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# Central Bank Digital Currency with Collateral-constrained Banks

Hanfeng Chen  Maria Elena Filippin[1]${}^{\,,\,}$

(April 18, 2024)

###### Abstract

We analyze the risks to financial stability following the introduction of a
central bank digital currency (CBDC). The CBDC competes with commercial bank
deposits as the household’s source of liquidity. We revisit the result in the
literature regarding the equivalence of payment systems by introducing a
collateral constraint for banks when borrowing from the central bank. When
comparing two equilibria with and without the CBDC, the central bank can
ensure the same equilibrium allocation and price system by offering loans to
banks. However, to access loans, banks must hold government bonds as
collateral at the expense of extending credit to firms, and the central bank
assumes part of the credit-extension role. Thus, in the equivalence analysis,
while the CBDC introduction has no real effects on the economy, it does not
guarantee full neutrality as it affects banks’ business models. We also
analyze the dynamics of an increase in the CBDC and show that the CBDC not
only does not cause bank disintermediation and financial instability but may
foster an expansion of bank credit to firms.
JEL codes: E42, E58, G21
Keywords: Central bank digital currency, collateral-constrained bank,
equivalence, credit extension††This paper previously circulated under the
title “Unveiling the Interplay between Central Bank Digital Currency and Bank
Deposits”. We are grateful to Mikael Carlsson, Daria Finocchiaro, Ulf
Söderström, and Karl Walentin for their invaluable guidance and continuous
support. For their useful feedback and comments, we also thank Itai Agur,
Ragna Alstadheim, Darrel Duffie, Rod Garratt, Ricardo Lagos, Cyril Monnet,
Dirk Niepelt (discussant) and Joshua Weiss (discussant), as well as conference
and seminar participants at Uppsala University, New York University, the 2nd
PhD Workshop in Money and Finance at Sveriges Riksbank, the 2nd Conference on
the Economics of CBDC by the Bank of Canada and Sveriges Riksbank and the
CEPR-ECB Conference on the Macroeconomic Implications of CBDCs.

##  1 Introduction

Digital currencies have been around for a while, but their potential
significance in the global economy has increased recently due to a growing
demand for digital payment methods for retail purposes and the gradual decline
of the use of cash for transactions in many economies [see, e.g., [13]].
Besides the private digital means of payment currently in circulation, many
central banks have been investigating the possibility of launching a central
bank digital currency (CBDC). CBDCs are central bank liabilities denominated
in an existing unit of account that serve as a medium of exchange and a store
of value.111As defined by the Committee on Payments and Market Infrastructures
of the Bank of International Settlements [[9]]. Were CBDCs to be issued to
retail customers (i.e., households), they would likely be a digital form of
cash that share features with banknotes, as they are universally accessible
but in a digital form.

Alongside an intense policy debate, a growing academic literature on the
broader economic implications of CBDCs has emerged. A primary concern for
central banks when considering the issuance of a CBDC is the risk to financial
stability, intended as the risk of the CBDC disintermediating the banking
sector as households substitute the CBDC for bank deposits, potentially
leading to financial instability. Introducing a CBDC will likely alter the
equilibrium in the real economy, as it will represent a novel payment
alternative to cash and commercial bank deposits. The macroeconomic
consequences of introducing a CBDC will impact individuals and financial
institutions. This paper analyzes the implications of introducing a retail
CBDC, particularly concerning its relationship with bank deposits.

The recent literature establishes an equivalence result between different
payment systems. [6] consider a simplified scenario without reserves and
resource cost of providing liquidity, while [16] includes a reserves layer and
shows that introducing CBDC has no real effects on the economy if the private
and the public sectors are equally efficient in operating payment systems. For
this to happen, the central bank must refinance the bank at a lending interest
rate that supports the bank’s original portfolio position so that central bank
funding exactly replaces the lost deposits for the bank. [16] assumes central
bank loans are extended against no collateral. However, the collateral
requirement imposed by central banks when lending to commercial banks is
potentially important for how introducing a CBDC may affect the banking sector
and the real economy.222See, e.g., [7] and [21]. In practice, central banks
lend to commercial banks (i.e., discount window lending) against collateral to
support the liquidity and stability of the banking system. The liquidity
provided by central banks helps financial institutions to manage their
liquidity risks efficiently. These loans are issued at an administered
discount rate and must be collateralized to the satisfaction of the issuing
central bank. In the euro area, banks can make use of the marginal lending
facility, which enables banks to obtain overnight liquidity from the European
Central Bank against sufficient eligible assets. In the United States, the
Federal Reserve offers different types of discount window credit, which must
be collateralized. The discount window mechanism has become increasingly
important after the Global Financial Crisis. In this paper, we will revisit
this equivalence result in the literature and explore its implication in terms
of financial disintermediation by introducing a financial friction for central
bank lending to banks (i.e., the collateral requirement).333In Appendix C, we
will revisit the equivalence result by considering different degrees of
substitutability between CBDC and bank deposits (i.e., imperfect
substitutability).

This paper addresses the potential risk to financial stability following the
introduction of a CBDC. Specifically, we investigate if the issuance of the
CBDC leads to disruptions in financial markets, thereby positing a risk to
financial stability due to bank disintermediation rather than bank runs. To
address this concern, we build on [16] and develop a model with a CBDC and
bank deposits, adding a collateral requirement for central bank lending to
banks. The framework is an extension of the model by [19] that embeds a
banking sector, bank deposits, government bonds, reserves, and a CBDC into the
baseline real business cycle model. Households value goods, leisure, and the
liquidity services that deposits and CBDC provide. Non-competitive banks
invest in capital, reserves, and government bonds and fund themselves through
either deposits or borrowing from the central bank. Firms produce using labor
and physical capital. Finally, the consolidated government collects taxes,
pays deposit subsidies, invests in capital, lends to banks against collateral,
and issues CBDC and reserves.

After building our model, we use it to analyze two different perspectives: a
static study comparing two equilibria with and without CBDC and a dynamic
analysis of the economy’s responses to a temporary increase in the CBDC.

In the first part of the paper, we revisit the result in the literature
regarding the equivalence between payment systems when introducing a
collateral constraint for central bank lending to banks. Our findings reveal
that the introduction of CBDC has no real effects on the economy as long as
(1) CBDC and deposits are perfect substitutes, (2) the resource cost per unit
of effective real balances is the same for CBDC and deposits and (3) the
central bank offers a loan rate that renders the non-competitive banks
indifferent to the introduction of the CBDC. Our equivalent central bank loan
rate is lower than the one obtained in [16] because of the collateral
requirement the bank must respect when borrowing from the central bank. In
particular, when the collateral constraint becomes more restrictive, the
central bank loan rate must be lower. However, different from the results in
the literature, we find that while the central bank compensates for the lower
deposit funding by lending to the bank, it assumes part of the credit
extension role from the bank, which now holds government bonds as collateral
and less capital to extend credit. In other words, the central bank insulates
bank profits but not the bank’s business model. It follows that, although the
new policy has no real effects on the economy, it does not guarantee full
neutrality as it affects the bank’s business model.

In the second part of the paper, we explore the potential threat to financial
stability should CBDC crowd out deposits. We depart from the equivalence
analysis, where we compared two equilibria with and without CBDC, and study
the economy’s responses to a temporary increase in the CBDC, where the central
bank issues CBDC equal to a fraction of steady-state output. We find that the
increased amount of CBDC in circulation boosts banks’ capital. This result
suggests that banks expand credit intermediation to firms. Interestingly, the
CBDC does not lead to bank disintermediation or crowd out of deposits, but it
expands bank activity.

Related literature. Our work contributes to the recent literature examining
the impact of the introduction of CBDC on commercial banks. For instance, [8]
develop a micro-founded general equilibrium model calibrated to the U.S.
economy and find that a CBDC expands bank intermediation when the price of
CBDC falls within a certain range while leading to disintermediation if its
interest rate exceeds the upper limit of that range. In another study, using a
dynamic banking model, [20] assume that banks rely on deposits and wholesale
funding and that the latter can potentially substitute deposit loss. Depending
on whether the CBDC pays interest or not, the synergies between deposits and
lending can attenuate the impact of CBDC. In another work, [12] consider a
competitive market and show that a deposit-like CBDC tends to crowd out bank
deposits but, at the same time increases the aggregate stock of liquid assets
in the economy, promoting more efficient levels of production and exchange and
ultimately raising welfare. Specifically, our study is closer to the
literature examining the CBDC introduction when banks borrow from the central
bank subject to a collateral requirement. In recent work, [7] construct a
quantitative euro area dynamic stochastic general equilibrium (DSGE) model,
where banks must post government bonds as collateral to borrow from the
central bank. They investigate the transmission channels of the issuance of
CBDC to bank intermediation, finding a bank disintermediation effect with
central bank financing replacing deposits, and government bonds displacing
reserves and loans. Along similar lines, [3] use a DSGE model to investigate
the macroeconomic effects of CBDC when the central bank administrates the CBDC
rate and collateral and quantity requirements. Their findings indicate that a
more ample supply of CBDC reduces bank deposits, while stricter collateral or
quantitative constraints reduce welfare but can potentially contain bank
disintermediation. The latter effect is particularly true when the elasticity
of substitution between bank deposits and CBDC is low. [21], on the other
hand, explores the effects of the introduction of CBDC using a model of
multiple means of payment. In his model, the CBDC is a more efficient payment
instrument than cash, but it lengthens the central bank’s balance sheet,
creating collateral scarcity in the economy. Differently from these works, our
study investigates the implications of CBDC issuance on bank intermediation
using a real business cycle model that is closely connected to the baseline
macroeconomic workhorse model, building on [16] and embedding a collateral
requirement for central bank lending to banks. Our findings suggest that
introducing CBDC does not cause bank disintermediation but may foster an
expansion of credit extension to firms.

Our study also contributes to the literature on the equivalence of payment
systems. Existing work by [6] and [16] propose a compensation mechanism where
the household’s shift from deposits to CBDC can be offset by central bank
lending to banks. However, these models abstract from the collateral
constraint for central bank lending that is common in practice. Notably, [18]
show that when banks are required to hold liquid assets to back their deposits
and face asset management costs, the equivalence between alternative payment
instruments breaks down, even if banks can be refinanced directly by the
central bank. In light of this, we revisit the equivalence result by
incorporating a collateral constraint for banks. We derive a new central bank
lending rate that depends on the restrictiveness of the collateral
requirement. Our findings reveal that the more restrictive the collateral
constraint, the lower the loan rate the central bank must post. Moreover, to
access central bank loans, banks must hold government bonds as collateral at
the expense of extending credit to firms, and the central bank assumes part of
the credit-extension role. Thus, while the CBDC introduction has no real
effects on the economy, it does not guarantee true neutrality as it affects
banks’ business models.

The rest of the paper is organized as follows. Section 2 describes the model.
Section 3 revisits and discusses the equilibrium analysis of the equivalence
between operating payment systems. Section 4 characterizes the general
equilibrium in which the household holds CBDC and deposits and discusses the
dynamic effects of a temporary increase in the CBDC. Section 5 concludes.

##  2 Model with CBDC and collateral-constrained banks

The model is based on [16] and describes an economy with a banking sector and
CBDC in the absence of nominal rigidities. The CBDC and deposits provide
direct utility. We depart from that framework by considering a collateral
constraint for banks when borrowing from the central bank. There is a
continuum of mass one of homogeneous infinitely-lived households who own a
succession of two-period-lived banks and of one-period-lived firms. The
consolidated government determines monetary and fiscal policy.

###  2.1 Households

The representative household wants to maximize the discounted felicity
function $\mathcal{U}$, which is increasing, strictly concave and satisfies
Inada conditions. Subject to its budget constraint, equation (1), the
household takes wages, $w_{t}$; returns on asset $i$, $R^{i}_{t}$; profits,
$\Pi_{t}$; and taxes, $\tau_{t}$ as given and solves

$\displaystyle\max_{{\\{c_{t},x_{t},k_{t+1}^{h},m_{t+1},n_{t+1}\\}}_{t=0}^{\infty}}\mathbb{E}_{0}\sum_{t=0}^{\infty}\beta^{t}\mathcal{U}(c_{t},x_{t},z_{t+1})$
s.t. $\displaystyle
c_{t}+k_{t+1}^{h}+m_{t+1}+n_{t+1}+\tau_{t}=w_{t}(1-x_{t})+\Pi_{t}+k_{t}^{h}R^{k}_{t}+m_{t}R^{m}_{t}+n_{t}R^{n}_{t},$
(1) $\displaystyle k_{t+1}^{h},m_{t+1},n_{t+1}\geq 0,$

where $\beta\in(0,1)$ is the positive discount factor, $c_{t}$ and $x_{t}$
denote household consumption of the good and leisure at date $t$,
respectively; $k_{t+1}^{h}$ is capital at date $t+1$; and
$z_{t+1}=z(m_{t+1},n_{t+1})$ are effective real balances carried from date $t$
to $t+1$. Effective real balances are a function of both CBDC, $m_{t+1}$, and
bank deposits, $n_{t+1}$.444The household values liquidity, as suggested by
the money in the utility function specification. In this setting, it only
matters that the household demands liquidity services, not why they do. The
household consumes, pays taxes, invests in capital, and has real balances, out
of wage income, distributed profits and the gross return on the portfolio.

We focus on interior solutions for capital, CBDC, and deposits. To express the
Euler equations for CBDC and deposits in a more compact form, we define the
risk-free rate as

$\displaystyle R^{f}_{t+1}=\frac{1}{\mathbb{E}_{t}\Lambda_{t+1}},$ (2)

where $\Lambda_{t+1}$ is the household’s stochastic discount factor:

$\displaystyle\Lambda_{t+1}=\beta\frac{\mathcal{U}_{c}(c_{t+1},x_{t+1},z_{t+2})}{\mathcal{U}_{c}(c_{t},x_{t},z_{t+1})}.$
(3)

Also, we define the liquidity premium, or interest spread, on asset $i$ as

$\displaystyle\chi^{i}_{t+1}=1-\frac{R^{i}_{t+1}}{R^{f}_{t+1}},\qquad
i\in\\{m,n\\}.$ (4)

The spread on an asset $i$ denotes the household’s opportunity cost of holding
said asset. A positive deposit spread, for instance, shows the interest return
that the household forgoes by holding deposits. The household is willing to
accept a lower return on deposits due to the liquidity service they provide.
Assuming that the interest rates on CBDC and deposits are risk-free, we can
summarize the first-order conditions as

$\displaystyle x_{t}:\qquad$
$\displaystyle\mathcal{U}_{x}(c_{t},x_{t},z_{t+1})=\mathcal{U}_{c}(c_{t},x_{t},z_{t+1})w_{t},$
(5) $\displaystyle k_{t+1}^{h}:\qquad$ $\displaystyle
1=\mathbb{E}_{t}R^{k}_{t+1}\Lambda_{t+1},$ (6) $\displaystyle m_{t+1}:\qquad$
$\displaystyle\mathcal{U}_{c}(c_{t},x_{t},z_{t+1})\chi^{m}_{t+1}=\mathcal{U}_{z}(c_{t},x_{t},z_{t+1})z_{m}(m_{t+1},n_{t+1}),$
(7) $\displaystyle n_{t+1}:\qquad$
$\displaystyle\mathcal{U}_{c}(c_{t},x_{t},z_{t+1})\chi^{n}_{t+1}=\mathcal{U}_{z}(c_{t},x_{t},z_{t+1})z_{n}(m_{t+1},n_{t+1}),$
(8)

The household’s first-order conditions have standard interpretations. The
leisure choice condition (5) equalizes the marginal benefit of leisure,
$\mathcal{U}_{x}(c_{t},x_{t},z_{t+1})$, with its marginal cost in the form of
reduced consumption due to less labor income,
$\mathcal{U}_{c}(c_{t},x_{t},z_{t+1})w_{t}$. The Euler equation for capital
(6) dictates that the household saves in capital to the point where the
marginal cost of saving in terms of consumption,
$\mathcal{U}_{c}(c_{t},x_{t},z_{t+1})$, equals its expected discounted return,
$\beta\mathbb{E}_{t}\mathcal{U}_{c}(c_{t+1},x_{t+1},z_{t+2})R_{t+1}^{k}$. The
first-order conditions for CBDC and deposits, equations (7) and (8)
respectively, show that the household demands the liquid asset $i\in\\{m,n\\}$
to the point where its marginal benefit
$\mathcal{U}_{z}(c_{t},x_{t},z_{t+1})z_{i}(m_{t+1},n_{t+1})/\mathcal{U}_{c}(c_{t},x_{t},z_{t+1})$
equals its opportunity cost in terms of foregone interest, $\chi^{i}_{t+1}$.
Combining equations (7) and (8) we also see how the household trades off CBDC
and deposits:

$\displaystyle
z_{m}(m_{t+1},n_{t+1})\chi^{n}_{t+1}=z_{n}(m_{t+1},n_{t+1})\chi^{m}_{t+1}.$
(9)

Equation (9) shows that the household allocates between CBDC and deposits so
that the marginal rate of substitution,
$z_{m}(m_{t+1},n_{t+1})/z_{n}(m_{t+1},n_{t+1})$, equals the relative price,
$\chi^{m}_{t+1}/\chi^{n}_{t+1}$.

###  2.2 Banks

One of the often cited reasons in the literature for introducing a CBDC is
bank market power [see, e.g., [2], [11]]. Specifically, banks offer lower
deposit rates to extract rents, and households are willing to accept this
markdown as they value the liquidity service provided by deposits. A CBDC
could compete with bank deposits, reducing banks’ market power. Our set-up
follows [16] and assumes that each bank is a monopsonist in its regional
deposit market, such that the household in a region can only access the
regional bank. A bank lives for two periods, and at date $t$ issues deposits,
$n_{t+1}$, borrows from the central bank, $l_{t+1}$, and collects government
subsidies on deposits at rate $\xi_{t}$. It invests in reserves, $r_{t+1}$,
government bonds, $b_{t+1}$, and capital, $k^{b}_{t+1}$.555Bank’s capital is
defined as $k^{b}_{t+1}=n_{t+1}+l_{t+1}-r_{t+1}-b_{t+1}$. Alternatively, the
bank can invest in loans to firms that eventually fund physical capital
accumulation. Without loss of generality, we abstract from bank equity. We
follow [7] and assume that the bank is subject to a collateral requirement
such that the loans they get from the central bank can not exceed a fraction
$\theta_{b}$ of its government bond holdings. In this setting, government
bonds are the only asset that can be pledged as collateral. For
simplification, we abstract away from interbank loans with collateral. Holding
government bonds gives liquidity benefits to the bank since they can use their
holdings to obtain funding from the central bank. In other words, the bank is
willing to forego a spread on the risk-free rate because of the collateral
benefits of holding government bonds. This “convenience yield” of government
bonds reflects the additional benefits the bank derives from holding these
bonds beyond their financial yield. Therefore, government bonds are
remunerated at a slightly lower rate than the risk-free rate.

The operating costs in the retail payment system, $\nu$, are a decreasing
function of the bank’s reserve-to-deposit ratio, $\zeta_{t+1}$. This is
analogous to a binding minimum reserves requirement, as larger reserve
holdings relative to deposits lower the bank’s operating costs. We also allow
$\nu$ to decrease with the stock of reserves and deposits of other banks,
$\bar{\zeta}_{t+1}$, so as to capture positive externalities of reserve
holdings.666[16] uses a cost function in the form
$\nu+\omega_{t}(\zeta_{t+1},\bar{\zeta}_{t+1})$, where $\nu$ is the resource
cost per unit of deposit funding, and $\omega$ represents the bank’s resource
costs of liquidity substitution. To simplify the analysis, we make some
assumptions which imply that in equilibrium $\zeta_{t+1}=\bar{\zeta}_{t+1}$,
and reserves are strictly positive if and only if deposits are strictly
positive: When a bank holds no deposits, its operating costs are null, and
when all other banks have no deposits, the bank’s operating costs are large
but bounded. In this way, we rule out asymmetric equilibria in the bank’s
deposits and other banks’ deposits. Otherwise, the operating cost function,
$\nu(\zeta_{t+1},\bar{\zeta}_{t+1})$, is strictly decreasing in both
arguments, strictly convex, and satisfies $\nu_{\zeta\bar{\zeta}}=0$ and
$\nu_{\zeta\zeta}\geq\nu_{\bar{\zeta}\bar{\zeta}}$, as well as
$\lim_{\zeta_{t+1}\to 0}\nu_{\zeta}=\infty$.

The bank chooses the quantity of deposits and central bank loans subject to
the deposit funding schedule of the household.777In the model, the central
bank’s loan funding schedule replicates the household’s deposit funding
schedule. This assumption plays a crucial role in the context of the
equivalence analysis. Since the bank acts as a monopsonist in its regional
deposit market, it takes the deposit funding schedule (rather than the deposit
and the central bank loan rates) as given. The program of the bank at date $t$
reads

$\displaystyle\max_{n_{t+1},l_{t+1},r_{t+1},b_{t+1}}\Pi^{b}_{1,t}+\mathbb{E}_{t}\left[\Lambda_{t+1}\Pi^{b}_{2,t+1}\right]$
s.t.
$\displaystyle\Pi^{b}_{1,t}=-n_{t+1}\left(\nu(\zeta_{t+1},\bar{\zeta}_{t+1})-\xi_{t}\right),$
(10)
$\displaystyle\Pi^{b}_{2,t+1}=(n_{t+1}+l_{t+1}-r_{t+1}-b_{t+1})R^{k}_{t+1}$
$\displaystyle\qquad\qquad+r_{t+1}R^{r}_{t+1}+b_{t+1}R^{b}_{t+1}-n_{t+1}R^{n}_{t+1}-l_{t+1}R^{l}_{t+1},$
(11) $\displaystyle l_{t+1}\leq\theta_{b}\frac{b_{t+1}}{R^{l}_{t+1}},$ (12)
$\displaystyle R^{n}_{t+1},R^{l}_{t+1}\text{ perceived endogenous},$
$\displaystyle n_{t+1},l_{t+1},b_{t+1}\geq 0,$

where

$\displaystyle\zeta_{t+1}\equiv\frac{r_{t+1}}{n_{t+1}},\qquad\bar{\zeta}_{t+1}\equiv\frac{\bar{r}_{t+1}}{\bar{n}_{t+1}},$

and $\Pi^{b}_{1,t}$, $\Pi^{b}_{2,t+1}$ denote the cash flow generated in the
first and second periods of the bank’s operations, respectively.

We focus on interior solutions for deposits, loans, and government bonds, and
we make use of the risk-free rate and the household’s first-order condition
for capital, equations (2) and (6), respectively. Also, we define the
elasticity of the asset $i$ with respect to the rate of return on $i$ as

$\displaystyle\eta_{i,t+1}=\frac{\partial i_{t+1}}{\partial
R^{i}_{t+1}}\frac{R^{i}_{t+1}}{i_{t+1}},\qquad i\in\\{n,l\\},$

and the liquidity premia on central bank loans, reserves and government bonds
as in equation (4). Let $\gamma_{t}$ denote the Lagrange multiplier associated
with the collateral constraint. The collateral constraint is binding in
equilibrium, such that888See Appendix A for the conditions under which the
collateral constraint binds. The intuition is that, with a non-binding
collateral constraint, in equilibrium $\gamma_{t}=0$ and, from the collateral
constraint condition (12),
$0\leq\theta_{b}\frac{b_{t+1}}{R^{l}_{t+1}}-l_{t+1}.$ However, from the
government bonds optimality condition (not shown here), this violates the
condition that $R^{b}_{t+1}<R^{f}_{t+1}$, so the collateral constraint must
bind in equilibrium.

$\displaystyle\gamma_{t}>0,\qquad
l_{t+1}=\theta_{b}\frac{b_{t+1}}{R^{l}_{t+1}}.$

We can write the bank’s optimality conditions as

$\displaystyle n_{t+1}:\qquad$
$\displaystyle\chi^{n}_{t+1}-\left(\nu(\zeta_{t+1},\bar{\zeta}_{t+1})-\xi_{t}-\nu_{\zeta}(\zeta_{t+1},\bar{\zeta}_{t+1})\zeta_{t+1}\right)=\frac{1}{\eta_{n,t+1}}\frac{R^{n}_{t+1}}{R^{f}_{t+1}},$
(13) $\displaystyle r_{t+1}:\qquad$
$\displaystyle-\nu_{\zeta}(\zeta_{t+1},\bar{\zeta}_{t+1})=\chi^{r}_{t+1},$
(14) $\displaystyle l_{t+1}:\qquad$
$\displaystyle\chi^{l}_{t+1}-\gamma_{t}\left(1+\frac{1}{\eta_{l,t+1}}\right)=\frac{1}{\eta_{l,t+1}}\frac{R^{l}_{t+1}}{R^{f}_{t+1}},$
(15) $\displaystyle b_{t+1}:\qquad$
$\displaystyle\gamma_{t}\frac{\theta_{b}}{R^{l}_{t+1}}=\chi^{b}_{t+1},$ (16)

where the spreads on bonds and central bank loans, $\chi^{b}_{t+1}$ and
$\chi^{l}_{t+1}$, are defined in the same way as the spreads on deposits and
CBDC given by expression (4).

We first comment on the liability side of the bank’s balance sheet, starting
with deposits. The left-hand side of equation (13) represents the marginal
profit from issuing deposits, which is given by the difference between the
bank’s gain from the positive deposit liquidity premium and the marginal cost
associated with increased deposit issuance. The right-hand side equals the
marginal cost of inframarginal deposits, as higher deposit issuance is
associated with an increased interest rate on deposits. Similarly, the
condition for central bank loans, equation (15), states that the sum of the
bank’s marginal benefits of taking on more central bank loans and the gain
coming from the positive loan liquidity premium should be equal to the
marginal cost associated with central bank loans. In fact, higher loan
holdings are associated with an increase in the interest rate on the central
bank loans. Turning now to the asset side of the bank’s balance sheet,
equation (14) equalizes the marginal benefit of reserves in the form of
reduced operating costs with the bank’s opportunity cost of reserves. Looking
at equation (16), the optimal choice of government bonds is when the bank’s
marginal costs of bond holdings are equal to the loss coming from the bank’s
lower return with a positive spread on government bonds.

Combining equations (13) and (14) yield

$\chi^{n}_{t+1}-\big{[}\left(\nu(\zeta_{t+1},\bar{\zeta}_{t+1})-\xi_{t}\right)+\chi^{r}_{t+1}\zeta_{t+1}\big{]}=\frac{1}{\eta_{n,t+1}}\frac{R^{n}_{t+1}}{R^{f}_{t+1}}.$
(13a)

This implies that the bank’s net benefit of issuing more deposits must equal
the inframarginal cost of deposits. Combining equations (15) and (16) results
in the relation

$\chi^{l}_{t+1}-\chi^{b}_{t+1}\frac{R^{l}_{t+1}}{\theta_{b}}\left(1+\frac{1}{\eta_{l,t+1}}\right)=\frac{1}{\eta_{l,t+1}}\frac{R^{l}_{t+1}}{R^{f}_{t+1}}.$
(15a)

The marginal cost of taking on more central bank loans must equal the bank’s
net benefit of taking on more loans. This is given by the difference between
the liquidity benefits given by the central bank loans and the marginal cost
associated with the collateral constraint.

###  2.3 Firms

Neoclassical firms live for one period and rent capital, $k_{t}$, and labor,
$\ell_{t}$, to produce the output good to maximize the profit, $\Pi^{f}_{t}$.
The representative firm takes wages, $w_{t}$; the rental rate of capital,
$R^{k}_{t}+\delta-1$; and the good price as given and solves

$\begin{split}&\max_{k_{t},\ell_{t}}\Pi^{f}_{t}\\\ \text{s.t.
\qquad}&\Pi^{f}_{t}=f(k_{t},\ell_{t})-k_{t}(R^{k}_{t}+\delta-1)-w_{t}\ell_{t},\end{split}$

where $f$ is the neoclassical production function. The first-order conditions
read

$\displaystyle k_{t}:\qquad$ $\displaystyle
f_{k}(k_{t},\ell_{t})=R^{k}_{t}+\delta-1,$ (17) $\displaystyle\ell_{t}:\qquad$
$\displaystyle f_{l}(k_{t},\ell_{t})=w_{t}.$ (18)

###  2.4 Consolidated government

The consolidated government collects taxes and subsidies deposits; lends to
the bank against collateral, $l_{t+1}$; invests in capital, $k^{g}_{t+1}$; and
issues CBDC and reserves. The government budget constraint reads

$\displaystyle k^{g}_{t+1}+l_{t+1}-b_{t+1}-m_{t+1}-r_{t+1}=$ $\displaystyle
k^{g}_{t}R^{k}_{t}+l_{t}R^{l}_{t}-b_{t}R_{t}^{b}-m_{t}R^{m}_{t}-r_{t}R^{r}_{t}$
$\displaystyle+\tau_{t}-n_{t+1}\xi_{t}-m_{t+1}\mu-r_{t+1}\rho,$ (19)

where $\mu$ and $\rho$ are the unit resource costs of issuing (and managing)
CBDC and reserves payments, respectively.

###  2.5 Market clearing

Market clearing in the labor market requires that the firm’s labor demand
equals the household’s labor supply:

$\displaystyle\ell_{t}=1-x_{t}.$ (20)

Market clearing for capital requires that the firm’s demand for capital equals
capital holdings of the household, the bank, and the government:

$\displaystyle k_{t}=k_{t}^{h}+(n_{t}+l_{t}-r_{t}-b_{t})+k_{t}^{g}.$ (21)

Profits distributed to the household must equal the sum of the bank and firm
profits:

$\displaystyle\Pi_{t}=\Pi^{b}_{1,t}+\Pi^{b}_{2,t}+\Pi^{f}_{t}.$ (22)

By Walras’ law, market clearing on labor and capital markets and the budget
constraints of the household, bank, firm, and consolidated government imply
market clearing on the goods market.

To derive the aggregate resource constraint for the economy, we plug equation
(22) into the household’s budget constraint, equation (1), and we impose
market clearing conditions (20) and (21). Then, in combination with the
government’s budget constraint, equation (19), the resulting expression is the
aggregate resource constraint:

$\displaystyle
k_{t+1}=f(k_{t},1-x_{t})+k_{t}(1-\delta)-c_{t}-\left(m_{t+1}\mu+n_{t+1}\left(\nu(\zeta_{t+1},\bar{\zeta}_{t+1})+\zeta_{t+1}\rho\right)\right).$
(23)

##  3 Revisiting the equivalence of payment instruments

In this section, we use our model in a static study comparing two equilibria
with and without CBDC and revisiting the result in the literature regarding
the equivalence of payment instruments. Existing works by [6] and [16] suggest
a compensation mechanism where the households’ shift from deposits to CBDC can
be offset by central bank lending to banks. We build upon these insights by
incorporating a collateral constraint for central bank lending to banks, as
specified in equation (12) from the bank’s problem. We assume perfect
substitutability of CBDC and deposits in the household’s real balances, such
that effective real balances are a weighted sum of the two instruments:

$\displaystyle z_{t+1}=\lambda_{t}m_{t+1}+n_{t+1},$ (24)

where $\lambda_{t}\geq 0$ represents the liquidity benefits of CBDC relative
to deposits.999See Appendix C for the equivalence study in the case of CBDC
and deposits as imperfect substitutes. The assumption that an interest-bearing
CBDC and deposits are close substitutes is common in the literature [see,
e.g., [2] and [20]] and is consistent with the CBDC experiments central banks
are currently running.

The following proposition is formally proved in Appendix B.

Proposition 1. Consider a policy that implements an equilibrium with deposits,
reserves, central bank loans, and government bonds, where the latter are used
as banks’ collateral to obtain central bank loans. The central bank introduces
a new payment instrument, CBDC, which is a perfect substitute for deposits.
There exists another policy and equilibrium with less deposits and reserves, a
positive amount of CBDC, central bank loans, and government bonds, a different
ownership structure of capital, additional taxes on the household, and
otherwise the same equilibrium allocation and price system. In this
equilibrium, the public and private sectors are equally efficient in providing
liquidity to the household:

$\displaystyle\frac{\mu}{\lambda_{t}}=\nu(\zeta_{t+1},{\zeta}_{t+1})+\zeta_{t+1}\rho,$
(25)

and the central bank lends to the bank at a loan rate equal to

$\displaystyle
R^{l}_{t+1}=\frac{R^{n}_{t+1}+\left(\nu(\zeta_{t+1},\zeta_{t+1})-\xi_{t}\right)R^{f}_{t+1}-\zeta_{t+1}R^{r}_{t+1}}{(1-\zeta_{t+1})\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)}.$
(26)

$\blacksquare$

Condition (25) stipulates that the resource cost the government pays to
provide one unit of real balances through CBDC equalizes the corresponding
cost for deposits, which is the sum of the operating cost incurred by the
bank, and the resource cost associated with reserves incurred by the
government. This condition is realistic since one of the options central banks
are considering for issuing a CBDC to the public involves using the existing
commercial banks’ deposit distributing systems.101010See, e.g., [14] for the
results of the 2022 BIS survey on central bank digital currencies and crypto.
A central bank’s loan rate equal to the value in equation (26) ensures that
(i) the market values of taxes on the household and of changes in bank profits
are zero; (ii) the government budget constraint is unaffected by the new
policy; and (iii) the bank chooses loans to make up for the reduction in
funding from the household.

The loan rate we derive is lower than the one in [16] due to the extra term
$\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)>0$ in equation
(26).111111From the household’s problem, we know that $R^{k}_{t+1}\leq
R^{f}_{t+1}$, assuming that the rate of return on capital is not risky, we can
approximate $R^{k}_{t+1}\simeq R^{f}_{t+1}$. We also know that for the bank
there is a collateral premium associated with holding government bonds, thus
$R^{b}_{t+1}<R^{f}_{t+1}$. It follows that the extra term is positive. It
follows that when the bank is subject to a collateral constraint for central
bank lending, the central bank must post a lower rate than in a no-collateral
constraint scenario. The intuition is that when the bank is not collateral-
constrained, it can borrow as much as it wants from the central bank. With a
collateral requirement, the central bank needs to offer a lower lending rate
to incentivize the bank to borrow the same quantity as in the absence of the
constraint, such that it remains indifferent to the introduction of the
CBDC.121212We abstract from any social cost associated with central bank
lending to banks. Notice that, since [16] does not account for the central
bank’s collateral requirement, it is like considering
$\theta_{b}\rightarrow+\infty$. Calling the central bank loan rate obtained in
[16] as $\tilde{R}^{l}_{t+1}$, we can conclude that
$\lim_{\theta_{b}\to+\infty}R^{l}_{t+1}=\tilde{R}^{l}_{t+1}$.

The central bank loan rate we derive in equation (26) depends on how
restrictive is the collateral constraint: the tighter the constraint is (the
lower $\theta_{b}$), the lower the lending rate the central bank needs to
offer (the lower $R^{l}_{t+1}$). The central bank loan rate is higher than 1
for reasonable values of $\theta_{b}$ and is at its highest when all bonds
held by the bank can be pledged, i.e., $\theta_{b}=1$.

A final remark is worth noting. A central bank’s loan rate value as in
equation (26) insulates the bank’s profits. However, when the bank is
collateral-constrained, to access central bank loans, banks must hold
government bonds as collateral at the expense of extending credit to firms,
and the central bank assumes part of the credit-extension role. Figure 1
reports the bank’s balance sheet breakdown comparing the two equilibria before
and after the introduction of the CBDC.131313In the banks’ balance sheet
before the CBDC introduction, government bonds and central bank loans are
normalized to zero.

Figure 1: Bank’s balance sheets before and after the CBDC introduction

The following corollary follows from Proposition 1:

Corollary 1. The central bank loan rate as in equation (26) insulates bank
profits but not the bank’s business model. Although the new policy has no real
effects on the economy, it does not guarantee full neutrality as it changes
the bank’s business model.
$\blacksquare$

##  4 Dynamic effects of an increase in CBDC

In Section 3, we revisited the result regarding the equivalence of payment
instruments by considering a collateral constraint for central bank lending to
banks. We compared two equilibria, one with no CBDC and one when the central
bank introduces a CBDC that competes with deposits. Next, we use our model to
study the dynamic effects of a temporary increase in CBDC, more specifically,
the potential threat to financial stability should the CBDC crowd out
deposits.

###  4.1 Functional forms and equilibrium conditions

The functional form for real balances is represented by equation (24). We
assume that the household has utility function of the form

$\displaystyle\mathcal{U}(c_{t},x_{t},z_{t+1})=\frac{\left((1-\iota)c_{t}^{1-\psi}+\iota
z_{t+1}^{1-\psi}\right)^{\frac{1-\sigma}{1-\psi}}}{1-\sigma}x_{t}^{\upsilon},$
(27)

where $\iota>0$ is the utility weight of liquidity; $\sigma>0$ is the inverse
intertemporal elasticity of substitution between bundles of consumption and
real balances across times; $\psi>0$ is the inverse intratemporal elasticity
of substitution between consumption and real balances; and $\upsilon$ is the
exponent of the power function for leisure. The bank’s operating cost function
has the following form:

$\displaystyle\nu(\zeta_{t+1},\bar{\zeta}_{t+1})=\phi_{1}\zeta_{t+1}^{1-\varphi}+\phi_{2}\bar{\zeta}_{t+1}^{1-\varphi},$
(28)

where $\phi_{1},\phi_{2}\geq 0$ are the relative weights assigned to the
bank’s reserves-to-deposit ratio and to the other bank’s ratio; and
$\varphi>1$. Lastly, the firm has the standard Cobb-Douglas production
function:

$\displaystyle f(k_{t},\ell_{t})=k_{t}^{\alpha}\ell_{t}^{1-\alpha},$ (29)

where $k_{t}$ and $\ell_{t}$ are the firm’s demand for capital and labor,
respectively, and $\alpha$ is the capital share of output.

Given the functional form assumptions, we characterize the general
equilibrium. First, knowing the household’s utility functional form, we can
rewrite the stochastic discount factor, equation (3) as

$\displaystyle\Lambda_{t+1}=\beta\frac{c_{t+1}^{-\sigma}x_{t+1}^{\upsilon}\Omega_{t+1}^{c}}{c_{t}^{-\sigma}x_{t}^{\upsilon}\Omega_{t}^{c}},$

so that the risk-free rate, equation (2), is given by

$\displaystyle
R_{t+1}^{f}=\left(\mathbb{E}_{t}\left[\beta\frac{c_{t+1}^{-\sigma}x_{t+1}^{\upsilon}\Omega_{t+1}^{c}}{c_{t}^{-\sigma}x_{t}^{\upsilon}\Omega_{t}^{c}}\right]\right)^{-1}.$

The household’s capital Euler equation (6) and leisure choice condition (5),
and the aggregate resource constraint (23) become, respectively,

$\displaystyle
c_{t}^{-\sigma}x_{t}^{\upsilon}\Omega_{t}^{c}=\beta\mathbb{E}_{t}\left[c_{t+1}^{-\sigma}x_{t+1}^{\upsilon}\Omega_{t+1}^{c}R_{t+1}^{k}\right],$
(30) $\displaystyle\frac{c_{t}^{1-\sigma}}{1-\sigma}\upsilon
x_{t}^{\upsilon-1}\Omega_{t}^{x}=w_{t}c_{t}^{-\sigma}x_{t}^{\upsilon}\Omega_{t}^{c},$
(31) $\displaystyle
k_{t+1}=k_{t}^{\alpha}(1-x_{t})^{1-\alpha}+k_{t}(1-\delta)-c_{t}-\left(m_{t+1}\mu+n_{t+1}\left(\nu(\zeta_{t+1},\zeta_{t+1})+\zeta_{t+1}\rho\right)\right),$
(32)

where

$\displaystyle\Omega_{t}^{c}$
$\displaystyle=(1-\iota)^{\frac{1-\sigma}{1-\psi}}\left(1+\left(\frac{\iota}{1-\iota}\right)^{\frac{1}{\psi}}\left({\chi^{n}_{t+1}}\right)^{1-\frac{1}{\psi}}\right)^{\frac{\psi-\sigma}{1-\psi}},$
$\displaystyle\Omega_{t}^{x}$
$\displaystyle=(1-\iota)^{\frac{1-\sigma}{1-\psi}}\left(1+\left(\frac{\iota}{1-\iota}\right)^{\frac{1}{\psi}}\left({\chi^{n}_{t+1}}\right)^{1-\frac{1}{\psi}}\right)^{\frac{1-\sigma}{1-\psi}}.$

These equilibrium conditions closely parallel those of a standard real
business cycle model. Unlike the standard model, however, the auxiliary
variables $\Omega_{t}^{c}$ and $\Omega_{t}^{x}$, summarize the impact of the
household’s preference for liquidity on consumption/savings and leisure
choices. Moreover, the term
$m_{t+1}\mu+n_{t+1}\left(\nu(\zeta_{t+1},\zeta_{t+1})+\zeta_{t+1}\rho\right)$
in the resource constraint shows the societal costs of providing liquidity to
the household incurred by the government and banks.

We combine the household’s first-order condition for deposits, equation (8),
with the expression for real balances, equation (24) to derive the deposit
demand:

$\displaystyle
n_{t+1}=c_{t}\left(\frac{\iota}{1-\iota}\frac{1}{\chi^{n}_{t+1}}\right)^{\frac{1}{\psi}}-\lambda_{t}m_{t+1},$
(33)

where we combine the first-order conditions for deposits and reserves from the
bank’s problem, (13) and (14), to derive the expressions for the equilibrium
deposit spread, $\chi_{t+1}^{n}$:

$\displaystyle\chi_{t+1}^{n}=\frac{(\phi_{1}\varphi+\phi_{2})\zeta_{t+1}^{1-\varphi}-\xi_{t}}{1-\psi\frac{n_{t+1}}{z_{t+1}}},$
(34)

and the bank’s optimal reserves-to-deposits ratio, $\zeta_{t+1}$, depends on
the spread on reserves, $\chi_{t+1}^{r}$,

$\displaystyle\zeta_{t+1}=\left(\frac{\chi_{t+1}^{r}}{\phi_{1}(\varphi-1)}\right)^{-\frac{1}{\varphi}}.$
(35)

Given the real balances functional form assumption, equation (24), from the
household’s problem equation (9) we derive the CBDC spread as

$\displaystyle\chi_{t+1}^{m}=\lambda_{t}\chi_{t+1}^{n}.$ (36)

Note that the spread on reserves is derived from equation (4).

From the binding collateral constraint, equation (12), we derive the bank’s
demand for government bonds as

$\displaystyle b_{t+1}=\frac{l_{t+1}R_{t+1}^{l}}{\theta_{b}}.$ (37)

Combining the bank’s optimality conditions for central bank loans and
government bonds, equations (15) and (16), we derive the bank’s demand for
central bank loans. We restrict our attention to the case where this demand is
non-negative, i.e.,141414 To derive equation (38), we assume that the central
bank’s loan supply schedule replicates the household’s deposit demand
schedule, i.e.,  $\displaystyle\frac{\partial l_{t+1}}{\partial
R^{l}_{t+1}}=\frac{\partial n_{t+1}}{\partial R^{n}_{t+1}}.$

$\displaystyle
l_{t+1}=\begin{cases}\left(\chi^{l}_{t+1}-\chi^{b}_{t+1}\frac{R^{l}_{t+1}}{\theta_{b}}\right)\left(\frac{\theta_{b}}{\chi^{b}_{t+1}R^{f}_{t+1}+\theta_{b}}\right)\frac{z_{t+1}}{\psi\chi^{n}_{t+1}}&\text{if
$\chi^{l}_{t+1}-\chi^{b}_{t+1}\frac{R^{l}_{t+1}}{\theta_{b}}\geq 0$}\\\
0&\text{if
$\chi^{l}_{t+1}-\chi^{b}_{t+1}\frac{R^{l}_{t+1}}{\theta_{b}}<0$}\end{cases}.$
(38)

In the baseline, we assume that the interest rates on central bank loans,
government bonds and reserves are fixed at their steady state level, i.e.
$R_{t+1}^{l}=R^{l},R_{t+1}^{b}=R^{b},R_{t+1}^{r}=R^{r}.$

Finally, from the firm’s optimality conditions, equations (17) and (18), we
derive the return on capital and the real wage:

$\displaystyle R_{t}^{k}$
$\displaystyle=1-\delta+\alpha\left(\frac{k_{t}}{1-x_{t}}\right)^{\alpha-1},$
(39) $\displaystyle w_{t}$
$\displaystyle=(1-\alpha)\left(\frac{k_{t}}{1-x_{t}}\right)^{\alpha}.$ (40)

###  4.2 Shock

After characterizing the general equilibrium, we aim to investigate the
potential threat to financial stability should the CBDC crowd out deposits. We
address this concern by studying the economy’s responses to a temporary
increase in the CBDC. We follow [7] and assume that the central bank issues
CBDC according to a policy rule which stipulates that CBDC is equal to a
fraction of steady-state output, $y$:

$\displaystyle m_{t+1}=\theta^{m}_{t}y.$ (41)

The CBDC share, $\theta^{m}_{t}$, follows an AR(1) process of the form

$\displaystyle\theta^{m}_{t}=\rho^{\theta}\theta^{m}_{t-1}+e_{t},$

where $\rho^{\theta}$ is the persistence parameter and $e_{t}$ is the
exogenous one-time shock.

###  4.3 Calibration

The model is quarterly, and we calibrate it to the U.S. economy. We use
variables without subscripts to denote their steady-state values. Table 1
summarizes the baseline calibration and Appendix D describes the calibration
of the model parameters.

####  4.3.1 Households

The household’s discount factor, $\beta$, is set to the standard value of
$0.99$. We assume that the household perceives CBDC and deposits as equally
useful for liquidity purposes, i.e., $\lambda=1$. We set the inverse
intertemporal elasticity of substitution, $\sigma$, to $0.5$. The leisure
function coefficient, $\upsilon$, is set to $0.85$ to match a steady-state
labor supply of approximately $1/3$. We assume that consumption and liquidity
services are complements. Therefore, the inverse intratemporal elasticity of
substitution between the two, $\psi$, is set higher than $\sigma$ and equal to
$0.6$. We set the utility weight of liquidity, $\iota$ to 0.009 to match a
consumption velocity of 1.147, in line with the estimates of [10] for the U.S.

####  4.3.2 Banks

We calibrate the banks’ operating cost parameters $\varphi$ and $\phi_{1}$ to
$2.893$ and $4.632*10^{-5}$, respectively, to match a reserves-to-deposits
ratio of 0.1945, in line with the estimate of [17]. For simplicity, we assume
that $\phi_{2}=\phi_{1}$.

####  4.3.3 Firms and government

The production sector is standard. The capital share of output, $\alpha$, and
the rate of capital depreciation, $\delta$, are set to $1/3$ and $0.025$,
respectively.

We follow [17] and set the government’s marginal cost of providing reserves,
$\rho$, to $0.0001$. In line with our discussion in the equivalence section,
we assume that the government is equally efficient in providing liquidity to
the household as the banking sector. Thus, we set the government’s cost of
issuing CBDC, $\mu$, in line with condition (25). Similarly, the interest rate
on central bank loans is set to reflect the central bank loan rate (26)
derived in section 3. For simplicity, we assume that CBDC, reserves and bonds
are non-interest bearing (in real terms). We assume in the baseline that the
government does not extend subsidies to the bank, i.e., $\xi=0$. Lastly, we
set the haircut on government bonds to 0.5% which implies a collateral haircut
$\theta_{b}=0.995$.

Parameter | Value | Source/Motivation
---|---|---
Households
$\beta$ | $0.99$ | Standard
$\lambda$ | $1$ | Assumption
$\sigma$ | $0.5$ | Assumption
$\upsilon$ | $0.85$ | $\ell_{t}\approx 1/3$
$\psi$ | $0.6$ | $\psi>\sigma$
$\iota$ | $0.009$ |  $c/z=1.147$ ([10])
Banks
$\varphi$ | $2.893$ |  $\zeta=0.1945$ ([17])
$\phi_{1}$ | $4.632*10^{-5}$ |  $\zeta=0.1945$ ([17])
$\phi_{2}$ | $4.632*10^{-5}$ | Assumption
Firms
$\alpha$ | $1/3$ | Standard
$\delta$ | $0.025$ | Standard
Government
$\rho$ | $0.0001$ | [17]
$\mu$ | $0.002$ | Condition (25)
$R^{l}$ | $0.993$ | Condition (26)
$R^{m}$ | $1.0$ | Assumption
$R^{r}$ | $1.0$ | Assumption
$R^{b}$ | $1.0$ | Assumption
$\theta_{b}$ | 0.995 | Haircut on bonds 0.5%
$\xi$ | $0$ | Assumption
AR(1) process
$\rho^{\theta}$ | $0.9$ | Standard
Table 1: Model Parameters

###  4.4 Impulse responses

In this section, we keep the interest rates on reserves, central bank loans
and government bonds constant. The impulse responses are reported as
percentage or basis point deviations.151515CBDC deviations are reported in
absolute values. This is because in the steady state there is no CBDC. Figure
2 illustrates the impulse responses to an increase in the CBDC share of
steady-state output, $\theta^{m}_{t}$, from zero to 5%.

Figure 2: Impulse responses to increase in $\theta^{m}_{t}$ from zero to 5%

As the share of CBDC increases, there is now a positive amount of CBDC in
circulation. The equilibrium deposit spread, pinned down by equation (34),
depends on the reserve spread which represents the marginal cost of issuing
deposits, and the deposit share of total liquidity services, which affects the
markup the bank can charge on its prices over the marginal cost. Since both
quantities decrease, the deposit spread drops. Even if the CBDC in circulation
increases, the cheaper deposits effect, stemming from the decrease in the
deposit spreads, is stronger. Therefore, deposits in the economy increase. In
the equilibrium steady state, we assumed that the household perceives CBDC and
deposits as equally useful for liquidity purposes (i.e., $\lambda=1$) and the
rates of returns on CBDC and deposits are the same.161616 The intuition behind
this latter assumption is that the government, upon introducing CBDC, wants to
ensure that both instruments circulate. It follows that the spreads on the two
instruments are equal, and the CBDC spread decreases as the deposit spread.
With equal spreads, the average cost of liquidity is equal to the individual
instrument spread. A drop in the average cost of liquidity induces the
household to hold more liquidity services and increase consumption. This is
because a lower cost of liquidity increases the household’s current marginal
utility of consumption. In other words, the opportunity cost of savings has
increased, incentivizing the household to save less and consume more. At the
same time, the household’s marginal benefit of leisure is now higher than the
marginal cost, inducing them to decrease labor supply.

Upon an increase in the CBDC share, the central bank loans to the bank
increase. This follows from the increase in liquidity services and the drop in
the deposit spread. These contemporaneous effects imply that it is becoming
cheaper to get liquidity. To obtain loans from the central bank, the bank must
post collateral, and government bonds increase as a result.

The aggregate capital in the economy is the sum of capital held by households,
banks, and the government. Before the shock, the capital was mainly held by
households. After the shock, banks and the government hold more, and aggregate
capital drops because of the decrease in household capital (see Figure 3 in
Appendix E for the aggregate capital breakdown by components). The expansion
in the bank’s capital follows from the liability side increasing less than the
change in the sum of reserves and government bonds. Interestingly, the
intuition behind the bank’s capital increase is that the bank expands credit
intermediation to firms. In other words, introducing CBDC not only does not
cause bank disintermediation but expands bank activity.

###  4.5 Robustness checks

Uncertainty regarding the household’s perception of CBDC’s usefulness would be
important for any practical implementation of CBDC. Therefore, we first test
the robustness of our results by changing the liquidity benefit of CBDC.
Secondly, the household’s willingness to substitute between CBDC and
consumption would be uncertain as we have few empirical results to rely on.
Thus, we also test for different values for the intratemporal elasticity of
substitution between consumption and liquidity services. Lastly, given the
central role of the pledgeability of bonds for our equivalence results, we
test for different magnitudes for the haircuts on bonds.

First, we change the liquidity benefit of CBDC, $\lambda$. Figures 4 and 5 in
the Appendix E.1 show the impulse responses to an increase in $\theta_{t}^{m}$
when $\lambda$ is $0.5$ and $1.5$, respectively. Comparing these responses to
the main specification in Figure 2, we see that the results remain
qualitatively the same. The shapes of the impulse responses are identical, and
the magnitudes of changes in consumption, hours worked, and capital are
largely unchanged. At the same time, there are some differences in the
magnitudes of variables in the financial sector. Next, we test for different
elasticities of substitution between consumption and liquidity services.
Figures 6 and 7 show impulse responses when the inverse elasticity, $\psi$, is
$0.8$ and $0.5$, respectively. Similar to the previous case, the qualitative
conclusions we can draw from the responses remain the same. However, when the
elasticity of substitution is lower ($\psi=0.8$), the size of the responses is
larger than in the baseline, and vice versa when the elasticity of
substitution is higher ($\psi=0.5$). Lastly, Figures 8 and 9 show the impulse
responses when bond haircuts are at $0.1\%$ ($\theta_{b}=0.999$) and $1.5\%$
($\theta_{b}=0.985$), respectively. We see that the changes in bond
pledgeability do not alter the responses in shape or magnitude.

##  5 Conclusion

We investigate the potential risk to financial stability when introducing a
CBDC. We revisit the results on the equivalence of payment instruments when
introducing a collateral constraint for central bank lending to banks. We find
that, when CBDC and deposits are perfect substitutes, as long as they have the
same resource cost per unit of effective real balances, the central bank can
offer bank loans at a loan rate that renders the non-competitive banks
indifferent to the introduction of the CBDC and CBDC has no real effects on
the economy. Additionally, it is crucial to account for the collateral
requirement that the bank must respect when borrowing from the central bank,
as the central bank’s lending rate depends on how restrictive the collateral
constraint is. The tighter the constraint is (the lower the fraction of the
bank’s bond holdings that can be pledged as collateral), the lower the central
bank’s loan rate should be to keep the equilibrium allocations unchanged when
introducing a CBDC. However, different from the results in the literature, we
find that while the central bank compensates for the lower deposit funding by
lending to the bank, it assumes part of the credit extension role from the
bank, which now holds government bonds as collateral and less capital to
extend credit. In other words, the central bank insulates bank profits but not
the bank’s business model. It follows that, although the CBDC introduction has
no real effects on the economy, it does not guarantee full neutrality as it
affects the bank’s business model. Our dynamic analysis suggests that an
increase in CBDC does not lead to bank disintermediation or crowding out of
deposits, but it expands bank activity as banks expand credit to firms.

Overall, our findings can help policymakers and central bankers design and
implement CBDCs to minimize the risk of financial instability. A possible
extension in the analysis of the equivalence result is to investigate the
transition from the equilibrium with no CBDC to the equilibrium after CBDC has
been introduced and identify the driving forces governing the transition
between the two.

##  Appendix A Condition under which the collateral constraint binds

Assuming interior solutions, the bank’s optimality conditions for loans and
bonds are, respectively:

$\begin{split}\mathbb{E}_{t}\left[\Lambda_{t+1}(R^{k}_{t+1}-R^{l}_{t+1}-l_{t+1}\frac{\partial
R^{l}_{t+1}}{\partial
l_{t+1}})\right]&=\gamma_{t}\left(1+\theta_{b}\frac{b_{t+1}}{{R^{l}}^{2}_{t+1}}\frac{\partial
R^{l}_{t+1}}{\partial l_{t+1}}\right),\\\
\mathbb{E}_{t}\left[\Lambda_{t+1}(R^{k}_{t+1}-R^{b}_{t+1})\right]&=\gamma_{t}\frac{\theta_{b}}{R^{l}_{t+1}},\end{split}$

where $\gamma_{t}$ denotes the Lagrange multiplier associated with the
collateral constraint. Subtracting the condition for bonds from the one for
loans:

$\mathbb{E}_{t}\left[\Lambda_{t+1}(R^{b}_{t+1}-R^{l}_{t+1}-l_{t+1}\frac{\partial
R^{l}_{t+1}}{\partial
l_{t+1}})\right]=\gamma_{t}\left(1-\frac{\theta_{b}}{R^{l}_{t+1}}+\theta_{b}\frac{b_{t+1}}{{R^{l}}^{2}_{t+1}}\frac{\partial
R^{l}_{t+1}}{\partial l_{t+1}}\right).$ (42)

To define the sign of the RHS, recall that $\theta_{b}\in[0,1]$, and since the
rate of return on reserves is positive, and we assumed interior solutions, all
the terms are positive.

We define the elasticity of central bank’s loans with respect to their rate of
returns as

$\eta_{l,t+1}=\frac{\partial l_{t+1}}{\partial
R^{l}_{t+1}}\frac{R^{l}_{t+1}}{l_{t+1}},$

such that we can rewrite the last term on the LHS as

$\frac{1}{\eta_{l,t+1}}R^{l}_{t+1}.$

Expression (42) says that the collateral constraint is binding if:

$R^{b}_{t+1}-R^{l}_{t+1}>\frac{1}{\eta_{l,t+1}}R^{l}_{t+1}.$

Rearranging

$R^{b}_{t+1}>R^{l}_{t+1}+\frac{1}{\eta_{l,t+1}}R^{l}_{t+1}.$

We can conclude that the collateral constraint is binding if the sum of the
cost of borrowing from the central bank and the bank’s cost of taking on more
loans is cheaper than the return the bank gets from holding government
bonds:171717We replicated the same analysis in the setting by [7], and we got
an analogous result.

$\displaystyle\gamma_{t}>0,\qquad
l_{t+1}=\theta_{b}\frac{b_{t+1}}{R^{l}_{t+1}}\qquad\text{ iff
}R^{b}_{t+1}>R^{l}_{t+1}+\frac{1}{\eta_{l,t+1}}R^{l}_{t+1}.$

##  Appendix B Equivalence

For convenience, we repeat Proposition 1 as in Section 3 and prove it
formally.

Proposition 1. Consider a policy that implements an equilibrium with deposits,
reserves, central bank loans, and government bonds, where the latter are used
as banks’ collateral to obtain central bank loans. The central bank introduces
a new payment instrument, CBDC, which is a perfect substitute for deposits.
There exists another policy and equilibrium, indicated by circumflexes. with
less deposits and reserves, a positive amount of CBDC, central bank loans, and
government bonds, a different ownership structure of capital, additional taxes
on the household, and otherwise the same equilibrium allocation and price
system.
$\blacksquare$

Suppose that deposit holdings decrease by a magnitude of $\Delta$ from the
initial equilibrium, i.e. $\hat{n}_{t+1}-n_{t+1}=-\Delta$. Suppose also that
real balances, the aggregate capital stock and the reserves-to-deposits ratio
remain unchanged in the new equilibrium, i.e.,

$\displaystyle\hat{z}_{t+1}=z_{t+1},\quad\hat{k}_{t+1}=k_{t+1},\quad\hat{\zeta}_{t+1}=\zeta_{t+1}.$

The above implies the following changes in the other equilibrium quantities:

$\displaystyle\hat{m}_{t+1}-m_{t+1}=\frac{1}{\lambda_{t}}\Delta,$
$\displaystyle\qquad\hat{r}_{t+1}-r_{t+1}=-\zeta_{t+1}\Delta,$
$\displaystyle\hat{l}_{t+1}-l_{t+1}=(1-\zeta_{t+1})\Delta,$
$\displaystyle\qquad\hat{b}_{t+1}-b_{t+1}=\frac{\hat{l}_{t+1}R^{l}_{t+1}}{\theta_{b}},$
$\displaystyle\hat{k}_{t+1}^{h}-k_{t+1}^{h}=\left(1-\frac{1}{\lambda_{t}}\right)\Delta,$
$\displaystyle\qquad\hat{k}^{g}_{t+1}-k^{g}_{t+1}=-\left(1-\frac{1}{\lambda_{t}}\right)\Delta+\hat{b}_{t+1},$

where $l_{t+1}$ and $b_{t+1}$ will be normalized to zero in what
follows.181818To guarantee the non-negativity of deposits, capital holdings,
and reserves, $\Delta$ must not be too large. Specifically, we impose
$\Delta\leq n_{t+1},\qquad\zeta_{t+1}\Delta\leq
r_{t+1},\qquad\left(1-\frac{1}{\lambda_{t}}\right)\Delta\leq
k^{g}_{t+1},\qquad\left(1-\frac{1}{\lambda_{t}}\right)\Delta\geq-k_{t+1}^{h}.$

First, we show that the new policy has no real effects on the economy, given
an appropriate level of interest rate on central bank loans. Note that, before
the implementation of the new policy, the cash flows generated in the first
and second periods of the bank’s operations are given by equation (10) and
(11), respectively. Recalling that in equilibrium
$\zeta_{t+1}=\bar{\zeta}_{t+1}$, the changes in bank profits at dates $t$ and
$t+1$ are, respectively:

$\displaystyle\hat{\Pi}^{b}_{1,t}-\Pi^{b}_{1,t}$
$\displaystyle=\Delta\left(\nu(\zeta_{t+1},\zeta_{t+1})-\xi_{t}\right),$ (43)
$\displaystyle\hat{\Pi}^{b}_{2,t+1}-\Pi^{b}_{2,t+1}$
$\displaystyle=\Delta\left(R^{n}_{t+1}-\zeta_{t+1}R^{r}_{t+1}-(1-\zeta_{t+1})\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)R^{l}_{t+1}\right).$
(44)

Let $\hat{T}_{1,t}$ be a tax on the household at date $t$ that compensates for
the reduced bank losses:

$\displaystyle\hat{T}_{1,t}$
$\displaystyle=\hat{\Pi}^{b}_{1,t}-\Pi^{b}_{1,t}=\Delta\left(\nu(\zeta_{t+1},\zeta_{t+1})-\xi_{t}\right).$
(45)

We denote $\hat{T}_{2,t+1}$ as a tax at date $t+1$ that compensates for the
change in the household’s portfolio return as well as for the change in bank
profits that the household collects at date $t+1$:191919Given the household’s
trades-off between CBDC and deposits, expression (9), it follows that:
$\frac{\lambda_{t}}{R_{t+1}^{m}}=R_{t+1}^{n}-(1-\frac{1}{\lambda_{t}})R_{t+1}^{f}$.

$\displaystyle\hat{T}_{2,t+1}$
$\displaystyle=(\hat{k}_{t+1}^{h}-k_{t+1}^{h})R^{k}_{t+1}+(\hat{n}_{t+1}-n_{t+1})R^{n}_{t+1}+(\hat{m}_{t+1}-m_{t+1})R^{m}_{t+1}+\hat{\Pi}^{b}_{2,t+1}-\Pi^{b}_{2,t+1}$
$\displaystyle=\Delta\left[\left(1-\frac{1}{\lambda_{t}}\right)R_{t+1}^{k}+\frac{R_{t+1}^{m}}{\lambda_{t}}-\zeta_{t+1}R^{r}_{t+1}-(1-\zeta_{t+1})\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)R^{l}_{t+1}\right].$
(46)

Let $\mathcal{T_{t}}=\hat{T}_{1,t}+\mathbb{E}_{t}\Lambda_{t+1}\hat{T}_{2,t+1}$
denote the market value of taxes at date $t$. Substituting the two expressions
for taxes, equations (45) and (46), and using conditions from the household’s
optimization problem, we can rewrite $\mathcal{T_{t}}$ as

$\displaystyle\mathcal{T_{t}}=\Delta\left[\left(\nu(\zeta_{t+1},\zeta_{t+1})-\xi_{t}\right)+\frac{R^{n}_{t+1}-\zeta_{t+1}R^{r}_{t+1}-(1-\zeta_{t+1})\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)R^{l}_{t+1}}{R^{f}_{t+1}}\right].$

In order for the new policy to have no real effects on the economy, the market
value of taxes must be zero. This is true if the central bank posts a loan
rate equal to equation (26):

$\displaystyle
R^{l}_{t+1}=\frac{R^{n}_{t+1}+\left(\nu(\zeta_{t+1},\zeta_{t+1})-\xi_{t}\right)R^{f}_{t+1}-\zeta_{t+1}R^{r}_{t+1}}{(1-\zeta_{t+1})\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)}.$

We denote the market value of the changes in bank profits at date $t$ as
$\mathcal{P_{t}}=(\hat{\Pi}^{b}_{1,t}-\Pi^{b}_{1,t})+\mathbb{E}_{t}\Lambda_{t+1}(\hat{\Pi}^{b}_{2,t+1}-\Pi^{b}_{2,t+1}).$
Plugging in the expressions for changes in bank profits, equations (43) and
(44), and using the definition of the risk-free rate from the household’s
problem, $\mathcal{P_{t}}$ reads

$\displaystyle\mathcal{P_{t}}\ $
$\displaystyle=\Delta\left(\nu(\zeta_{t+1},\zeta_{t+1})-\xi_{t}\right)$
$\displaystyle+\frac{1}{R^{f}_{t+1}}\Delta\left(R^{n}_{t+1}-\zeta_{t+1}R^{r}_{t+1}-(1-\zeta_{t+1})\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)R^{l}_{t+1}\right),$

which is equal to zero given equation (26). It follows that if the central
bank offers an interest rate on central bank loans according to equation (26),
the market values of the taxes and of the changes in bank profits are zero.

Next, we show that the government’s dynamic and intertemporal budget
constraints continue to be satisfied with the new policy. Before the
implementation of the new policy, the government budget constraint at time $t$
reads:

$\displaystyle
k^{g}_{t+1}-m_{t+1}-r_{t+1}=k^{g}_{t}R^{k}_{t}-m_{t}R^{m}_{t}-r_{t}R^{r}_{t}+\tau_{t}-n_{t+1}\xi_{t}-m_{t+1}\mu-
r_{t+1}\rho.$ (47)

The government budget constraint at time $t$ with the new policy and changes
is

$\displaystyle\hat{k}^{g}_{t+1}+\hat{l}_{t+1}-\hat{m}_{t+1}-\hat{r}_{t+1}-\hat{b}_{t+1}$
$\displaystyle=k^{g}_{t}R^{k}_{t}-m_{t}R^{m}_{t}-r_{t}R^{r}_{t}+\tau_{t}$
$\displaystyle-\hat{n}_{t+1}\xi_{t}-\hat{m}_{t+1}\mu-\hat{r}_{t+1}\rho+\hat{T}_{1,t}.$

Rearranging, simplifying, and collecting terms:

$\displaystyle
k^{g}_{t+1}-m_{t+1}-r_{t+1}+\Delta\left(\frac{\mu^{m}}{\lambda_{t}}-\left(\nu(\zeta_{t+1},\zeta_{t+1})+\rho\zeta_{t+1}\right)\right)$
$\displaystyle=k^{g}_{t}R^{k}_{t}-m_{t}R^{m}_{t}-r_{t}R^{r}_{t}+\tau_{t}$
$\displaystyle\quad-n_{t+1}\xi_{t}-m_{t+1}\mu-r_{t+1}\rho.$ (48)

The government budget constraints before and after the intervention at time
$t$, equations (47) and (48), are identical as long as

$\displaystyle\frac{\mu}{\lambda_{t}}=\nu(\zeta_{t+1},{\zeta}_{t+1})+\zeta_{t+1}\rho.$

That is, when the public and private sectors are equally efficient in
providing liquidity to the household, the government’s budget constraint at
date $t$ is unaffected by the changes in allocation

Similarly, before the new policy, the government budget constraint at time
$t+1$ reads

$\displaystyle k^{g}_{t+2}-m_{t+2}-r_{t+2}$
$\displaystyle=k^{g}_{t+1}R^{k}_{t+1}-m_{t+1}R^{m}_{t+1}-r_{t+1}R^{r}_{t+1}$
$\displaystyle\quad+\tau_{t+1}-n_{t+2}\theta_{t+1}-m_{t+2}\mu-r_{t+2}\rho.$
(49)

With the new policy and changes, the government budget constraint at time
$t+1$ becomes

$\displaystyle k^{g}_{t+2}-m_{t+2}-r_{t+2}$
$\displaystyle=\hat{k}^{g}_{t+1}R^{k}_{t+1}+\hat{l}_{t+1}R^{l}_{t+1}-\hat{m}_{t+1}R^{m}_{t+1}-\hat{r}_{t+1}R^{r}_{t+1}-\hat{b}_{t+1}R^{b}_{t+1}$
$\displaystyle\quad+\tau_{t+1}-n_{t+2}\theta_{t+1}-m_{t+2}\mu-
r_{t+2}\rho+\hat{T}_{2,t+1}.$

Rearranging, simplifying, and collecting terms:

$\displaystyle k^{g}_{t+2}-m_{t+2}-r_{t+2}=$
$\displaystyle\hat{k}^{g}_{t+1}R^{k}_{t+1}+\hat{l}_{t+1}R^{l}_{t+1}-\hat{m}_{t+1}R^{m}_{t+1}-\hat{r}_{t+1}R^{r}_{t+1}-\hat{b}_{t+1}R^{b}_{t+1}$
$\displaystyle+\tau_{t+1}-n_{t+2}\theta_{t+1}-\hat{r}_{t+1}R^{r}_{t+1}-m_{t+2}\mu-
r_{t+2}\rho+\hat{T}_{2,t+1}.$ (50)

Using the expression for the central bank loan rate we derived, equation (26),
it follows that the government budget constraints before and after the
intervention at time $t+1$, equations (49) and (50), are the same. In other
words, the central bank loan rate ensuring that the market values of taxes and
changes in bank profits are zero, also ensures that the government budget
constraint at time $t+1$ is unaffected by the changes in allocation.

We claimed initially that the proposed intervention does not change the price
system. In this case, the firm’s optimal production decisions and profits are
unchanged. Lastly, we must show that the modified bank’s portfolio is still
optimal. Before the intervention, the bank’s choice set is determined by the
cost function, the subsidy rate, the household’s stochastic discount factor,
rates on returns on capital and reserves, and the deposit funding schedule.
The new policy leaves unchanged the cost function, the subsidy rate, the
stochastic discount factor, and the rates on returns on capital and reserves.
After the intervention, as the household holds more CBDC, there is a modified
deposit funding schedule, together with a central bank loan funding schedule.
The central bank needs to post an appropriate loan funding schedule to induce
the non-competitive bank to go along with the equivalent balance sheet
positions as before the intervention. Subject to this schedule, the bank
chooses loans that make up for the reduction in funding from the household,
net of reserves, at the same effective price. The central bank chooses a loan
funding schedule which mirrors the deposit funding schedule and posts the loan
rate as in equation (26).

##  Appendix C Equivalence with imperfect substitutability between payment
instruments

In Section 3 we consider CBDC and deposits as perfectly substitutable for the
household. However, some works in the literature consider imperfect
substitutability between the two instruments [see, e.g., [1], [4], [5], [7]
and [15]]. We now assume a constant elasticity of substitution (CES)
functional form for the household’s real balances:

$\displaystyle
z_{t+1}(m_{t+1},n_{t+1})=\left(\lambda_{t}m^{1-\epsilon_{t}}_{t+1}+n^{1-\epsilon_{t}}_{t+1}\right)^{\frac{1}{1-\epsilon_{t}}},$

where $\lambda_{t}\geq 0$ represents the liquidity benefits of CBDC relative
to deposits, and $\epsilon_{t}\geq 0$ is the inverse elasticity of
substitution between payment instruments. We study the equivalence of payment
systems assuming that CBDC and deposits are imperfect substitutes, such that
$\epsilon_{t}>0$ for all $t$.

Proposition 2. Consider a policy that implements an equilibrium with deposits,
reserves, central bank loans, and government bonds, where the latter are used
as banks’ collateral to obtain central bank loans. The central bank introduces
a new payment instrument, CBDC, which is an imperfect substitute for deposits.
There does not exist another policy and equilibrium, indicated by
circumflexes, that guarantees the same equilibrium allocation and price
system.
$\blacksquare$

Suppose again that deposit holdings decrease by a magnitude of $\Delta$ from
the initial equilibrium and that real balances, the aggregate capital stock
and the reserves-to-deposits ratio remain unchanged in the new equilibrium.
This implies the same changes in equilibrium quantities as we have seen in
Appendix B, except for the CBDC. Due to the imperfect substitutability between
CBDC and deposits, in order for real balances to remain unchanged, the
quantity of CBDC must change according to

$\displaystyle\hat{m}_{t+1}-m_{t+1}=\left[\frac{1}{\lambda_{t}}\left(n_{t+1}^{1-\epsilon_{t}}-\hat{n}_{t+1}^{1-\epsilon_{t}}\right)+m_{t+1}^{1-\epsilon_{t}}\right]^{\frac{1}{1-\epsilon_{t}}}-m_{t+1}.$

We define taxes at dates $t$ and $t+1$, equations (45) and (46) respectively,
as in Appendix B, as well as the market values of taxes,
$\mathcal{T_{t}}=\hat{T}_{1,t}+\mathbb{E}_{t}\Lambda_{t+1}\hat{T}_{2,t+1}$.
The central bank loan rate ensuring that the market value of taxes is zero
is:202020When deriving the central bank loan rate, we use expression (9) for
the household’s trade-off between CBDC and deposits. Given the CES functional
form assumption, equation (24) it follows that:
$\frac{m_{t+1}}{n_{t+1}}=\left(\lambda_{t}\frac{\chi_{t+1}^{n}}{\chi_{t+1}^{m}}\right)^{\frac{1}{\epsilon_{t}}}$.

$\displaystyle
R_{t+1}^{l}=\frac{\mathcal{A_{t}}R_{t+1}^{n}-\zeta_{t+1}R_{t+1}^{r}+\left(\nu(\zeta_{t+1},\zeta_{t+1})-\xi_{t}+1-\mathcal{A_{t}}\right)R_{t+1}^{f}}{(1-\zeta_{t+1})\left(1+\frac{R^{k}_{t+1}-R^{b}_{t+1}}{\theta_{b}}\right)},$
(51)

where

$\displaystyle\mathcal{A_{t}}=\lambda_{t}\left(\frac{\hat{m}_{t+1}-m_{t+1}}{\Delta}\right)\left(\frac{n_{t+1}}{m_{t+1}}\right)^{\epsilon_{t}}.$

Consider the changes in bank profits at dates $t$ and $t+1$ as given from
equations (43) and (44), respectively. We can check whether the market value
of the changes in bank profits,
$\mathcal{P_{t}}=(\hat{\Pi}^{b}_{1,t}-\Pi^{b}_{1,t})+\mathbb{E}_{t}\Lambda_{t+1}(\hat{\Pi}^{b}_{2,t+1}-\Pi^{b}_{2,t+1})$,
also reduces to zero given the central bank loan rate in expression (51). It
turns out this is not true. In particular, after making the appropriate
substitutions, the market value of changes in bank profits reads:

$\displaystyle\mathcal{P_{t}}=\mathbb{E}_{t}\frac{1}{R^{f}_{t+1}}\Delta\left(R_{t+1}^{n}-\mathcal{A_{t}}R_{t+1}^{n}-\left(1-\mathcal{A_{t}}\right)R_{t+1}^{f}\right).$

Notice that if there were perfect substitutability between CBDC and deposits
(i.e., $\epsilon_{t}=0$), as in the case studies in Section 3,
$\mathcal{A_{t}}$ equals 1, and the market value of the changes in bank
profits reduces to zero. It follows that, in case of imperfect
substitutability between CBDC and deposits, the central bank lending rate that
renders the market value of taxes zero does not result in changes in bank
profits being zero. In other words, the central bank cannot make the bank
indifferent to the competition from CBDC. In fact, a change in the bank’s
profitability implies that the new policy does not guarantee the same
equilibrium allocation as before, implying that the introduction of CBDC has
real effects on the economy. In [6], one condition for equivalence to hold is
that CBDC and deposits are minimally substitutable, such that their marginal
liquidity contribution is unchanged. When the two instruments are imperfect
substitutes, as in the case under study, the marginal rate of substitution is
not constant, and equivalence is not guaranteed.

##  Appendix D Calibration

Given the desired steady state consumption velocity, we derive the utility
weight of liquidity, $\iota$, using condition (33)

$\displaystyle\iota=\frac{\left(\frac{z}{c}\right)^{\psi}\chi^{n}}{1+\left(\frac{z}{c}\right)^{\psi}\chi^{n}}.$

The bank operating cost function parameter $\varphi$ is derived using the
bank’s expression for deposit spread (34)

$\displaystyle\varphi=\frac{\chi^{n}(1-\psi)+\chi^{r}\zeta}{\chi^{n}(1-\psi)-\chi^{r}\zeta}.$

Note that $\chi^{n}$ is known here because the CBDC spread, $\chi^{m}$, is
known, and the two spread relate to each other according to equation (36).
Given $\varphi$, the other bank operating cost parameter, $\phi_{1}$, is
derived using the expression for the bank’s reserves-to-deposits ratio (35)

$\displaystyle\phi_{1}=\frac{\chi^{r}}{\zeta^{-\varphi}(\varphi-1)}.$

As explained in the main text, the government’s marginal cost of issuing CBDC
and the interest rate on its loans are set according to their respective
expressions in the equivalence Section 3:

$\displaystyle\mu$
$\displaystyle=\left(\nu(\zeta,{\zeta})+\zeta\right)\lambda,$ $\displaystyle
R^{l}$ $\displaystyle=\frac{R^{n}+\left(\nu(\zeta,\zeta)-\xi\right)R^{f}-\zeta
R^{r}}{(1-\zeta)\left(1+\frac{R^{k}-R^{b}}{\theta_{b}}\right)}.$

##  Appendix E Additional figures

Figure 3: Capital response breakdown by components

###  E.1 Robustness

Figure 4: Lower liquidity benefit of CBDC ($\lambda=0.5$) Figure 5: Higher
liquidity benefit of CBDC ($\lambda=1.5$) Figure 6: Lower elasticity of
substitution between $c_{t}$ and $z_{t+1}$ ($\psi=0.8$) Figure 7: Higher
elasticity of substitution between $c_{t}$ and $z_{t+1}$ ($\psi=0.5$) Figure
8: Lower haircut on bonds ($\theta_{b}=0.999$) Figure 9: Higher haircut on
bonds ($\theta_{b}=0.985$)

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