

Banks

Households own many competitive banks. We describe the problem of a typical bank which
maximizes shareholder valueµ −1 ¶
∞
P
P
exp −
(3)
̂   
=0

 =0

Here bank dividends  are discounted at the rate ̂   as are payoﬀs from trees owned byhouseholds. Dividends can be positive or negative; the latter corresponds to recapitalization of
the bank.
Liquidity management
The typical bank enters the period with deposits  , outstanding credit lines  and reserves
 . We want to capture the fact that enduser payment instructions may lead to payments
between banks. For example, a payment made by debiting a deposit account or drawing acredit line may be credited to an account holder at a diﬀerent bank. We thus assume that the
typical bank receives an idiosyncratic withdrawal shock: an amount ̃ ̄( +  ) must be sent
to other banks, where ̃ is iid across banks with mean zero and cdf  We also assume that ̃is bounded above: the cdf  is increasing only up to a bound ̄ with ̄̄  1.
payments, while otherIn the cross section, some banks draw shocks ̃  0 and mustih makebanks draw shocks ̃  0 and thus receive payments. Since  ̃ = 0, any funds that leave
one bank arrive at another bank; there is no aggregate flow into or out of the banking system.
The scale of the required gross interbank payments depends on velocity in the enduser layer.
In particular, if households almost never use their payment instruments, ̄ is close to zero andfew interbank payments are needed. The distribution of ̃ depends on the structure of thebanking system as well as the pattern of payment flows among endusers.3
Banks that need to make a transfer ̃ ̄( +  )  0 can send reserves they have brought
into the period, or they can borrow from other banks. The bank liquidity constraint is̃ ̄( +  ) ≤ (1 + ) ( + +1 ) (4)where +1 ≥ 0 is overnight borrowing and  ≥ 0 is a parameter that captures the eﬃcacy of
netting arrangements among banks.
If  = 0, there is no netting: all interbank transfers must be made using either reserves
 or overnight credit +1 . More generally,   0 allows banks to make more than one
dollar of transfers per dollar of liquid funds  + +1 . We can think of those liquid funds as
downpayments in an intraday credit system. Below we describe a timing protocol to capture
this idea and derive implications for the observed volume of intraday and overnight credit.
Suppose the marginal cost of overnight credit is larger than other sources of funding available
to the bank. In this case — which we consider below — it is always better to take out as little
overnight credit as necessary. Banks thus optimally choose a threshold rule: do not borrow
3

The likelihood of payment shocks is the same across banks, regardless of bank size. Since the size distribution of banks is not determinate in equilibrium below, little is lost in thinking about equally sized banks.
Alternatively, one may think about large banks consisting of small branches that cannot manage liquidity jointly
but instead each must deal with their own shocks.

13overnight unless ̃ is so large that the withdrawal ̃ ̄( +  ) exhausts both reserves and theintraday credit limit  available by just paying down reserves.
For a bank that enters the period with reserves   deposits  and credit lines   theliquidity constraint (4) implies a threshold shock =(1 + ) ̄( +  )(5)

We refer to  as the liquidity ratio of a bank. It is inversely proportional to a money multiplierthat relates the total value of payment instruments to the quantity of reserves.
If the liquidity constraint (4) binds, then reserves provide a liquidity benefit, measured by
the multiplier on the constraint. A bank thus obtains a liquidity benefit from reserves if itdraws a suﬃciently large shock ̃   . The bank then borrows overnight+1 = ̃̄( +  )−  
1+

(6)

Since ̃ ≤ ̄, banks can choose a liquidity ratio high enough that they never have to borrow,
but can handle even the largest shock just out of reserves.
Portfolio and capital structure choice
Banks adjust their portfolio and capital structure subject to leverage costs. They invest
in reserves, overnight credit and trees while trading oﬀ returns, collateral values and liquidity
benefits. They issue payment instruments and adjust equity capital, either through positive
dividend payouts or negative recapitalizations   Capital structure choices trade oﬀ returns,
leverage costs and liquidity costs.
The bank budget constraint says that net payout to shareholders must be financed either
through interest on credit lines or through changes in the bank’s positions in reserves, deposits,
overnight credit, or trees:
¡
¢
¡
¢

  = −1  +  1 + 
−1 − +1 −  1 + −1 + +1
Z
¡¡ 
¢
¢
+( −  ) (1 + −1 ) − (+1 − +1 ) + +   −1 −   
Θ



−

 (−1 ) (( +  ) +  ) − −1  (7)

In the second line,  ≥ 0 represents a positive position in overnight credit. The last line collectsbank leverage costs and credit lines that banks use to pay those costs, both discussed in detailbelow.
The first line in (7) collects payoﬀs from payment instruments. Interest on reserves −1accrues to the bank that held the reserves overnight, regardless of whether those reserves wereused by the bank to make a payment. Similarly, deposit interest 
−1 is paid by the bank thatissued the deposits in the previous period, regardless of whether the deposits were used byendusers to make a payment. Both conventions could be changed without changing the main
points of the analysis, but at the cost of more cluttered notation.
In the second line of (7), banks’ tree holdings must come from a subset Θ of all the trees
available to households. One way to think about this restriction is as the result of unmodelled14contracting frictions: we could interpret the fruit from trees  as housing services, so the trees
represent all claims on housing, which consist of mortgage bonds, as well as — perhaps due to acommitment problem — housing equity. Banks and households both participate in the market
for mortgage bonds, whereas only households own housing equity. Alternatively the restriction
could be due to regulation, as banks cannot own stocks in some countries.
Leverage costs
If the last line in (7) were omitted, the Modigliani-Miller theorem would hold and bankcapital structure would be indeterminate. We assume instead that the commitment to makefuture payments is costly. It takes resources to convince overnight lenders that debt will be
repaid, as well as to convince customers that the bank will indeed accept and execute payment
instructions. We also assume that convincing lenders and customers is cheaper if the bank owns
more assets to back up the commitments, especially if those assets are safe.
To link the cost of commitment depends on banks’ choices, we define the bank leverage ratio :=(+1 + +1 ) + +1
R
+1 +  Θ    + +1

(8)

where  and  are fixed parameters. A key feature of this ratio is that money-market mutualfund shares and outstanding credit lines are included in +1 and +1 , respectively. Whileboth are oﬀ-balance sheet commitments of a bank, we include them in the numerator. Ourperspective here is that payment instruments are valuable only if they are reliable, and thepresence of collateral makes them more credible. The weight  ∈ [0 1] allows a distinction
between payment instruments and other debt.
Banks choose the leverage ratio −1 at date  − 1 through their choice of nominal positions
at that date. Banks then have to purchase real resources  (−1 ) (( +  ) +  )−1 in the
goods market at date  to support that leverage ratio. The cost function  in (7) is smooth,nonnegative, strictly increasing and concave in bank leverage. We further assume that  slopes
up suﬃciently fast that banks always choose   1.
The denominator of the leverage ratio (8) introduces a collateral value for bank assets:
the resources needed to convince customers about future commitments are smaller if the bankowns more assets. The weight  allows a distinction between safe assets (reserves and overnight
lending) and trees, which we will later assume to be uncertain. Both the weights and thepresence of outstanding credit lines  in the numerator implies that our leverage ratio (8) does
not generally correspond to accounting measures of leverage.
Since leverage costs take up real resources, we need to address how banks pay for them.
The details of this process are not essential and we choose an approach that simplifies formulas.
Resources that support leverage chosen at date  − 1 are purchased by banks in the goods
market at the price  . In order to pay for those goods, banks must arrange credit lines atother banks4 . Banks thus face an additional liquidity constraint that is analogous to that forhouseholds:  (−1 ) (( +  ) +  ) ≤ ̄ (9)Here the term  converts nominal debt at date  − 1 dollars into nominal expenditure on
4

Equivalently we could assume that banks must hold depposits at other banks.

15goods at date  As long as the interest rate on credit lines is positive, the constraint binds inequilibrium: banks arrange for a line that is just large enough to cover the leverage costs thatwill accrue next period. We use the same velocity parameter ̄ as for the household liquidity
constraint — in equilibrium this will lead to a quantity equation that ties the price level to total
payment instruments and total output.
Bank optimal choices
The bank first-order conditions are derived in the appendix. Since the bank problem exhibits
constant returns to scale, they only pin down the leverage ratio  and the liquidity ratio . As
long as those ratios are chosen optimally, banks are indiﬀerent between positions in all assets
and liabilities that have eﬀective rates of returns equal to the rate of return on equity ̂. Eﬀective
rates of returns take into account not only future payoﬀs, but also the eﬀects of the asset or
liability position on leverage cost and the liquidity constraint.
In particular, the presence of leverage costs leads to a determinate optimal leverage ratio.
Given households’ indiﬀerence (2) between the payment instruments, all instruments provide
a liquidity benefit to endusers and therefore require a lower pecuniary benefit. From the bankside, providing payment instruments thus taps a source of funds that is “cheap” in pecuniaryterms: both issuing deposits and issuing equity together with extending credit lines is cheaper
than just issuing equity. For small leverage, it thus always makes sense to provide payment
instruments. Banks increase provision until the marginal leverage is high enough that further
leverage is not profitable.

3.3

Government & equilibrium

We treat the government as a single entity that comprises the central bank and the fiscal
authority. The government issues reserves  , borrows  in the overnight market and chooses
the reserve rate   The government also makes lump sum net transfers to households so thatits budget constraint is satisfied every period. Below we further consider particular policies
that target endogenous variables such as the overnight interest rate. Such policies are still
implemented using the basic tools  ,  and  .
Just like financial firms, the government incurs a cost of issuing debt, above and beyond the pecuniary cost. The government diﬀers from firms in that it has the power totax and hence¡the (implicit)
collateral that is available to it. We define government lever ¢age as 
=

+


+1
+1 Ω+1 and denote the date  government leverage cost as

+1



  (−1 ) ( +  ) where  is increasing and convex, as is the bank leverage cost function  . The more real debt ( +  )  the government issues relative to the labor income
tax base Ω , the more resources it must spend to convince lenders that it will repay. In orderto pay leverage cost, the government is required to arrange a credit line from banks, denoted .
Market clearing
Equilibrium requires that markets clear at the optimal choices of banks and households,
taking into account government policy. Tree market clearing requires that banks or households
hold all trees. The overnight credit market clears if borrowing by banks  plus government

16

borrowing  equals aggregate bank lending  . Banks must hold all reserves. Since the crosssectional distribution of bank portfolios is indeterminate, we now use the symbols  ,  , 
etc to denote aggregate bank positions.
The goods market clears if households consume the endowment and all fruit from trees, netof any resources spent by banks and the government as leverage costs. We denote the total
quantity of goods sold at date  by  . In nominal terms, goods market clearing means¡¢  =   +   −1  +   (−1 ) ( ( +  ) +  )

Below we will assume that  = Ω +  is constant over time and only the composition oftransactions is determined in equilibrium. For example if banks and the government are morelevered, then consumption must be lower.

The market for payment instruments clears if deposits and credit lines oﬀered by banks
equal payment instruments demanded by households plus credit lines arranged by banks andthe government. The model does not introduce a functional diﬀerence between¡credit lines
¢ and


deposits. To emphasize their similarity, we focus on equilibria with  = ̂  −  −  +1 andtreat the two types interchangeably. Market clearing in the market for payment instruments
requires that  +  +  +  =  +  . If the three goods- market liquidity constraints (forhouseholds, banks and the government) all bind, the total real quantity of payment instruments
satisfies  = ̄ ( +  ) (10)In order to for society to handle the transactions  , banks must supply a positive amount of
payment instruments in real terms. Given a finite real value of amount of collateral, banks thus
incur leverage costs. As a result, payment instruments are costly to endusers and endusers’
liquidity constraints bind: (10) holds with equality and works like a quantity equation that
relates the price level to nominal supply of payment instruments.
While enduser liquidity constraints always bind, banks’ liquidity constraints may or may
not bind, depending on how many real reserves are available relative to transactions  as wellas other collateral. We say that reserves are scarce at date  if the threshold shock is smaller
than the upper bound of the shock distribution, +1  ̄, so that the bank liquidity constraint
binds with positive probability at date  + 1. In contrast, reserves are abundant if +1 ≥ ̄ sobanks are sure that the constraint will not bind. In principle, reserves could be scarce at date even though some banks are constrained at date  itself.
The appendix derives a system of equations that characterize equilibrium. It describes the
dynamics of the endogenous prices — the interest rates  and  , the price of bank trees  , thenominal price level  — and three variables on that describe bank balance sheets — aggregate
payment instruments  +  as well as the liquidity ratio  and the leverage ratio   both ofwhich are equal across banks in equilibrium. The appendix further derives an approximation
to the system that works as long as rates of return are small numbers, which can be guaranteedby assumptions on exogenous policy parameters. The following sections then study the steadystate of that system for diﬀerent exogenous parameters.
Neutrality of nominal government liabilities
Government policy determines the total amount of nominal liabilities  +  as well as
their composition. If the path of both reserves and government debt are increased by the same17

factor, then the path of the nominal price level  is also increased by that factor. Mechanically,
reserves and government debt appear in the equations characterizing equilibrium only in theform of real reserves  =   and real borrowing   . This is because our model assumes
flexible prices and no nominal rigidities in the private sector.
The neutrality property here diﬀers from that in cash-in-advance models in which reserves(or currency) directly provide liquidity services to households. In the latter models, an increasein the money supply alone increases the price level in the same proportion — the outstanding
amount of nominal government debt is not important. This is not true in our model because
banks can use government debt as an additional nominal collateral to produce nominal payment
instruments.5

3.4

Timeline

To interpret the model and connect to the motivating facts in Section 2, suppose time is divided
into days. At the beginning of the day, endusers have prearranged payment instruments withbanks: they have money in a deposit account or a credit line. Those funds are more liquid thansecurities — enduser would have to first sell their securities in exchange for payment instruments.
Enduser pay for their transactions  in the morning before selling any securities. They pay
by making payment instructions to their bank, either arranging a transfer out their deposit
account or asking to draw on their credit line.
Table 1: Bank balance sheet at beginning of day 
AssetsReserves Overnight
lending R
Trees Θ  −1

Liabilities
EquityOvernight borrowing 
Deposits oﬀ balance sheet: loan commitments 

Banks enter the period with a balance sheet as in Table 1. The asset side consists ofreserves, overnight lending, and trees. The liability side has equity, overnight borrowing, anddeposits. In addition, the bank has extended loan commitments (credit lines) that are formallyoﬀ-balance sheet. To clarify the role of these loan commitments, it is helpful to compare two
extreme versions of a bank. If  = 0, we have a textbook bank-balance sheet where payment
instruments are all in the form of on-balance sheet debt. However, we can also imagine a bankwith zero deposits,  = 0. If the bank has made loan commitments  , it has assembled
a portfolio of collateral that is funded with equity. Another oﬀ-balance sheet item that we
include in our definition of payment instruments are money-market mutual funds sponsored by
the bank. Shares in these funds are typically held in trust for the client. These items make ourconcept of leverage quite diﬀerent from an accounting leverage ratio.
5

The special role of government thus comes from the endogenity of deposits and their exclusive use as a
medium of exchange. It does not come from “non-Ricardian” fiscal policy. In our model, government surplus is
not exogenous but instead adjusts so as to satisfy the government budget constraint.

18

Consider the following timeline for the opening of markets. In the morning, after all shocks
are realized and payment instructions as well as required interbank transfers are known, thefirst market to open is the Federal Funds market — banks decide whether to lend or borrow
reserves, or keep holding them. Next, enduser payment instructions are executed. At this point,
an intraday credit facility opens — this could be a private netting system such as CHIPS that
lends at zero interest for repayment in the evening. Banks obtain intraday credit proportional
to unborrowed plus borrowed reserves  +  At mid-day, banks make payments to otherbanks: some of these involve transfers of reserves, whereas others are credited to other banks’
intraday credit positions. Finally, in the evening the intraday credit positions are settled andbanks adjust securities portfolios as well as capital structure.
Table 2: Bank balance sheet at mid-day on day : bank with ̃  0AssetsnoReserves max  − ̃ ̄ ( +  )  0
Overnight lending R
Trees Θ  −1
Intraday loans to endusers Liabilities
EquityOvernight borrowing³ ´
Deposits ( +  ) 1 − ̃ ̄
New borrowingno̄( + )+1 = max ̃ 1+ −  0
Intraday borrowingmin{̃ ̄( +  ) −    } + +1

Table 2 shows the balance sheet of a bank at mid-day after all payments have been executed.
We consider a bank that has received a positive shock ̃ and therefore experiences an outflow
of funds. There are several changes relative to the beginning of the day. First, credit lines havebeen drawn: they now show up on the asset side as intraday loans to endusers. On the liability
side, drawn credit lines add to deposits — lending creates inside money. However, a share ̃ ̄ ofall deposits then leaves to other banks. How the bank handles the outflow depends on the sizeof the shock. While the bank is indiﬀerent between reserves and intraday credit within somerange, we illustrate the mechanics assuming a “pecking order”: banks first exhaust reserves,then turn to intraday credit and finally tap the overnight market.
For a small outflow ̃ ̄ ( +  )   , the deposit outflow simply shortens the balance
sheet — fewer deposits are matched by fewer reserves. For an intermediate outflow,  ̃ ̄ ( +  )   (1 + ), unborrowed reserves are suﬃcient as collateral for intraday credit
to fund the outflow. There is no new overnight borrowing, but the drop in deposits outflow is
oﬀset partly by the reduction of reserves to zero and partly by intraday borrowing. Finally, fora large outflow ̃ ̄ ( +  )   (1 + ), the bank turns in addition to the overnight market,
in addition to the above position changes there is also an increase in new overnight borrowing.
A bank that receives an inflow of funds (̃  0) shows an increase of deposits on the liability
side of the balance sheet. On the asset side, three positions may increase to oﬀset this increase:reserves, new borrowing +1 and a positive position relative to the intraday credit system.
Lender banks are indiﬀerent between these options. All that is determinate in equilibrium isthat the Federal funds market clears and that the intraday credit system nets out to zero.
19

Because the shock ̃ is iid with mean zero, we can always choose lender positions with theseproperties.6

4

Steady state equilibrium

We now consider steady state equilibria with constant output Ω +  and constant rates ofreturn. We thus restrict attention to policies that imply constant growth rates for the nominal
quantities  and  so that the   is also constant. With constant rates of return, themarginal rate of substitution of wealth across dates is constant, we have ̂  = . Moreover,the key ratios chosen by banks, leverage  and the liquidity ratio , are constant over time.
With fixed output, payment instruments and the price level also grow at same rate as nominal
government liabilities. The nominal price level and the real quantity of reserves are endogenous
and depend on how many payment instruments banks produce for a given level of reserves.

4.1

Graphical analysis

The predictions of the model can be characterized by reducing the system of equations characterizing equilibrium to only two equations in two key variables: the bank collateral ratio,
which we defined as the inverse  = 1 of the leverage ratio, and the liquidity ratio  The firstequation — the capital structure curve — describes the amount of collateral  that banks needto handle transactions  for given liquidity . It depends on policy because the government
can change the mix of collateral available to banks.
The second equation — the liquidity management curve — describes liquidity management:
banks’ optimal choice of reserves  that equates the opportunity costs of holding reserves to its
collateral and liquidity benefits given the collateral ratio . The curve depends on policy because
the government can change the opportunity costs of holding reserves. Properties of equilibrium
can be studied by plotting the two curves in the ( )-plane, as in Figure 3. Equilibrium is
described by their intersection and its welfare properties can also be studied graphically since
the welfare costs of leverage can be written as functions of  and .
A figure in ( )-plane helps think about not only the liquidity and collateral ratios, butalso a number of other variables that are related in simple ways to those ratios in equilibrium.
In particular, location of the equilibrium along the vertical axis provides insight on real asset
values, whereas location along the horizontal axis contains information about bank balance
sheets and the price level. We now explain these relationships. Derivations of all equations arecollected in the appendix.
The scarcity of reserves and the overnight credit market
The ( )-plane splits into two halves that correspond to scarce and abundant reserves,
respectively. The vertical pink line in Figure 3 marks the critical liquidity ratio ̄ such that
reserves are abundant if  ≥ ̄ (the yellow region shown to the right) and reserves are scarceif   ̄. If reserves are abundant, then banks do not borrow overnight. With scarce reserves,6

If we had written the model with an explicit Fed Funds market with its own interest rate, then the overnight
rate and the Fed Funds rate would be equated in equilibrium.

20

collateral ratio κλ̄liquidity ratio λFigure 3: The green line is the capital structure curve. The blue line is the liquidity management
curve. In the yellow shaded region, reserves are abundant,  ≥ ̄.
bank borrowing follows from integrating over all banks that receive a liquidity shock beyond :Z ̄ ³
´
³ ´

̄
̄
 ( ) ̃ −   ̃ :== + 
1 +  1+The function  is decreasing: if the banking system has more real reserves to handle a given
amount of transactions, then less overnight credit is needed. Overnight borrowing by banks
can therefore be read on the horizontal axis going left. Moreover, if more transactions can benetted within the day (higher ), then banks again run out of reserves less often and borrowless overnight.
Asset market participation and intermediary asset pricing
Except for the restriction that reserves must be held by banks, we have not made assumptions on asset market segmentation. Instead, participation patterns of intermediaries andhouseholds are determined endogenously in equilibrium. Throughout, the basic principle atwork is familiar from other models with short sale constraints: assets are held — and hence
priced — by the investor who likes them the most, either because that investor derives nonpecuniary (liquidity or collateral) benefits from them or later also because that investor has more
optimistic beliefs.
Consider participation in overnight credit and tree markets. Since the production of payment
instruments entails leverage costs, banks obtain a positive collateral benefit from assets they
are eligible to hold. As a result, households do not invest in any asset markets that banks caninvest in: banks bid up the price of any asset accessible to banks until its return is below the21

discount rate and the asset is unattractive to households. For overnight credit and bank trees,banks are thus the only marginal investors.
The real overnight interest rate satisfies the bank Euler equation
 − ( − ) = mb () (11)where mb() is the marginal benefit of a unit of collateral 0 (1) 2  Since the collateral ratio
 = 1 is inversely related to leverage, the condition resembles other “intermediary asset
pricing” equations in the literature, as it draws a connection between returns and leverage.
Importantly, our concept of leverage is special and reflects the provision of payment instruments
as opposed to book or market leverage in an accounting sense.
The real interest rate  −  is positively related to the collateral ratio . In Figure 3, the real
rate can be read on the vertical axis going up. Intuitively, less collateralized banks are willingto pay more for collateral and bid up prices, thus lowering returns. The same logic applies tobank trees, which are priced by a similar Euler equation. In steady state, cash flows on banktrees, denoted  are discounted at a low rate that accounts for the collateral benefits: theirsteady state value is   ( −  mb ()). In the figure, it can be read on the vertical axis going
down.
Payment instruments and the nominal price level
Location along the horizontal axis contains information about payment instruments andthe price level. The “money multiplier” ( + ) is inversely proportional to  and canbe read on the horizontal axis moving left. The nominal price level is determined by nominal
government liabilities. It is helpful to write the relationship between the price level and realreserves as(1 + ) (0 + 0 )0 = (1 + 0 0 )For given nominal liabilities 0 + 0 , we can read oﬀ the price level on the horizontal axis
moving left as long as the split of government debt into reserves and bonds 0 0 does notchange. For comparative statics that change that mix, for example when we look at the resultof open market operations, the price level no longer changes one for one with  but the figure
is still helpful to see the eﬀect on the price level, as explained below.
The capital structure curve
How much liquidity  is required in order for banks to manage transactions  with givenbank collateral ? Steady state liquidity and collateral are linked by
³
´
0
()


1 + 0 1+
+  (− mb())
+ 1+
=

(12)
()

+ 1+
̄We refer to pairs ( ) that satisfy this relationship it as the capital structure curve. Since0 0 is exogenous, the curve also tells us how much government leverage is required tohandle transactions with given bank leverage.
The capital structure curve slopes up in the ( )-plane, as shown as green line in Figure 3.
The basic intuition is that the real quantity of payment instruments  ̄ is pinned down by the22volume of transactions. Since reserves contribute to collateral, we have that the collateral ratio
increases with higher . As a stark example, consider a “narrow bank” which holds no trees orovernight credit — its only collateral is reserves or short term government debt. The collateral
ratio is then simply the ratio of reserves to payment instruments:  = (1+0 0 )̄ (1 + ).
For a narrow bank, the capital-structure curve is thus a straight line through the origin.
In general, two more subtle eﬀects further contribute to an upward slope. First, bankswith higher liquidity ratios run out of reserves less often, which results in lower outstanding
interbank credit and hence a higher collateral ratio. Indeed, since every dollar of interbank
credit is both an asset and a liability to the banking sector, a reduction in interbank credit also
increases overall collateral. Second, more collateralized banks compete less for trees, so tree
prices and hence collateral values fall. As the liquidity ratio increases, the collateral ratio has
to rise to compensate this eﬀect.
The capital structure curve shifts up if banks obtain access to more collateral other thanreserves. One example is an increase in the ratio of government bonds to reserves 0 0 : ifmore government bonds are available to back payment instruments, then banks require fewer
real reserves to achieve any given collateral ratio. An increase in the cash flow from bank trees
 has the same eﬀect: if more private sector assets are available to banks, then fewer reserves
are required. Both shifts also entail an increase in interbank credit for given leverage ratio: thisincrease — which contributes to leverage — is feasible because of the increase in collateral.
The capital structure curve is steeper if banks’ portfolio share of nominal collateral is larger.
Consider in particular the region with abundant reserves, where reserves and government bonds
are the only nominal assets. If a larger share of banks’ collateral consists of reserves as opposedto trees, then any given change in real reserves  and thus the liquidity ratio  implies a largerchange in collateral, that is, we have a steeper capital structure curve. Intuitively, an increasein  corresponds to a decrease in the nominal price level and an increase in the value of money.
The more nominal collateral banks hold relative to trees, the more an increase in the value ofmoney increases the value of collateral.
The location of the capital structure curve also depends on velocity as well as interbank
netting arrangements. For given real reserves, an economy with higher velocity requires lowerbank leverage — this is because fewer payment instruments are needed to handle a given amountof transactions. Similarly, for given real reserves, an economy with more intraday netting (higher) requires lower leverage. This because that economy requires less overnight credit. This eﬀect
is limited to the scarce reserves region — if reserves are abundant then netting is irrelevant forleverage.
The liquidity management curve
Optimal liquidity management equates the opportunity cost of reserves to their collateraland liquidity benefits: − ( − ) = mb () + (1 −  ()) (mc () − mb ()) (13)where mc() is the marginal cost of leverage  () + 0 ()  which is increasing in leverage and thus decreasing in collateral  = 1 For a given real return on reserves, this equation
represents pairs ( ) such that banks optimally choose their liquidity ratio given collateral .
We refer to it as the liquidity management curve.
23

The left hand side of (13) is the opportunity cost to banks of holding reserves, relative toshrinking the balance sheet and paying back shareholders who demand the return on equity
. In equilibrium, it must equal the benefit of reserves as collateral as well as for liquidity
purposes. The collateral benefit is simply mb() ; it is decreasing in the collateral ratio. The
second term on the right hand side is the liquidity benefit: with probability 1 −  (), the bankruns out of reserves and obtains the benefit mc() −mb().

The liquidity benefit of reserves is decreasing in the real quantity of reserves — in fact, once
reserves are abundant, the liquidity benefit shrinks to zero. As long as reserves are scarce, theliquidity benefit of reserves mc() −mb() is positive and decreasing in the collateral ratio. In
equilibrium, every dollar of interbank credit entails leverage cost for the borrowing bank but
adds collateral benefits for the lending bank. Since the former eﬀect is larger, having to borrow
overnight implies a penalty for running out of reserves that decreases with collateral.
Putting together these eﬀects, the liquidity management curve (13) behaves much like amoney demand function for banks. It consists of two pieces, as shown in Figure 3. As long as
reserves are scarce, it describes a downward sloping curve in the ( ) plane: more reserves lower
the liquidity benefit, and are consistent with the same opportunity cost only if the collateral
ratio is overall lower. Once reserves are abundant, the liquidity management curve is flat: theeconomy is at a collateral ratio such that the opportunity cost of reserves is just compensatedby the collateral benefit alone.
The location of the liquidity management curve depends on the real return on reserves
 − . In particular, a higher return on reserves shifts the curve up, that is, banks choose tohold more collateral for a given liquidity ratio. Intuitively, lower opportunity costs of holding
reserves make it cheaper for banks to handle transactions. As a consequence, banks can aﬀord
to hold more collateral.
The location of the liquidity management curve also depends on interbank netting arrangements, at least as long as reserves are scarce. Greater “netting eﬃciency” (higher ) shifts
the curve to the left: if intraday credit is more eﬀective at making payments, a lower liquidity
ratio is enough to handle them. A leftward shift implies that the curve remains unchangedin the abundant reserves region where netting is not important. At the same time, the mixof collateral available in the economy — for example how many trees are available relative togovernment liabilities — is not relevant for liquidity management and does not aﬀect the locationof the curve.

4.2

Equilibrium & changes in policy

Equilibrium real reserves and leverage are determined by the intersection of the capital structure
and liquidity management curves. Whether reserves are scarce or abundant in equilibrium isdetermined by the interaction of policy and the availability of collateral. In particular, thegovernment can set the real rate on reserves so as to select its preferred region. For example,
increasing the real rate on reserves shifts the liquidity management curve down towards anequilibrium with abundant reserves.
At the same time, holding fixed policy, an equilibrium with abundant reserves obtains whencollateral is relatively scarce: as discussed above, a decrease in the cash flow of trees accessible
24collateral ratio κλ̄liquidity ratio λFigure 4: Faster growth rate of nominal liabilities when reserves are scarce
to banks  or a decline in the relative amount of government bonds   shifts the capital
structure curve up, and reserves become relatively more abundant.
We now consider comparative statics of steady states. The time period in the model shouldbe thought of as very short, such as a day. We are therefore comfortable using the model forthinking about the behavior of asset prices, payments and credit over a sequence of years, suchas the recent boom bust cycle. In particular, we are interested in the eﬀect of shocks to agents’belief about future asset payoﬀs (such as those on claims to housing) as well as monetary policy
responses and study those eﬀects as a sequence of comparative statics.
Monetary/fiscal policy & interest on reserves
Since the steady state rate of inflation equals the growth rate of reserves, a government thatcommits to a growth rate of nominal liabilities eﬀectively controls the real rate on reserves.
This specification helps think about two policy regimes. One is traditional: the government
sets debt and reserves while interest on reserves is zero. The other — more recently popular —regime is one where the central bank has issued so many reserves that they are abundant, andthe spread between  and  is zero. The central bank then makes policy by moving aroundthe interest rate on reserves which can be positive or negative.
Consider how changes in monetary policy aﬀect equilibrium in the two regions. First,
suppose we start from an initial equilibrium with  = 0 and scarce reserves — the typicalsituation in many countries before the recent financial crisis. Suppose now the government
decides on faster growth of nominal liabilities. Mechanically, the capital structure curve remains
unchanged, whereas the liquidity management curve shifts down. Figure 4 shows the shifted25

collateral ratio κλ̄liquidity ratio λFigure 5: Higher interest on reserves  when reserves are abundantcurve as dotted line.
Banks optimally respond to higher inflation and hence higher opportunity costs of holding
reserves by holding less collateral for any given liquidity ratio. They provide more nominal
payment instruments relative to reserves (the “money multiplier” increases), which is inflationary in the short run. Since reserves are more scarce, overnight credit also increases. At thesame time, banks compete more for collateral and bid up asset prices: the price of bank trees
increases and the real overnight interest rate falls.
In comparison, consider raising the reserve rate when reserves are already abundant, as the
Fed did in December 2015. Assume further that the government continues to commit to thesame constant growth rate  of nominal liabilities. The mechanical eﬀect is the opposite of theabove: as the real rate on reserves increases, the liquidity management curve shifts up, as in
Figure 5. As the opportunity cost of holding reserves falls, banks find it profitable to handle
the same payments with more collateral. They produce fewer payment instruments relative
to reserves, the money multiplier shrinks and there is deflationary pressure in the short run.
Since reserves are already abundant and now become even more attractive, the overnight credit
market remains inactive. Banks also value additional collateral less, so that asset prices decline.
The two examples show that changes to the opportunity costs of holding reserves haveintuitive eﬀects. In particular, expansionary policy (such as faster money growth) lowers interest
rates and is inflationary in the short run, whereas contractionary policy (such as a higher
nominal rate on reserves) does the opposite. We emphasize that the eﬀects are permanent —they result from comparative statics across steady states. This is in contrast to many models

26collateral ratio κλ̄liquidity ratio λFigure 6: Open market operations when reserves are scarcewith sticky prices or segmented markets, where “liquidity eﬀects” on the real interest rate aretemporary phenomena. Permanent eﬀects arise in our model because the opportunity costs ofholding reserves lead banks to change the way they produce payment instruments, with eﬀects
on the cost of leverage and the value of collateral.
Open market operations & the collateral mix
By engaging in open market policy — for example, a swap of reserves for government bonds —the government can alter the tradeoﬀ between bank leverage and government leverage required
to handle transactions. In other words, it shifts the capital structure curve. This is diﬀerent
from changes to the real return on reserves that alters the tradeoﬀs in liquidity management.
To illustrate, consider a comparative static that increases reserves and oﬀsets this change by anequal change in bonds: we move from an initial equilibrium with (0  0 ) to a new equilibriumwith (̃0  ̃0 ), where ̃0 − 0 = 0 − ̃0  0. As before, reserves and bonds then grow at
the same rate  throughout.
The expansionary open market operation can be summarized by a decrease in the ratio  which shifts the capital structure curve to the right. If reserves are abundant in theinitial equilibrium, then the policy has no eﬀects: it does not change nominal liabilities so theprice level remains unchanged, as do leverage and interest rates. The horizontal shift in the
curve thus reflects the change in nominal reserves ̃0 − 0 .

In contrast, when reserves are scarce as in Figure 6, then a purchase of bonds with reserves is
inflationary in the short run and lowers the real interest rate. Mechanically, since the liquidity
management curve slopes down, the change in real reserves is smaller than ̃0 − 0 , indicating27

an increase in the price level. Intuitively, when more liquid reserves are available, competition
drives banks to produce more payment instruments, pushing the price level up. As a result,
the real value of nominal collateral falls — banks have less collateral and bid up the prices ofcollateral including short bonds, a permanent “liquidity eﬀect” on the real interest rate.

4.3

The role of the nominal interest rate

Some central banks conduct monetary policy by following a nominal interest rate rule. In
practice, the rule is typically implemented by open market policy. For example, during the
scarce reserves regime in place in the US until 2008, the New York Fed’s trading desk bought
and sold bonds of various maturities in exchange for reserves in order to move the overnightinterest rate (the Federal Funds rate) close to the Fed’s target. More recently, as reserves havebecome abundant, the Fed Funds rate and the interest rate on reserves have been essentially thesame, and the policy lever is the interest rate on reserves. It is then tempting to simply transfer
existing analysis of interest-rate rules to the abundant reserves environment even though thepolicy implementation is diﬀerent.
In many monetary models, the details of how the central bank implements the interest-rate
rule are indeed irrelevant — the nominal interest rate alone summarizes the stance of monetary
policy. In particular, many models use households’ optimal choice between currency and shortterm bonds to derive optimal real balances as a function of the nominal interest rates andconsumption. At the same time, intertemporal asset pricing equations — and possibly pricesetting equations — imply a path for inflation. The path for the money supply can then be
inferred ex post so as to generate the implied path for real balances, but is often omitted fromthe analysis altogether. In particular, it does not matter whether policy is implemented with
open market purchases or interest on reserves.
In our model, policy cannot be summarized by the nominal interest rate alone. As discussed
above, policy matters in two ways. Policy can change the nominal rate on reserves. Moreover,policy can change the collateral mix between reserves and government bonds, which matters
as long as reserves are scarce. Both policy actions aﬀect the nominal interest rate. First, withscarce reserves, the same nominal interest rate can be achieved with many combinations ofinterest on reserves and open market purchases that have diﬀerent implications for real interest
rates, inflation, and real reserves. Second, with abundant reserves when open market purchases
are irrelevant, interest on reserves is the key policy tool.
Interest rate policy with scarce reserves
Consider first the case of scarce reserves. We start from an initial equilibrium in that region;
it is generated by initial parameters    and 0 0 and implies some initial overnight rate.
Holding fixed  , we now choose a new target overnight rate that is above the reserve rate andask how  and 0 0 can change to implement it. To proceed graphically, we combine the firstorder condition for overnight lending (11) and the liquidity management curve (13) to trace
out all equilibrium pairs ( ) that are consistent with the spread  −  :
 −  = (1 −  ()) (mc () − mb ()) (14)

The spread is the opportunity cost of holding reserves rather than lending overnight. It must28

collateral ratio κλ̄liquidity ratio λFigure 7: The red curve is equation (14)be equal to the liquidity benefit of reserves on the right hand side since the collateral benefitsof the two assets are the same.
The key properties of the curve in ( ) plane described by equation (14) are illustrated in
Figure 7. First, the red curve is downward sloping: if banks have less collateral, then overnight
credit is more costly; in order to maintain the same opportunity cost of reserves  −  , there
must be more liquidity  to lower the probability 1 −  () with which banks run out ofreserves. Second, the curve never enters the abundant reserves region. As reserves become
more abundant, collateral must fall to maintain a positive spread. Third, the new curve lies
above the liquidity management curve at the initial equilibrium. In order for the new overnight
rate to be lower requires a lower liquidity benefit, which requires more collateral for given
liquidity . Finally, the curve is independent of  and 0 0 , the two parameters describing
policy.
The equilibrium pair ( ) not only satisfies (14), but must also lie on the capital structure
and liquidity management curves. How can the government change policy to move to the newlower overnight rate? There are two stark options. The first option is for the government toannounce a lower growth rate of nominal liabilities  shifting the liquidity management curve
up until all three curves intersect. This policy leaves the collateral mix unchanged but only
lowers the opportunity cost of reserves. The second option is for the government to engage
in expansionary open market operations, shifting the capital structure curve to the right until
all three curves intersect. This policy leaves the opportunity costs on reserves unchanged butchanges the collateral mix.

29

The two extreme policies produce the same change in the overnight nominal rate. At thesame time, they have very diﬀerent implications for the real interest rate and inflation, as
well as for real reserves and overnight credit. In the first option, lower money growth reduces
inflation which contributes to the decline in the nominal interest rate. The eﬀect is less than
one-for-one however — the equilibrium real interest rate actually increases. The reason is thatlower opportunity costs of reserves increase bank collateral and thus lower the collateral benefitof overnight lending. In the other option, open market purchases leave inflation unchanged.
However, fewer bonds implies less collateral.
In addition to the two extreme policies just sketched, many other policies are also consistent
with the new target nominal overnight rate. Indeed, we can combine open-market purchaseswith announcement of future growth of liabilities: we then shift both curves at once, rather
than one at a time as for the extreme policies. The only requirement on the shifts is thatthe new equilibrium ends up on the curve described by equation (14). In particular, the same
nominal interest rate is compatible with an entire range of bank liquidity ratios in equilibrium.
A key diﬀerence between our model and other models of scarce outside money is that the
only medium of exchange for endusers is payment instruments produced by banks. Outsidemoney — here reserves — is only one input into the production of payment instruments. In
particular banks also use government bonds as collateral to back payment instruments. As
a result, the spread between the overnight rate and the reserve rate measures the scarcity ofreserves for banks; it does not measure the scarcity of payment instruments in the economy asa whole. In particular, there is not a unique amount of reserves implied by a given volume oftransactions and a spread.
Interest rate policy with abundant reserves
Consider policy in the abundant reserve regime. Can the government describe policy onlyby the single interest rate  =  ? Equilibrium is described by (11) and (12), which determine and  for a given real return on reserves. As a result, a nominal reserve rate alone cannot pin
down bank liquidity and collateral. Similarly, a feedback rule that relates the rate on reserve
to inflation, for example  =  (), does not uniquely determine , , inflation and the realinterest rate. This is true even if we directly select a rule for the real rate as a function of ,
thus eliminating possible multiplicity coming from the shape of  that has been discussed in
the literature.
Our model diﬀers from other monetary models in what happens once outside money becomesabundant. Consider first models with bonds and currency only. At the zero lower bound insuch models, bonds and outside money become perfect substitutes to endusers, so the medium
of exchange (currency) loses its liquidity benefit. In the current model, endusers hold neither
bonds nor outside money — both are held only by banks. Equating  and  makes bonds andoutside money perfect substitutes for banks, but does not remove the liquidity benefit of themedium of exchange, namely payment instruments produced by banks.
There are also models in which reserves, bonds and currency coexist. In such models, =   0 makes bonds and reserves perfect substitutes. At the same time, currency remains ascarce medium of exchange that is relevant for some transactions. The reserve rate represents
the spread between reserves and currency; it measures the scarcity of currency and relates thedemand for real balances to real variables such as consumption. The tradeoﬀ between currency
30

collateral ratio κλ̄liquidity ratio λFigure 8: The cyan line is an indiﬀerence curve implied by equation (15)and reserves is what enables those models to work with interest rate rules in the usual way evenwhen reserves are abundant.

4.4

Optimal policy

The optimal payment system minimizes the loss of resources due to leverage. Our technological assumptions say that a given volume of transactions requires a fixed amount of payment
instruments supplied by banks, as well as some outside money supplied by the government that
can in turn serve as collateral for banks. Total consumption lost every period in steady state
can expressed as a function of leverage and liquidity:
µµ
¶
¶
¶µ
¶
µ
¡ −1 ¢

 


 ()

1+
1+
 +  
+

(15)


 1 +  Ω 1 + ̄ 1 + Provided that the leverage cost of the government slopes up fast enough, the indiﬀerence curves
are upward sloping and convex, as shown in Figure ??.

We consider the best steady state equilibrium that the government can select by choiceof its two policy instruments, the collateral mix represented by   and the real return on
reserves  −  which determines the opportunity costs of holding reserves. The optimal policy
problem is to choose those instruments together with  and  to minimize losses (15) subject
to the capital structure curve (12) and the liquidity management curve (13).
If the government can freely choose the ratio of bonds to reserves, it is optimal to set  = 0. Indeed, while bonds and reserves provide the same collateral services, reserves31also provide liquidity services, which lowers the need for interbank borrowing and hence costlybank leverage. Since the model focuses on the provision of payment services, there is no benefitof government bonds per se, nothing is lost by just issuing reserves. More generally, the way
fiscal policy is conducted independently of monetary policy may imply that there is a constraint
on   . We can then view the welfare costs as a function in  and  with   a fixedparameter.7
The return on reserves directly aﬀects neither the welfare cost nor the capital structure
curve. We can therefore find the optimal solution in two steps. We first find a point ( ) on
the capital structure curve (for given   ) that minimizes (15). The optimal real return on
reserves is then whatever return shifts the liquidity management curve so that the equilibrium
occurs precisely at that optimal point. If the indiﬀerence curves are convex and the capital
structure curve is concave — which is a reasonable assumption if the eﬀects of interbank credit
are relatively small — then we obtain an interior solution as shown in Figure ??.
Should reserves be abundant? The figure suggests that this is not necessarily the case.
Indeed, if the government leverage cost curve slopes upward very steeply, then it may be optimal
to run a system with scarce reserves, in which real government leverage is much lower than thedebt required to run the payment system. It is better to have banks rely on other collateral inorder to back payment instruments. However, if the government can borrow cheaply at will,
so that its leverage cost is close to zero, then it makes sense to move towards narrow banking
where reserves make up the lion’s share of bank portfolios.

5

Securities markets and the payment system

In this section we consider the interplay between securities markets and the payment system.
We maintain throughout our focus on steady states. We first introduce uncertainty premia,
the key source of fluctuations in asset prices. This allows to discuss the eﬀect of uncertainty
shocks on the supply of payment instruments as well as unconventional monetary policy — thegovernment buys trees that carry uncertainty premia. These questions can be studied even if
the only link between tree (that is, securities) markets and the payment system is that banks
invest in trees.
We then extend the model to introduce two additional links: banks lend overnight to institutional investors and institutional investors use payment instruments to trade assets. To
clarify the eﬀect of each link in isolation, we introduce two types of asset management firms:
carry traders buy trees on margin, whereas active traders face liquidity constraints for someasset purchases. Both types of firms otherwise work like banks: they are competitive firms
owned by households that have access to a subset of trees and maximize shareholder value.
7

Alternatively, we could capture fiscal policy by a given real amount of bonds, say . The welfare cost canthen be written with total debt equal to  +  as opposed to  (1 +   ). The basic tradeoﬀ remains the
same.

32

5.1

Introducing uncertainty and collateral quality

We capture a change in uncertainty as a change in beliefs about output: we assume thathouseholds behave as if the component of output — that is tree dividends  and labor incomeΩ are expected to fall by  percent per period in the future. Actual output and tree payoﬀ
remains constant throughout. One way to think about these beliefs is that households are
simply pessimistic. Our preferred interpretation is ambiguity aversion: households contemplatea range of models for payoﬀs, and evaluate consumption plans using the worst case model. In
either case, the key eﬀect of pessimistic valuation is to generate premia on assets: an observer(such as an econometrician measuring the equity premium) will observe low prices relative to
payoﬀs and hence high average returns.
To define equilibrium, we must take a stand not only on beliefs about exogenous variables,
but also about endogenous variables such as the nominal price level and asset prices. We follow
Ilut and Schneider (2014) who also capture the presence of uncertainty with low subjective
mean beliefs about exogenous variables: beliefs about endogenous variables follow from agents’knowledge of the structure of the economy. In particular, agents know the policy rule of thegovernment and that banks maximize shareholder value given the households’ discount factor.
Households’ worst case beliefs thus also aﬀect bank decisions; shareholder value (3) is replacedby its worst case expectation.
We characterize equilibria in which households expect the government to implement a constant inflation rate . This approach focuses on the eﬀect of uncertainty on collateral values.
Since households expect output to fall at the rate , nominal governmment liabilities must growat the rate  − . Since actual output does not fall, actual inflation is lower in the steady statewith uncertainty. However, all agent decisions and hence asset prices are based on the perceived
inflation rate .
Working with mean beliefs allows us to capture the essential features of uncertainty withminimal changes to the model equations. The household first-order condition relates the pricehouseholds observe in the market in the current period with their expected future price. Sincehouseholds behave as if they live in a steady state in which tree payoﬀs fall at the rate , thefirst-order condition is now
¶
µ


−−
≥
+ 
(16)



In other words, payoﬀs on trees held by households are now discounted at the rate  + .

To see how ambiguity generates premia on assets, consider a tree  held by households so that(16) holds with equality. The equilibrium price of the tree is  ( + ). An econometrician
who observes prices and payoﬀs measures a return  +  which is higher than the discountrate. If there was also a second “safe” tree held by households that earns exactly the discountrate, the econometrician would measure an equity premium on the uncertain tree. In termsof comparative statics, an increase in uncertainty captured by an increase in  leads to higher
premia and lower prices.
Uncertainty and collateral quality
It is natural to assume that trees that are more uncertain also represent worse collateral.
In this section, we make the weight that trees receive in the aggregation of collateral explicitly
33

a decreasing function  () of the spread. A change in uncertainty thus has two eﬀects onbanks tree portfolios. There is a direct eﬀect on prices. In addition, the fact that the trees are
uncertain makes them worse collateral per dollar of funds invested in them.
The bank first order condition for trees accessible by banks is now =  −  +  () mb () In the presence of uncertainty, banks still hold all assessable trees in equilibrium because of
their collateral benefits. Returns on assets held by banks are aﬀected by two opposing forces:
compensation for uncertainty born by shareholders tends to increase returns, while the collateralbenefit tends to lower returns. The presence of uncertainty thus also puts an additional wedge
between the return on trees held by banks and the return on safer overnight credit.
In terms of our graphical analysis, the only change is to the value of bank trees in the capital
structure curve. In particular, the first term in the denominator on the right hand side of (12)should be replaced by
 +  −  () mb ()the value of bank trees after discounting at the higher rate that includes the uncertainty premium. An increase in the uncertainty premium  lowers the value of bank trees and thus thevalue of banks’ collateral. For the collateral ratio  to remain the same, the value of reserves
must increase, which is deflationary in the short run. It follows that the capital structure curve
shifts to the right, as in Figure 9. The liquidity management curve is not aﬀected by changes
in .

5.2

An increase in uncertainty and policy responses

The value and collateralizability of trees aﬀects the scarcity of reserves even if policy (describedby  − and   ) does not change. Indeed, starting from an equilibrium with scarce reserves,
an increase in uncertainty lowers the real interest rate and the price level. This is because thedrop in collateral values makes it more expensive for banks to create payment instruments.
As a result, the supply of payment instruments declines and generates deflation. In the new
equilibrium, bank collateral ratios are lower, bank liquidity ratios are higher and the ratio ofinterbank credit to payment instruments is lower.
If the increase in uncertainty is large enough, the capital structure curve shifts so far to theright as to push the economy into the abundant reserves region, as shown in Figure ??. At thispoint, further increases in uncertainty no longer change the overnight interest rate. The realquantity of reserves is now so high relative to the real quantity of transactions that all liquidity
shocks can be handled with reserves and intraday credit alone. The interbank overnight market
shuts down entirely. Further increases in uncertainty do still lower the nominal quantity of
payment instruments and the price level.
An increase in uncertainty is an attractive candidate for a shock that could have occurredat the beginning of the recent financial crisis. It is consistent with an increase in asset premia, adrop in uncertain asset prices, a decline in the overnight interest rate all the way to the reserve
rate as well as a decline in bank collateral and an eﬀective shutdown of interbank Federal Funds
34

collateral ratio κλ̄liquidity ratio λFigure 9: An increase in uncertainty shifts the solid green capital structure curve to the right,
resulting in the dotted line.
lending. However, we did not see a large deflation — after an initial small drop in late 2008 theprice level remained quite stable over time.
An expansion of reserves
According to our model, a candidate for the absence of deflation is monetary policy. Suppose
the Treasury issues a lot of new debt that is then purchased by the central bank in exchange
for reserves. Suppose further that this is perceived as a one time change, with a stable path of
nominal liabilities thereafter. In terms of the model, this policy corresponds to an increase in
the outstanding nominal quantity of reserves.
In the abundant reserve regime, where reserves and other debt are perfect substitutes, anincrease in reserves is neutral. The real interest rate and leverage do not change, and theeconomy remains in the abundant reserves regime. The only variable that changes is the pricelevel which rises in proportion to the increase in reserves. We conclude that an increase inuncertainty coupled with a large injection of reserves can move the economy into a period
of abundant reserves with low asset prices, collateral and real rates, and the move is notaccompanied by drop in the price level.
Unconventional monetary policy
An alternative response by central banks to a decline in asset values — a collateral shortage— has been to purchase low quality collateral, such as risky mortgage backed securities. We
now consider what happens when the government purchases risky trees instead. We set up an
experiment analogously to the open market purchase above. We start from an initial equilibrium
35

collateral ratio κλ̄liquidity ratio λFigure 10: Central bank purchase of trees when reserves are abundant.
with abundant reserves 0 , price level 0 and collateral ratio 
We assume that the government injects reserves to purchase all trees from banks’ balance
sheet, that is, new reserves are chosen such that, at the new equilibrium with reserves ̃0 , pricelevel ̃0 and collateral ratio ̃, we have̃0 − 0 =̃0 
 +  − mb (̃)The eﬀect of the purchase are just like in Figure 10. As trees are removed from bank balance
sheets, the capital structure curve moves to the right.
After the additional injection, reserves continue to be abundant, so the policy has no eﬀect
on collateral (so ̃ = ) and real asset values. However, the policy does help stabilize the pricelevel — it counteracts the deflationary eﬀect of higher uncertainty. Indeed, collateral in the new
equilibrium is ̃0 ̃0 which equals the real value of old reserves 0 ̃0 plus the full real valueof trees. In contrast, collateral in the initial equilibrium was given by the real value of reserves
0 0 as well as the value of trees multiplied by the collateral quality weight  ()  1. Since
the collateral ratio is the same in the two equilibria, it must be that ̃0  0 .
Unconventional policy thus works by replacing low quality real collateral on bank-balance
sheets with high quality nominal collateral. Since reserves continue to be abundant and the real
return on reserves stays the same, this does not actually lead to an increase in real collateral.
However, backed by the new reserves, banks provide more nominal payment instruments, whichis inflationary in the short run. Compared with an outright increase in reserves, the inflationaryeﬀect of tree purchases is smaller since at the same trees are removed from the collateral pool.
36

Tree purchases by the central bank have two additional, more subtle, eﬀects. First, like otherincreases in banks’ portfolio share of nominal assets discussed above, it makes the capitalstructure curve steeper. The slope in turn matters for the inflation response to changes in
the interest rate on reserves: indeed, the steeper the capital-structure curve, the less does anincrease in the return on reserves push the price level down.
Second, removing trees from bank balance sheets reduces banks’ exposure to further shocks
to asset quality. In particular, suppose that after all trees have been bought by the government,
the uncertainty shock is reversed and asset prices increase. The payment system would notreact to this shock as trees no longer serve as collateral to produce payment instruments. The
economy would remain in an abundant reserves environment even though the asset-market
turbulence that has sent it there in the first place has actually subsided.

5.3

Carry traders

So far, the eﬀect of asset values on bank balance sheet is direct: it requires bank investment intrees. In this section, we introduce institutional investors who borrow short term from banks
in order to invest in trees. We call these investors “carry traders” — they do not actively trade
trees but roll over their debt. This creates an additional link between securities markets andbanks that operates even if banks only engage in short term lending. Monetary policy canaﬀect carry traders’ funding cost.
Carry traders are competitive firms that issue equity, borrow overnight and invest in thesubset of Θ∗ , which is distinct from the subset accessible to banks. Like banks, carry traders
face leverage costs, captured by an increase concave function ∗ that could be diﬀerent from thecost function  assumed for banks.8 The leverage ratio of carry trader  is defined as overnight∗
credit divided by the market value of the tree portfolio∗
+1∗ = R ∗ Θ∗Carry traders’ marginal collateral benefit is mb∗ () and their marginal cost of leverage ismc∗ (), respectively.
We assume further that carry traders are more optimistic about the payoﬀ of trees inΘ than households: they perceive uncertainty  whereas households perceive ∗  . The
idea here is that the firm employs specialized employees who households trust to make assetmanagement decisions. As a result, the spread relevant for investment in carry trader trees
indirectly through investment by carry traders carries the uncertainty premium  that makesthese trees as desirable as other trees held directly by households.
∗

Optimal investment and borrowing
Carry traders’ first order condition for overnight credit resembles that of banks in (A.6),
except that overnight borrowing does not provide liquidity benefits: the return on equity must8

We do not consider welfare eﬀects of leverage for carry traders — instead we focus on the positive implicationsof margin trading. We thus assume for simplicity that leverage costs of carry traders are paid lump sum tohouseholds so that they have no impact on welfare.

37

be smaller than the real overnight rate plus the marginal cost of leverage We focus on steadystates only and drop time subscripts. Since we already know that the real rate is lower than in equilibrium, it is always optimal for carry traders to borrow and we directly write thecondition as an equality: =  −  + mc∗ (∗ ) (17)
It follows that all carry traders choose the same collateral ratio and we drop subscripts  fromnow on. Moreover, carry trader collateral is higher in equilibrium when interest rates are high.
Like banks, carry traders hold all trees accessible to them. This is due not only to thecollateral benefit conveyed by trees, but also to carry traders’ relative optimism. The firstorder condition for tree  is
 =  −  + mb∗ (∗ ) When interest rates or uncertainty is low, carry traders apply a lower eﬀective discount rate totrees, which results in higher tree prices.
The amount of carry trader borrowing in steady state equilibrium is ∗ = ∗∗
 +  − mb∗ (∗ )(18)Here ∗ represents total dividends on carry trader trees. Lower interest rates increase bothleverage and the value of collateral, and therefore increase borrowing. Moreover, an increase inuncertainty (that is, an increase in ) lowers collateral values and borrowing.
Equilibrium with carry traders
Our graphical analysis of equilibrium remains qualitatively similar when carry traders areadded to the model. The only change is that carry trader borrowing now enters on the asset
side of the banking sector. We can thus add in the numerator of the collateral ratio (12) a term ∗ () that expresses carry trader borrowing as a function of bank collateral. We obtain thefunction  ∗ by substituting for ∗ in (18) from (17) and then substituting for the interest ratefrom the bank first order condition from (A.10). The function  ∗ is decreasing: if banks havemore collateral, the interest is higher and carry traders borrow less.
We can now revisit the eﬀect of changes in beliefs and monetary policy in an economy withcarry traders. Suppose first that, starting from an equilibrium with scarce reserves, there is anincrease in uncertainty. The new eﬀect is that, as carry traders value trees less, they demand
fewer loans from banks. This lowers bank collateral and shifts the capital structure curve tothe right. The liquidity management curve does not change. In the new equilibrium, bankcollateral is even lower and the interest rate is lower, as is the price level. The deflationaryeﬀect therefore amplifies the increase in uncertainty about bank trees considered earlier. The
additional prediction is that we should see a decline in funding of institutional investors withshort-term credit from payment intermediaries, such as a decline in repo extended by moneymarket mutual funds to broker-dealers.
It is also interesting to reconsider the eﬀect of monetary policy. Suppose policy engineersa change in the mix of bank assets or their value that lowers the real overnight interest rate.
Carry traders borrow more and bid up the prices of the trees they invest in. As one segmentof the tree market thus increases in value, the aggregate value of trees also rises: there is a38

tree market boom. Importantly, this is not a real interest rate eﬀect: the discount rate ofhouseholds, which is used to value trees held by households, is unchanged. The eﬀect comessolely from the eﬀect of monetary policy on the overnight rate and hence on carry traders’funding costs.

5.4

Active traders

Carry traders provide collateral to banks and thus aﬀect the supply of payment instruments.
We now introduce another group of institutional investors who demand payment instruments.
“Active traders” periodically rebalance their portfolios as their view of tree payoﬀs changes, andthey require cash to pay for new trees. The demand of active traders for payment instruments
counteracts the supply side eﬀects due to changes in the collateral values or the borrowing by
carry traders. At the same time, monetary policy also aﬀects their funding costs.
Active traders are competitive firms that issue equity and invest in payment instruments
as well as a subset of trees Θ̂. There are many active traders and each is optimistic about
one particular “favorite” tree: the trader perceives uncertainty  about this tree. In contrast,
households and other active traders perceive uncertainty ̂  . Active traders also perceive̂   about all other trees. Every period, the identity of the favorite tree within the subset Θ̂changes with probability ̂ ≤ 1 to some other tree in the subset.
To generate a need for payment instruments, we assume that active traders who change
their tree position must pay for the new tree purchases with prearranged payment instruments
or intraday credit. Active trader  faces the liquidity constraintZ ̂  =  + (̂ + ̂ )
Θ̂where  is a random variable that indicates whether or not the favorite tree of active trader 
has changed in the current period,  is the intraday credit position, ̂ are deposits that thefund keeps at its bank together with credit lines ̂ 
Like banks, active trader  faces a limit on intraday credit ≤ ̂(̂ + ̂ )
where ̂ is a parameter that governs netting in tree transactions. It is generally diﬀerent fromthe parameter  that governs netting among banks, since it captures netting by a clearing andsettlement system for the securities that active traders invest in.
Active traders choose payment instruments, trees and their shareholder payout. We focus
on equilibria in which every active trader always holds only his favorite tree — we can assume
that the perceived uncertainty on other trees is high enough. Since payment instruments arecostly — the real rate on deposits is below the discount rate — active traders arrange as few
payment instruments as necessary in order to purchase the entire outstanding amount of their
new favorite tree in case the identity of their favorite tree changes. It follows that the intraday
credit limit binds in equilibrium, a form of “cash-in-the-market pricing”.
Optimal investment and deposits
39

To make the marginal benefit of payment instruments equal to the discount rate, payment
instruments must provide a liquidity benefit. The latter comes from traders’ ability to invest
in their favorite tree, which carries a return that compensates them for the opportunity cost of
deposits. Indeed, in steady state, the first order conditions for deposits and trees simplify to =  −  + (1 + ̂) ( −  − ) 
The equilibrium return  adjusts to provide an excess return  −  −  on the favorite trees.
Since every trader perceives a lower spread on his own favorite tree compared to any other tree,
this excess return can persist in equilibrium.
Equilibrium payment instruments are proportional to the market value of active traders’favorite trees:̂(̂ + ̂) (1 + ̂) = −)
 +  + −(1+̂
where ̂ represents dividends on active trader trees. The market value of trees, as well as
deposit holdings, respond to the opportunity cost of trading: if the deposit rate is higher, thenactive traders earn lower returns on their trees, prices are higher and active traders hold more
deposits. Moreover, an increase in uncertainty lowers asset values and the demand for deposits.
Equilibrium with active traders
We focus on local changes to equilibria with abundant reserves. Our graphical analysis
of equilibrium then remains qualitatively similar when active traders are added to the model.
What changes is that transactions now include carry traders’ asset purchases: = Ω +  + (̂ + ̂)
Since we know that the equilibrium deposit rate is decreasing in banks’ leverage and liquidity
ratio, we can replace the exogenous ratio  in the leverage ratio (12) by a function  () thatis increasing in collateral.
We can now revisit the eﬀect of changes in beliefs and monetary policy in an economy withactive traders. Suppose first that there is an increase in uncertainty. As active traders value
trees less, they demand fewer payment instruments. This increases the collateral ratio of banks
and shifts the capital structure curve to the left. The liquidity management curve does notchange. In the new equilibrium, bank collateral and the interest rate are higher, as is theprice level. In other words, active traders are a force that generate the opposite response to achange in uncertainty from banks and carry traders. Since in the typical economy all traders arepresent to some extent, we can conclude that their relative strength is important. An additional
prediction is that we should see a decline in payment instruments — either deposits or credit
lines — provided to institutional investors.
We can also reconsider the eﬀect of monetary policy. Suppose once more that policy lowers
the real overnight interest rate. The opportunity cost of holding deposits falls and active tradersdemand more payment instruments. At the same time, they bid up the prices of the trees they
invest in. Again a segment of the tree market increases in value and the aggregate value of trees
also rises: there is a tree market boom. Again the eﬀect is not due a change in the discountrate, but instead a change in the funding cost: here it aﬀects active traders’ strategy which
requires payment instruments in order to wait for trading opportunities.
40

6

Related literature

In this section we discuss how our results relate to existing work in monetary economics.
Balance sheet eﬀects and government liabilities
In our model, welfare costs derive from "balance sheet eﬀects" and policy matters by changing the asset mix in the economy. This theme is familiar from other work on unconventionalmonetary policy. Several papers study setups where banks are important to channel funds tocertain borrowers. By purchasing the bonds of these borrowers, policy can eﬀectively substitutepublic credit when weak balance sheets constrain private credit (e.g. Cúrdia and Woodford
2010, Christiano and Ikeda 2011, Gertler and Karadi 2011, Gertler, Kiyotaki and Queralto2012).9 Our model diﬀers from this literature in how banks add value — their special ability is
not lending, but the handling of payment instructions.
Since the price level depends on the supply of payment instruments, shocks to bank assets
have deflationary eﬀects in our model. If all payment instruments are taken to be deposits,
we obtain a collapse of the money multiplier along the lines of Friedman and Schwartz (1963).
Brunnermeier and Sannikov (2016) also consider the link between asset values and the supply
of inside money by banks. In their model, banks’ special ability is to build diversified portfolios
and deposits are a perfect substitute to outside money as a store of value. In contrast, in ourmodel inside money is a medium of exchange for endusers, and outside money works like anintermediate good for producing inside money, rather than a substitute.
Asset pricing and money
In our model, collateral benefits generate market segmentation. We thus arrive endogenously
at "intermediary asset pricing" equations that are reminiscent of those in He and Krishnamurthy(2013) or Bocola (2016). Unlike our banks, banks in those models are investors with limited
net worth who are assumed to hold some assets because they have special investment abilities.
The interaction of liquidity and collateral benefits in our model also generates permanentliquidity eﬀects. In contrast, the literature on monetary policy with partially segmented asset markets (for example, Lucas 1990, Alvarez, Atkeson and Kehoe 2002) obtains temporary
liquidity eﬀects; collateral benefits play no role there.
Asset values in our model depend on the cost of payment instruments to institutional investors. A permanent eﬀect of monetary policy on asset values also obtains in Lagos and Zhang(2014) where the inflation tax discourages trade between heterogeneous investors; this alters
which investor prices assets in equilibrium. The eﬀect we derive is diﬀerent because the cost of
liquidity to endusers is not captured by the inflation tax; instead the cost of payment instruments depends on banks’ cost of leverage. In particular, our mechanism is also operative whenreserves are abundant.
Bank liquidity management and monetary policy
With scarce reserves, bank liquidity management matters for asset valuation, policy impact
and welfare in our model. The liquidity management problem arises because banks cannot perfectly insure against liquidity shocks due to customer payment instructions, as in Bhattacharya
9

Buera and Nicolini (2014) also consider the eﬀect on monetary policy on balance sheets in a model ofentrepreneurs who face collateral and cash-in-advance constraints.

41

and Gale (1987).10 Recent work has discussed the interaction of monetary policy and liquidity management with scarce versus abundant reserves (for example, Whitesell 2006, Keister,Monnet and McAndrews 2008)11 . While these papers consider more detail that is useful tounderstand the cross section of banks, our stylized model tries to capture the main tradeoﬀ and
its interaction with other features of the economy.
Several papers have incorporated bank liquidity management into DSGE models. Cúrdia
and Woodford (2011) study optimal monetary policy in a New Keynesian model.12 In theirsetup, reserve policy can be stated in terms of a rule for the overnight interest rate and thereserves rate; there is no need to formulate policy in terms of the quantity of reserves. Our
setup is diﬀerent because of market segmentation: the nominal interest rate is not directly
connected to a household marginal rate of substitution, but rather to bank leverage. Rulesfor interest rates are then not enough to characterize the behavior of inflation — the supply of
nominal government liabilities is also relevant.
Bianchi and Bigio (2014) study a quantitative model in which banks have a special ability tolend and face an inelastic demand for debt as well as idiosyncratic liquidity shocks. Monetary
policy changes the tradeoﬀ between reserves and interbank credit and hence the willingness ofbanks to make loans. In contrast, the demand for payment instruments in our model comesfrom their role as a medium of exchange for goods and securities; monetary policy aﬀects thecost of payment instruments to endusers, not only to banks.
In Drechsler, Savov and Schnabl (2016), banks are investors with relatively low risk aversion
who issue debt subject to aggregate liquidity shocks. Monetary policy changes the cost of selfinsurance via reserves and thereby aﬀects banks’ willingness to take leveraged positions in risky
assets as well as the risk premium on those assets. In our model, monetary policy aﬀects not
only the funding cost of banks, but also that of banks’ institutional clients; the two channels
have opposite eﬀects on uncertainty premia.
The role of interest on reserves as a policy tool has recently received renewed attention. A
number of papers ask when the price level remains determinate (Sargent and Wallace 1985,
Hornstein 2010, Ennis 2014). Woodford (2012) and Ireland (2014) consider macroeconomic
eﬀects of interest on reserves in a New Keynesian framework. Kashyap and Stein (2012) consider
a model with a financial sector; they emphasizes the presence of quantity and price tools for
macroeconomic and financial stability, respectively.
Multiple media of exchange and liquidity premia
While our model allows for both deposits and credit lines in the enduser layer, we assume
10

A related literature asks whether government supplied liquidity is useful when firms cannot perfectly insure
shocks to investment opportunities (for example, Woodford 1990, Holmstrom and Tirole 1998). An alternative
approach to bank liquidity, following Diamond and Dybvig (1983), considers optimal contracts oﬀered by banks
to endusers. This approach typically abstracts from interbank transactions; the focus is instead on optimal
dependence of contracts on enduser liquidity needs, given information problems as well as the scope for multiple
equilibria that include bank runs.
11
Similar tradeoﬀs have been developed in the literature on the dynamics of the Federal Funds market (e.g.
Ho and Saunders 1985, Hamilton 1996, Afonso and Lagos 2015.)
12
Other notions of bank liquidity have also been explored in DSGE settings. Gertler and Kiyotaki (2010)
consider a model in which bank borrowing not only depends on bank net worth but also is fragile and subject
to runs. Del Negro, Eggertsson, Ferrero and Kiyotaki (2013) study a model in which assets become illiquid in
the sense of being harder to sell, as in Kiyotaki and Moore (2008).

42

that those instruments are perfect substitutes. An interesting related literature asks which
instruments are used in which transactions. In particular, Telyukova and Wright (2008) consider a model in which both credit and money are used and explain apparently puzzling costdiﬀerences with convenience yields. Lucas and Nicolini (2015) distinguish currency and interest
bearing accounts and show that a model that makes this distinction can better explain the relationship between interest rates and payment instruments. Nosal and Rocheteau (2011) survey
models of payment systems.
In our model, asset values reflect collateral benefits to banks and hence indirectly benefits
to the payments system. Moreover, government policy can matter by changing the scarcity ofcollateral that eﬀectively backs payment instruments. Similar themes appear in "new monetarist" models with multiple media of exchange.13 In models based on Lagos and Wright (2005),
assets that are useful in decentralized exchange earn lower returns. Several papers have recently
studied collateralized IOUs as media of exchange, following Kiyotaki and Moore (2005).
For example, in Williamson (2012, 2014) some payments are made with claims on bankportfolios that contain money, government bonds or private assets; banks moreover provide
insurance to individuals against liquidity shocks. Rocheteau, Wright and Xiao (2015) consider
payment via money or government bonds (or, equivalently in their setup, IOUs secured bybonds). These models give rise to regimes of scarcity or abundance for each medium of exchange.
The real eﬀects of scarcity can be diﬀerent for, say, bonds and money because money is used
to purchase a diﬀerent set of goods.
While we also study how policy aﬀects the scarcity of diﬀerent assets like bonds and reserves,the mechanisms we emphasize as well as our welfare conclusions diﬀer in important ways.14
Indeed, in our model only one medium of exchange helps in enduser transactions — payment
instruments supplied by banks. Since any bank commitment is costly, payment instruments
are never abundant and collateral to back them is always scarce — only the degree of scarcity
changes and aﬀects welfare. In contrast, reserves can be scarce or abundant depending on theirrole in bank liquidity management.
In our two layer setup, whether scarcity aﬀects asset prices or welfare thus depends cruciallyon features of the banking system. For example, scarcity of bonds has diﬀerent eﬀects fromscarcity of reserves because reserves change banks’ liquidity management problem and theleverage costs of tapping the overnight market. In addition, the price level in our model isrelated to the supply of nominal payment instruments by banks and hence the nominal collateralthat banks hold. For example, the share of nominal collateral available to banks shapes theprice level response to policy.
13

See Lagos, Rocheteau and Wright (2014) for a recent survey.
In addition, the new monetarist literature considers explicit models of decentralized exchange, rather than
reduced form liquidity constraints as we do. At the same time, Andolfatto and Williamson (2015) consider acash-in-advance model with bonds and money and show that several key eﬀects from the more complex papers
can be seen already there.
14

43

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47

A

Appendix

In this appendix we derive the main equation characterizing equilibrium. Section A.1 derives
banks’ first order conditions. Section A.2 derives a system of equations characterizing equilibrium.

A.1

Bank optimization

This section studies the optimal choice of banks that maximize (3) subject to (7), 4) and (9).
Paying for leverage costs, marginal leverage costs and marginal collateral benefit
Consider first banks’ choice of credit line to pay for leverage costs. Credit lines granted byother banks do not contribute to collateral and hence do not show up in the leverage ratio (8).
As long as the interest rate on credit lines is positive, the constraint then binds in equilibrium:banks arrange for a line that is just large enough to cover the leverage costs that will accrue
next period.
Using the budget constraint and the binding liquidity constraint, the last two terms in thebank budget equation (7) describe the cost of leverage chosen in the previous period and canbe written as− (1 + −1 ̄) (−1 ) (( +  ) +  )
(A.1)
Since banks have to arrange a credit line, their eﬀective cost of leverage thus also includes thecost of the line.
To derive bank first-order conditions, it is helpful to define the marginal cost of leverage as
the derivative of the discounted eﬀective leverage cost (A.1) with respect to total commitments(the numerator in (8)) and the marginal benefit of collateral as the (discounted) derivative withrespect to the denominator:©
ªmc ( ) = −̂ ( ( ) + 0 ( )  ) 1 +  ̄ 
ª
©
(A.2)
mb ( ) = −̂ 0 ( ) 2 1 +  ̄ 
where  = 1 is the collateral ratio.

An extra unit invested in assets at date  that contributes to collateral earns not only
a pecuniary return, but also the collateral benefit mb( ). Similarly, committing to a unit
payable entails the extra leverage cost mc( ). Concavity implies that both mb() and mc()are decreasing in collateral since 20 () + 00 ()   0. Since the cost function  slopes up
suﬃciently fast that banks always choose   1, we also have that for a given level of leverage,
the marginal cost of leverage is higher than the marginal collateral benefit: mc() mb().
Bank first-order conditions for assets
The typical bank’s first-order conditions describe the key trade-oﬀs of portfolio and capital
structure choice. Since shareholders are risk neutral, the expected marginal benefits or costs of
all assets and liabilities are compared to the required return on equity ̂. If the portfolio of thebank is chosen optimally, the marginal benefit from any asset cannot be larger than ̂: if not,then the bank would choose to invest more. Moreover, the marginal benefit is equal to ̂ if the1bank optimally holds a positive position in the asset. If the marginal benefit is below ̂ thenthe bank does not invest: it is better to pay dividends instead.
Consider the first-order condition for overnight lending which not only earns the real interest
rate, but also the collateral benefit. The return on equity must be higher than the marginal
benefit:̂  ≥  − +1 + mb ( ) (A.3)
with equality if the bank lends overnight. In the latter case, low real interest rates imply thatovernight credit is costly, so banks optimally choose lower collateral. Put diﬀerently, highly
levered banks obtain a high benefit from overnight lending as collateral and thus require alower return on credit.
The first-order condition for trees is similar. The real rate of return on tree  held by banksis +1= log(+1 + +1 +1 ) −  +1 . Since trees also deliver collateral benefits, we musthave̂  ≥ +1
+  mb ( ) with equality for trees held by the bank. Since the collateral benefit lowers the return on
trees held by the bank, it raises the price of the trees relative to payoﬀ. Indeed, holding fixed the+ +1 +1 , the price of a tree held by the bank is  = (+1 +³ payoﬀ +1 ´

+1 +1 ) ̂  −  mb ( ) . In particular, if banks holding tree  are more levered, then those
trees are more valuable collateral, their cash flows are discounted at a lower rate and their priceis higher.

Reserves diﬀer from overnight lending and trees in that they not only provide returns  − +1 and collateral benefits, but also liquidity benefits — they can be used for payments. The
liquidity benefit depends on the Lagrange multiplier on the liquidity constraint (4). Writing 
for that Lagrange multiplier divided by the price level, the first order condition for reserves is£¤−̂  − +1
 +1 (A.4)
̂  ≥  −  +1 + mb ( ) + (1 + ) 
The liquidity£ benefit¤ (the second term) is higher the higher the expected discounted Lagrange
multiplier  +1 and the more payments can be made per dollar of reserves (higher ).

Bank first-order conditions for liabilities & credit lines

Consider now banks’ choice to finance themselves with deposits or overnight credit. If
the capital structure of the bank is chosen optimally, the marginal cost of any liability typecannot be lower than ̂  , the cost of issuing equity: if not, banks would choose to borrow more.
Moreover, the marginal cost is equal to ̂  if the bank holds a positive position in that particularliability type. If the marginal cost is above ̂  then the bank does not issue the liability: it isbetter to issue equity instead.
The first-order condition for deposits says that the equity return must be smaller than thesum of the real deposit rate plus the marginal leverage and liquidity costs of deposits£¤−̂ − +1(1 + )  +1 +1 (A.5)
̂  ≤  −  +1 +  mc ( ) + with equality if the bank issues deposits. Leverage costs increase with overall leverage throughmc( ) and are also scaled by the parameter  which makes deposits cheaper than other2borrowing. Liquidity costs arise in those states next period when positive liquidity shocks
+1  0 coincide with a binding intraday credit limit +1  0.
The first-order condition for overnight borrowing says that the equity return must be smaller
than the sum of the real overnight rate plus the marginal leverage cost, less the liquidity benefit
provided by overnight credit:̂  ≤  −  +1 + mc ( ) −  (1 + ) 

(A.6)

For banks that borrow overnight, the condition holds with equality. This can happen because
for those banks the intraday credit limit binds and   0. In contrast, banks with suﬃcient
reserves have  = 0 and do not borrow.
Finally, consider banks’ choice to extend credit lines, a form of implicit liability. The fee
earned for extending the line must at least compensate the bank for the leverage cost occurred
as well as the expected liquidity costs
£
¤
(A.7)
 ≥  mc ( ) + ̄−̂ −+1  +1 +1 As discussed above, we focus on equilibria with  −  +1 = ̂  −  so that deposits and credit
lines are equivalent from the perspective of households. Comparing (A.5) and (A.7), the two
payment instruments are then also equivalent from the perspective of banks.

A.2

Characterizing equilibrium

In this section, we derive a diﬀerence equation that summarizes the equilibrium dynamics. The
variables are the interest rates  and  , tree prices  , the nominal price level  , aggregate
payment instruments  +  , the liquidity ratio  (which is inversely proportional to the“money multiplier” ( +  ) ), the leverage ratio  , the marginal cost of leverage mc( ),the marginal benefit of collateral mb( ) and the marginal benefit of liquidity ̃ which is also
equated across banks in equilibrium.
We use the notation ̃ to denote the liquidity benefit that obtains per real dollar for any
bank for which the liquidity constraint binds. The notation  is the Lagrange multiplier in thefirst-order condition of an individual bank (A.4). Of course, in the cross section of banks, there
are banks for whom the constraint does not bind and for those banks we have  = ̃ = 0.
Asset pricing
Consider participation in overnight credit and tree markets. In an equilibrium with positive consumption, banks must supply payment instruments. Since payment instruments entail
leverage costs, banks obtain a positive collateral benefit from assets they are eligible to hold.
As a result, households do not invest in any asset markets that banks can invest in: at least one
bank will bid up the price of any asset accessible to banks until its return is below the discountrate and the asset is unattractive to households.
The collateral ratio is the same for all active banks in equilibrium15 . The equilibrium rateof return on a tree  accessible to banks is then related to the aggregate collateral ratio by the15

Indeed, suppose bank 1 has higher leverage than bank 2, and hence a higher collateral benefit  (). In
equilibrium, bank 2 does not invest in illiquid assets (trees or overnight credit); due to the lower collateral3

bank “Euler equation”̂  = +1
+  mb ( ) 

(A.8)


The flip side of lower returns is higher prices. Indeed,
R let  denote the sum of dividends
on all trees accessible to banks at date  and let  = Θ    denote the real value of
those trees. In equilibrium, the value of these bank trees reflects not only the present value thefuture payoﬀ on the tree, but also the collateral value: = −̂ (+1+ 
+1 +  mb ( )  )

(A.9)

Using our conventions, pricing eﬀectively works as if payoﬀs are discounted at the lower rate̂  −  mb( ) Liquidity benefits and reserve scarcity

Bank liquidity management and their activity in the overnight credit market depends crucially on the scarcity of reserves. Indeed, with abundant reserves, the first order condition forreserves (A.4) implies that the real rate on reserves is equal to ̂  −mb( )  From the first ordercondition for overnight lending (A.3), this is the same overnight interest rate that obtains if
banks lend in the overnight market — if reserves are abundant, they are a perfect substitute toovernight paper and must earn the same interest rate.16
The overnight interest rate is connected to collateral holdings through a bank Euler equation
analogous to (A.8),
̂  − ( −  +1 ) = mb ( ) (A.10)

As long as there is some government debt  , this equation holds in any equilibrium, whether
reserves are abundant or not. At the same time, the inequality (A.3) implies that households
never participate in the overnight market — as with banks trees, banks bid up the price ofovernight paper to the point where the asset is unattractive to households. In particular, aregime of abundant reserves has the property that reserves and overnight credit are perfect
substitutes from the perspective of banks, but neither asset is ever held by households.
Consider a bank that must borrow at the current date, that is, it receives a liquidity shock̃ beyond the liquidity ratio  =  (1 + ) ̄( +  ) from (5). The bank’s liquidity benefit
from overnight credit follows from (A.3) and (A.6):
̃ (1 + ) = mc ( ) − mb ( )  0

(A.11)

Tapping the overnight market entails both a leverage cost, and an opportunity cost in terms offoregone collateral value. A banking system with less collateral thus faces a larger penalty ofrunning out of reserves.
The spread between overnight and reserve rate reflects the expected liquidity benefit ofholding reserves. Substituting into the first-order condition for reserves, we obtain−̂ − +1
 − 
̃+1  = (1 −  (+1 )) (A.12)

benefit it is willing to pay strictly less than bank 1 for those assets. To be active, bank 1 must hold some assets,so suppose it holds only reserves. However, the liquidity benefit it earns from reserves cannot be higher thanthat of the more leveraged bank 2, a contradiction.
16
Of course, reserves still flow across banks. In particular an amount of reserves ̄̄ ( + )  (1 + ) stillserves to buﬀer liquidity shocks. However reserves beyond this amount are equivalent to overnight paper thatcannot be used to handle payments instructions.

4

A liquidity benefit obtains only when the withdrawal shock exceeds +1 , that is, with probability 1 −  (+1 ). In this case, a bank holding an extra dollar of reserves saves the excess costof overnight lending relative to equity.
In equilibria with scarce reserves   
 , the spread (A.12) shows how banks choose thereserve-deposit ratio as a function of interest rates and leverage. The interest rate  plays a dualrole here. On the one hand, it aﬀects the opportunity cost of reserves. Indeed, holding fixed
the liquidity benefit, the equation works much like a money demand equation: if the overnight
rate is higher, then it is more costly to hold reserves and banks choose smaller  , or fewer
reserves per dollar of deposits and credit lines. On the other hand, the interest rate aﬀects theliquidity benefit itself: holding fixed the collateral ratio  and the spread, a higher interest
rate increases the liquidity benefit of reserves and leads banks to choose more reserves to avoid
the higher penalty of running out.
Liquidity and leverage
Bank leverage depends on how much of each source of collateral is available, and how muchbanks must borrow apart from issuing deposits. In particular, interbank credit contributes bothto collateral (for lender banks) and raises costs (for borrower banks). Given collateral  and aliquidity ratio  ≤ ̄, the equation for interbank borrowing (6) delivers the ratio of outstanding
interbank credit to transactions
Z ̄ ³
´
³ ´
̄
̄

 ( ) ̃ −   ̃ =:=(A.13)
 + 
1 +  1+The function  is decreasing in  : if interest rates are such that banks hold a lot of reserves,then  is high and banks rarely run out of reserves, so outstanding interbank credit is low. In
this sense, reserves and overnight are substitutes in liquidity management.

Collecting promises due to payment instruments, real reserves, trees and interbank credit,
as well as collateral in the form of reserves, government debt, trees, and interbank credit,
equilibrium collateral satisfies
 =


̄
+   + ( +  ) 1+ ( )
+1 + +1̄ (+1 + +1 ) + ( +  ) 1+ ( )



(A.14)

Holding fixed the value of collateral from outside the banking system — reserves, government
debt and trees — scarcity of reserves requires more leverage to support a given quantity oftransactions. Indeed, we have assumed a cost function  such that the economy operates in
the range −1 =  ≤ 1; the presence of an interbank market adds an equal amount of debt andcollateral and hence increases leverage.
The cost of payment instruments
From banks’ first-order condition, the equilibrium rate on credit lines satisfiesZ ̄³ ´̄ −̂ −+1 =  mc ( ) +̃+1
̃  ̃ 

1+


(A.15)The two terms represent a leverage and a liquidity component. When reserves are abundant,
we have   ̄ so the liquidity component is zero and the cost of a credit line simply reflects
banks’ cost of leverage.
5

When reserves are scarce, banks incur additional costs when they run out of reserves. Those
costs are larger if velocity is higher and there is less netting among banks. They further dependon the marginal benefit of liquidity as well as on the liquidity shock the bank receives in thenext period. The costs banks incur when providing payment instruments also lower the deposit
17
rate 
 −  +1 by the same amount.

Equilibrium

An equilibrium is characterized by the seven equations (10) and (A.9)-(A.12) and (A.14)(A.15) together with equation (5) that defines the liquidity ratio as well as the two equations(A.2) that define marginal leverage cost and collateral benefit. If policy is described by a pathfor the money supply, then no additional equations are needed. The ten equations determine
ten endogenous variables        +   mc() mb() and ̃. The diﬀerence equation
characterizing equilibrium has only two state variables: aggregate nominal reserves  andaggregate nominal payment instruments  +  .
Alternatively, we can consider equilibria that obtain when policy is described by an interest
rate rule. Since output is exogenous, we focus on rules that depend on inflation,  =  (  )for some function . The interest rate rule then provides an additional equation, and  isan additional endogenous variable. Since the interest rate rule conditions on inflation at theprevious date, the past price −1 also becomes part of the initial conditions of the diﬀerence
equation.
In order to compute equilibrium consumption and welfare at date 0, we need to know additional predetermined variables: initial leverage ratios −1 and −1 for banks and the government,
respectively as well as banks’ outstanding overnight0 and lending 0 . Those vari¡ borrowing¢
ables are needed to compute the real resources  −1 0 −1 and  (−1 ) (0 + 0 + 0 )−1
purchased by the government and banks to pay leverage costs at date 0. At the same time,they do not aﬀect any endogenous variables other than consumption which does not enter thediﬀerence equation; as a result there is no need to treat them explicitly as state variables.
Consider equilibria in which reserves are abundant at all times, so  ≥ ̄ for all . The
three liquidity management equations (A.11)-(A.13) are then redundant, the variables  and 
can be removed from the system and the overnight interest rate achieves its lower bound  =  .
The remaining six equations then determine      mc() and mb().
Approximating equilibria with small rates of return
We have simplified formulas above by assuming that rates of return are small decimals sotheir products can be ignored. To guarantee that small rates to obtain in equilibrium, we scale
the leverage cost function and bound the real rate on reserves. Let ̄ denote a lower boundon the real rate on reserves, a small decimal number. In what follows, we guarantee this boundby appropriate assumptions on monetary policy.
The highest possible leverage ratio that can obtain in any equilibrium then satisfies
¡ ¡ ¢
¡ ¢
¡ ¢ ¢
̂ − ̄ = 0 ̄ ̄2 (1 +   ̄ + 0 ̄ ̄ This leverage ratio corresponds to an equilibrium where the reserve rate hits the bound and17

The term “deposit rate” should be interpreted broadly here: it is the only cost endusers pay for payments
serives in our model, since we do not explicitly model other costs such as account and transaction fees.

6reserves are abundant. If reserves were scarce, the benefit of reserves would include liquidity
components and leverage would have to be lower.
¡ ¢We now choose the cost function  such that ̄  1 and 0 ̄ ̄ is a small decimal number,much like ̂ − ̄ . It follows that the marginal cost of leverage mc() and the marginal benefitof collateral mb() are also small decimal numbers in any possible equilibrium. Moreover, theeﬀect of the interest rate on credit lines on mc() and mb() is second order. We thus use theadditional approximationmc () =  () + 0 () 
mb () = 0 () 2 

(A.16)

where both mc and mb are decreasing function of the collateral ratio  = 1. We thus abstractfrom eﬀects of the interest rate  on banks’ cost of leverage. Those eﬀects are small and noteconomically interesting; omitting them altogether makes for cleaner formulas below.
The fact that equilibrium mc and mb are small has several convenient implications. First,
we can omit the factor −̂− on the right hand sides of (A.12) and (A.15). The timing of
the liquidity benefit has only a second-order eﬀect. Second, combining (A.16) and (A.10), we
obtain a one-to-one relationship between the real overnight interest rate and the collateral ratio:banks increase collateral if interest rates are higher. We use both properties when characterizing
equilibria further below.

A.3

Steady stateThis section derives the equations characterizing steady state. We assume the same exogenous
growth rate  for the nominal quantities  and  so the ratio   is constant over time.
With constant rates of return, the marginal rate of substitution of wealth across dates equals
the discount rate, that is ̂  = . Moreover, the key ratios chosen by banks, collateral  andliquidity , are constant over time.
Since output is fixed, payment instruments and the price level also grow at the rate .
Using the quantity equation (10), the constant liquidity ratio  = (1 + )  ̄ ( +  ) canbe written as
1+
=

(A.17)
The steady state real amount of overnight credit is also linked directly to transactions andreserves. Combining (A.13) and (A.17), we have̄
 () = + 
1+
The Euler equation 11 for the overnight real interest rate follows directly from (A.10). The
steady value of bank trees is found from (A.9) as
 =



 − mb ()7Consider now the capital structure curve relationship (12). Substituting (10), (A.9) and(??) into (A.14), we find that steady state real reserves and collateral are linked by
³
´

()


1 +  1+
+  (− mb())
+ 1+
=

 ()

+
̄
1+
In a steady state with nonzero inflation , overnight credit  should be premultiplied by  .
We omit this factor here in line with our focus on small rates of return.
Consider now the equations describing liquidity management. To derive the liquidity management curve (13), we substitute (A.10) and (A.11) into (A.12) and obtain − ( − ) = mb () + (1 −  ()) (mc () − mb ()) Finally, to derive equation (14) — used in the text to discuss interest rate policy — we substitute(A.10) and (A.11) into (A.12) to get −  = (1 −  ()) (mc () − mb ()) 

8