1.1 Theoretical Literature Review

The literature on CBDCs has been growing rapidly, both in the academia and in policy-making institutions. While most of the theoretical literature has focused on macro-financial implication in a domestic economy setting, more recently a number of papers have started exploring also the open-economy context. In general, the literature finds that while there are various risks associated with CBDC issuance, most prominently regarding financial stability, they can in principle be well mitigated through adequate design features, as described in more detail below.

A large part of the literature is concerned with the implications of CBDC introduction for the banking sector. The issuance of a deposit-like CBDC may crowd out private banking and shift deposits away from the banking system, reducing bank lending and generating a risk of structural disintermediation of banks. For example in Fernández-Villaverde et al. (2021) – to which this paper is most closely related – the introduction of a CBDC allows the central bank to compete with commercial banks for deposits, which may lead to the centralization of the credit allocation process within the central bank (e.g. the central bank attracting all deposits in the system), when depositors internalize that the central banks is not subject to run risk). A CBDC may also have negative effects on financial stability by increasing the likelihood of commercial bank panics (Kim and Kwon (2019)). Because a CBDC provides a safe and easily accessible alternative to traditional demand deposits held in private banks, it can facilitate systemic runs on banks in crisis situations (Schilling et al. (2020)).

Other papers suggest that CBDCs can also trigger important efficiency gains for the financial sector which may improve financial stability. This may happen if the reduction in bank profits is outweighed by expansion in bank lending, itself made possible by the expansion in deposit funding through greater financial inclusion and desired saving (Andolfatto (2018)). In Chiu et al. (2019), a deposit-like CBDC with a proper interest rate would encourage banks to pay higher interest or offer better services to keep their customers, thus limiting banks’market power, and improve the efficiency of bank intermediation. Bitter (2020) also finds that while a CBDC reduces the net worth in the banking sector in normal times, it mitigates the risk of a bank run in times of crisis, by impeding the emergence of bank runs. Garratt and Zhu (2021) finds that the interest rate on CBDC puts a lower bound on banks’ deposit interest rates and may level the playing field between large banks and smaller banks.

Most papers in the literature tend to concur that various design features (such as accessibility, remuneration and integration with the monetary policy framework) are essential is making sure the benefits outweighed the risks. One key design feature concerns accessibility, namely the possibility to allow or restrict CBDC access to either certain (financial) institutions, or the broader public, or both, possibly through some tiering system. This may lead to widely different implications for financial stability. For example, Cukierman (2019) finds that a CDBC design in which the entire private sector is allowed to hold digital currency deposits at the central bank may lead to a narrow banking system. However, Bindseil (2020) finds that the design of a two-tier remunerated CBDC helps address both the risk of financial disintermediation of the banking sector and the risks of systemic runs on banks. Kumhof and Noone (2018) discuss several design options which can mitigate large-scale run out of bank deposits into CBDC (which include remuneration and no guaranteed, on-demand convertibility of bank deposits into CBDC at commercial banks nor at the central bank).

A number of other papers also highlight the importance of CBDC design to integrate in an efficient manner with the existing monetary policy framework. This would require in general CBDCs to be interest-bearing, and that interest rates be adjustable according to the objectives of the central bank. Bordo (2021) argue for a CBDC which should be universally accessible and interest-bearing, with the central bank adjusting its interest rate to foster true price stability. Agur et al. (2022) discuss how a CBDC can be designed with attributes similar to either cash or deposits, including being interest-bearing, with the implications being that a CBDC that closely competes with deposits depresses bank credit and output, while a cash-like CBDC may even lead to the disappearance of cash.

If CBDCs allow for another instrument of monetary policy and can be used to implement the LOLR function, this may reduce the risk of systemic banking sector runs (Williamson (2019)). For example, the central bank can lend all the CBDC deposits back to commercial banks, thus a CBDC may actually enhance financial stability and even increase supply of private credit (Brunnermeier and Niepelt (2019), Kim and Kwon (2019)). A CBDC may be a useful additional policy tool also in the context of lowering the effective lower bound and implementing negative interest rates (Beniak (2019), Jia (2020)). Barrdear and Kumhof (2016) discuss how countercyclical CBDC price or quantity rules, as a second monetary policy instrument, could substantially improve the central bank’s ability to stabilize the business cycle.

A few papers assess broader implication of CBDC issuance on output and welfare. Keister and Sanches (2019) find that while a digital currency tends to promote efficiency in exchange, it can also crowd out bank deposits, raise banks’ funding costs, and decrease investment, yet despite these effects it can often raise welfare. In a DSGE model calibrated to match the pre-crisis United States, Barrdear and Kumhof (2016) find that CBDC issuance could permanently raise GDP due to reductions in real interest rates, distortionary taxes, and monetary transaction costs

Relatively few papers consider the international implications of CBDCs, where substantially more issues may arise due to the presence of additional frictions in cross-border payments and overall financial systems. A key concern is that CBDCs of reserve-currency countries available across borders could increase currency substitution (“dollarization”) in other countries, in particular those suffering from high inflation, exchange-rate volatility and weak fundamentals more broadly. However, if a foreign CBDC were to be widely adopted internationally, it could affect also economies with stable currencies if it leads to the domestic currency losing some of its functions and traction, which could ultimately impair domestic monetary policy. Such an argument is made in Ferrari et al. (2022), who show how the presence of a CBDC with features of scalability, liquidity, safety, remuneration and international circulation amplifies the international spillovers of shocks and may reduce monetary policy autonomy in foreign economies, with the magnitude of these effects depending crucially on the CBDC design. From the perspective of an issuer of such a CBDC, in a open-economy model with trade and capital flows, George et al. (2020) show that a CBDC with an adjustable interest rate may be welfare-improving, as it offers a secondary monetary policy instrument, which is akin to arguments made in the closed-economy setting.

1.2 How to Model a CBDC ? Motivations and Design Features

Discussing an environment with a CBDC raises the question of modelling choices, given the potentially critical importance of various design features. The limited practical experience with CBDCs² leaves many open questions as to which particular elements might end up being most prevalent. We rely on the literature discussed above to identify several possible realistic features, focusing on the following key dimensions: form, access, remuneration and cross-border availability³. This paper will define a (retail) CBDC as a digital central bank liability which is account-based, accessible to the general public, interest-bearing and internationally available to nonresidents. These features are consistent with the common foundational principles and core features of a CBDC as outlined in the joint report by major central banks (see Group of Central Banks (2020)) and can be implemented technically within various architectures. We discuss the motivation behind these features in more detail below. The first set of considerations is whether the CBDC should be account-based or token-based. Whereas token-based CBDCs are similar to cash, with the difference that they take the form of cash cards or electronic media (wallets) which enable anonymous peer-to-peer payments, account-based CBDCs on the other hand constitute deposits generally envisioned to be held in a central bank account, to be used either for payment and/or store of value purposes. In addition, they could allow the verification of users’identity in different layers and thus offer advantages in terms of monitoring illicit activity in the payment system – a key concern of all central banks – making them a more realistic design choice. A related consideration is whether the CBDC would allow access only to selected users such as financial institutions or operators of payment systems (wholesale CBDC), or to the broader public (retail CBDC). Various policy objectives that central banks may have, such as improving financial inclusion, may suggest that retail CBDCs are the most impactful design option.

Key Dimensions of CBDC Design Options

Citation: IMF Working Papers 2022, 083; 10.5089/9798400209185.001.A001

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Key Dimensions of CBDC Design Options

Citation: IMF Working Papers 2022, 083; 10.5089/9798400209185.001.A001

• Download Figure
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Key Dimensions of CBDC Design Options

Citation: IMF Working Papers 2022, 083; 10.5089/9798400209185.001.A001

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• Download figure as PowerPoint slide

Another very important feature for a CBDC is whether it would offer any remuneration. Account-based CBDCs can easily be designed to pay an interest rate consistent with the monetary policy objectives of the central bank, including price stability and business cycle stabilization. The literature generally argues that an interest-bearing CBDC is the most desirable and thus realistic set-up, bringing important advantages such as allowing a potentially new powerful tool for the conduct of monetary policy, which could include a relatively easy way to implement negative rates or “helicopter money” if needed.

Let us note that while we model an account-based retail CBDC, which constitutes a liability of the central bank, this does not necessarily mean that all agents would hold directly accounts with the central bank. While this design is certainly possible, in general it can be expected that the central bank would not want to substitute itself completely or overwhelmingly to the private financial markets (see e.g. BIS (2021)). Such a “direct” CBDC could impose a significant operational burden on the central bank, and substitute the central bank to some front-end operations which the private sector (banks, fintechs) may do better, which generally is associated with more innovation and better outcomes for consumers. Instead, an “indirect/hybrid” (also called two-tiered) system could be envisioned, in which the private sector provides the majority of operational tasks and consumer-facing activities and the central bank retains its role operating the core of the CBDC system and focuses on its traditional mandates⁴. Such CBDC designs are the ones which appear most favored by the few central banks who have moved towards an actual CBDC launch.

However, regardless of the precise configurations and technologies, CBDCs could offer agents a closer access to the relative safety of the central bank’s balance sheet, in a manner which had previously generally not been possible. Thus they could offer advantages compared to other forms of domestic money, such as cash or commercial bank deposits, both in terms of digital features and ease of access, but also in terms of superior safety features.

Finally, we shall consider a CBDC that is available cross-border, e.g. allows the access of non-residents (not only it its own jurisdictions, but is also circulating abroad). There are numerous complex questions on system design and interoperability which need to be addressed before CBDCs can be widely used cross-border, which is why this step has currently not been taken in the CBDCs currently in circulation. Such an evolution is likely to require significant international cooperation. Interoperability would require consistent technical standards, oversight frameworks, private and public laws and requirements for anti-money laundering and counter terrorism financing, among others (Auer et al. (2021)). At the international level, an active collaboration has emerged to improve existing payment systems and interoperating across borders (CPMI (2021)) and more broadly lay the foundations for the CBDC systems of the future (Financial Stability Board (2020), IMF (2021))⁵.

2.1 Consumers

The domestic economy is populated by a [0, 1]-continuum of ex-ante identical consumers. There are three periods indexed by t = 0,1, 2. This is a real model, in which consumers are endowed in period 0 with one unit of a consumption-investment good, which can be saved domestically or abroad.

In period t = 1, consumers face idiosyncratic consumption shocks, in that they find out that they are either impatient with probability λ ∈ (0,1), namely that they value consumption in period 1, or they are patient with probability 1 – λ, implying that they value consumption in period t = 2. Preferences are thus described by:

The utility function u : ℝ₊ → ℝ₊ is strictly increasing, strictly concave and continuously differentiable. The realization of each agent’s type is private information to that agent. Type realizations are i.i.d. across agents and there is no aggregate uncertainty.

The domestic economy allows for a short-term and a long-term investment opportunity (asset). The short-term technology is similar to storage and simply takes one unit of good at date t = 0; 1 and returns one unit of the domestic good in period t+1, e.g. a rate of return r = 1. The long-term investment, however, is more productive, and returns R > 1 in period t = 2. The long-term technology can be liquidated in period 1 at a loss, namely it pays off a liquidation value l ∈ (0; 1) for each unit invested. If consumers would be able to anticipate their type in period 0, then all the impatient consumers would invest in the short-term technology and consume in period 1 and all the patient consumers would prefer the long-term technology, which yields them a higher return and consumption in their preferred period t = 2.

However, as consumers cannot anticipate their shock, if they acted in financial autarky they would bear the idiosyncratic risk, as costly liquidation of the long-term asset would occur with positive probability. This justifies the introduction of banks to achieve a superior allocation by pooling resources and risks. In our model, consumers will have access to either a domestic or a foreign investment option. Domestically, they can save via commercial banks offering savings deposits. Internationally, there is a foreign central bank offering CBDC deposits, in which agents can also invest up to a maximum limit (for simplicity, there is no international borrowing). We start first by describing the characteristics of the deposit contracts offered by domestic commercial banks.

2.2 Domestic Commercial Banks

The domestic economy is populated by banks who offer demand deposits. There are a large number of perfectly competitive domestic banks, who make investments on behalf of consumers. Banks offer consumers demand deposit contracts $(c_1,c_2)\in {ℝ}_{+}^2$. In return for deposit one unit of good with the bank in period 0, the bank promises to pay either c₁ units of the good in period t = 1 or the lower amount between c₂ units of the good and the resources available to the banker in period t = 2.

Banks have access to the same short-term and long-term productive investment technologies that would be available to consumers. They decide to invest a portion y ∈ (0; 1) of the goods in the short-term technology, yielding a return of r = 1, and the remaining portion 1 y in the long-term asset, which return R > 1 if held for 2 periods and l < 1 if liquidated prematurely. The banker achieves paying c₁ to the domestic consumers by using the returns from the short-term investment and if necessary, by liquidating the long-term investment until all resources are exhausted (resources will be pro-rated if necessary). Any left-over resources in period 1 are invested in the short-term technology. The banker pays the lower of c₂ and any leftover resources in period 2 to the patient consumers (again pro-rated as needed), and consumes any leftovers in period 2. We assume bankers are in Bertrand competition when offering demand deposit contracts to consumers, which forces them to maximize the expected utility of consumers:

subject to:

Constraint (3b) is the feasibility constraint in period 1. The consumption of impatient consumers c₁ is financed by the return on the short-term investment and possibly by liquidating some portion of the long-term asset.

Constraint (3c) is the feasibility constraint for period 2, which implies that each patient consumer receives c₂, which is derived after the non-liquidated long-term asset yields its return and also from any left-over period 1 investments carried over into period 2.

Constraint (3d) is the incentive-compatibility (truth-telling) constraint for patient agents. As banks cannot discriminate between consumers based on their types, this ensures that agents reveal their true type (namely no patient agent will pretend to be impatient). For this to happen, c₂ has to be greater than c₁. Thus if a patient agent lies about her her type, she will be given c₁ units of consumption. As she actually wants to consume in period 2, the best she can do is to invest this amount in the short-term technology (storage), which implies that she would only be able to consume c₁ in period 2. As this is inferior to consuming c₂, constraint (8d) ensures that patient depositors will not lie.

Before the introduction of the foreign central bank, the solution to this problem in the closed economy context admits an equilibrium in which agents reveal their true type and there is no early liquidation of the long-term asset, e.g. $\tilde{l}=0$ In this equilibrium, the expected utility of consumers is maximized, subject to the feasibility constraints and the incentive compatibility constrains (truth-telling constraint) and banks make zero profits. This is intuitive, since banks pool resources to prevent the inefficient liquidation and face no aggregate uncertainty. Banks are a welfare-improving solution to the financial autarky world, since they essentially perform the function of pooling the risks from patient and impatient consumers and thus allow for a superior optimum allocation. As consumers behave according to their type, the optimal deposit contract achieves an allocation which coincides with the efficient allocation derived from the planner’s problem (see Appendix for derivations).

2.3 A Foreign CBDC-Issuing Central Bank

The CBDC is modelled as an account-based and interest-bearing claim on the central bank. Thus, we assume that domestic residents can invest in direct CBDC deposits at the foreign central bank. The introduction of the foreign CBDC-issuing central bank is the key modification in this model, so we describe in detail the assumptions and features which characterize it.

First, the foreign central bank is assumed as perfectly credible, in the sense that consumers face no uncertainty or risk of loss to withdrawing their deposits (e.g. the central bank is not subject to runs). This “contract rigidity” can be justified in various ways. Diamond and Dybvig (1983) show that a combination of punishment and regulatory intervention can achieve the result of deterring runs from happening. Alternatively, the assumptions of no runs on the central bank can be theoretically modelled as in Fernández-Villaverde et al. (2021), as either deriving from a punishment that the central bank can impose on depositors who attempt to run or from a situation of equal treatment, in which the central bank treats depositors who contribute to a potential run as if they had rolled over their deposits (see Appendix).

The second key feature of this highly stylized foreign central bank is that CBDC deposits at the central bank are open to foreigners. The potential demand for perceived safe store-of-value foreign currencies has been exemplified by the spread of dollarization in numerous economies. Indeed it has been conjectured that compared to hoarding cash, the potentially high accessibility and ease of availability of CBDCs could incentivize further currency substitution in countries with weak credibility of the policy frameworks.

Third, we shall assume that the foreign central bank conducts its own monetary policy, which determines the interest rate offered on CBDC deposits (and is exogenous in the model). While we will not model explicitly the foreign central bank’s monetary policy decision, further work along making this more realistic appears a promising question for future research.

Overall, such assumptions capture the situation that a central bank with sound monetary policy and a robust track-record (possibly an issuer of a reserve currency) could offer a “safe-haven” international financial asset. More specifically, we will model the foreign central bank as offering a special type of deposit contract, where in exchange for one unit of good at time 0, it allows depositors to withdraw with certainty either $c_1^{*}\in {ℝ}_{+}$ units of the common good in period 1 or $c_2^{*}\in {ℝ}_{+}$ units in period 2. The fact that the central bank deposit are perfectly “safe” is a crucial difference compared to the demand deposits offered by commercial banks (which can be subject to runs, as we discuss next). The safe rates of return for period 1 and 2 $(c_1^{*}<c_2^{*})$ can be thought of as being determined by the term-structure of foreign interest rates.

2.4 The Consumer’s Problem

The consumer must decide whether to deposit with the commercial bank, consuming either c₂ upon withdrawal in period 1 or c₂ in period 2, or with the foreign CBDC, consuming either $c_1^{*}$ upon withdrawal in period 1 or $c_2^{*}$ in period 2. For simplicity, we assume that domestic residents can invest, but not borrow internationally. A more complex international financial market is an extension which deserves further analysis, yet we conjecture that international borrowing would not change the main implications of this model in terms of the volatility of capital flows.

Consumers choose the contract which delivers the highest ex-ante utility. If both contracts offer the same utility, then some fraction f ∈ [0,1] will pick the foreign central bank, and the remaining fraction will pick the domestic commercial banks. We denote by d_i ∈ {0,1} the deposit decision of consumer i ∈ [0,1], where d_i = 0 represents depositing with the commercial banks and d_i = 1 represents depositing with the foreign CBDC.

Consumers also face a strategic decision about when to withdraw their funds. We define a withdrawal strategy for the consumer i as the variable w_i ∈ {1,2} that denotes the time period when the consumer withdraws her deposit. An impatient consumer always has w_i = 1, that is it chooses to withdraw in period 1. A patient consumer has the option to withdraw early or late, a choice which will depend on her beliefs regarding the actions of others (denoted w_-i).

A further restriction on the consumers’ portfolio choice is given by the fact that domestic agents’ investment in the foreign asset is constrained by an investment cap k > 0, which we shall call a capital account constraint. While we treat this type of investment ceiling as exogenous, it can be justified as the result of domestic restrictions (e.g. regulated capital account or capital flow measures) or other frictions and biases (e.g. home bias) that limit the amount that domestic residents would invest in foreign assets.

2.5 Equilibrium with No Runs

We define now an equilibrium for this economy provided that depositors behave according to their type. An equilibrium is a description of the strategies of each depositor and aggregate outcomes as implied by the depositors’ strategies, given that each depositor’s strategy is optimal for her given the aggregate outcomes.

Definition 1 An equilibrium consists of: a demand deposit contract (c₁,c₂) for the representative domestic commercial bank, a deposit contract $(c_1^{*},c_2^{*})$ for the foreign central bank, deposit decisions d ∈ {0,1} in the initial period, a strategy profile w ∈ {0,1} for the withdrawal game in the intermediate period, a fraction f ∈ [0,1] of consumers depositing in the foreign CBDC and a fraction α ∈ [0,1] of depositors who withdraw in period 1 such that:

• 1. In period 0, given contracts (c₁,c₂) and $(c_1^{*},c_2^{*})$, each consumer i ∈ [0,1] optimally chooses where to deposit her unit of endowment so that it offers the highest expected utility. Total deposits in the CBDC cannot exceed the capital account constraint, i.e. f ≤ k.

• 2. The strategy profile w_i ∈ {1, 2} represents a Nash equilibrium of the withdrawal game in period 1.

• 3. Each commercial banks chooses the deposit contract to offer (c₁,c₂) to maximize profits in period 2, given $(c_1^{*},c_2^{*})$.

• 4. Withdrawals in period 1 satisfy satisfy: $\alpha =1-{\displaystyle \underset{\{i\in [0,1]:w_i=2\}}{\int }di}$.

• 5. Deposits in the CBDC in period 1 satisfy: f = ∫ d_idi.

We start by first noting that we can restrict attention to symmetric equilibria, in which all commercial banks use the same deposit contract.

Lemma 2 In equilibrium, all commercial banks that have depositors make zero profits and offer the socially optimal contract.

Let’s denote by π the commercial bank’s profit, by (y(π), c₁(π), c₂(π)) the solution to the depositor’s utility maximization problem, e.g. the value function describing the maximum expected utility for the consumer, contingent on the bank making profit π :

subject to: 0 ≤ y ≤ 1, λc₁ ≤ ry + (1 – y)l, c₂ > c₁ and

By the envelope theorem, it must be that V’(π) < 0 for any π > 0. Thus, all consumers will deposit with the banks which offer the highest expected utility contract, which may thus absorb the entire mass of deposits. As banks internalize the effects of the price competition, their incentive is to offer a contract with lower profits, say 0 < π’ < π, whose payouts (y(π’), c₁(π’), c₂(π’)) would offer the consumer a higher value. It follows that π > 0 and $(y\left(\pi \right),c_1\left(\pi \right),c_2\left(\pi \right))\to (\overline{y},{\overline{c}}_1,{\overline{c}}_2)$, which implies that all banks offer the socially optimal contract and make zero profits.

Next, we discuss the competition for deposits between domestic commercial banks and the CBDC-issuing central bank. First, it is trivial to see that there is an equilibrium in which, through its remuneration of CBDC deposits, the foreign central bank could replicate the socially optimum contract.

Lemma 3 The foreign central bank can replicate the socially optimum bank deposit contract, if the interest rates on the CBDC are such that $c_1^{*}={\overline{c}}_{1}and{c}_2^{*}={\overline{c}}_2=\frac{R(1-\overline{y})}{1-\lambda }$.

While the foreign central bank is capable of replicating this outcome, its objective function may be quite different. This gives rise to different outcomes, since effectively there will be a competition between the central bank and the commercial banking sector.

Proposition 4 If only domestic commercial banks offer the optimal contract, they absorb de entire deposit market and f = 0. If only the foreign central bank CBDC deposit offers the optimal contract, or if the foreign central bank offers a better contract than the optimal contract, then the CBDC absorbs all deposits up to the capital account constraint, e.g. f = k. If both the domestic commercial banks and the foreign central bank offer the optimal contract, then any f ∈ [0, k] is an equilibrium.

Proof. Assume that the foreign central bank offers a deposit contract $(c_1^{*},c_2^{*})$ which is offering a value $V=\lambda u\left(c_1^{*}\right)+(1-\lambda )u\left(c_2^{*}\right)$ to the consumer. Because V(.) is continuous and decreasing, there exists a profit level π^C so that the commercial bank can replicate the central bank contract. If π^C = 0, then the central bank has been offering the optimal contract, in which case consumers are indifferent where they deposit their savings, and any f ∈ [0,k] is an equilibrium. If π^C > 0, then the central bank is not offering the socially optimal contract, and the commercial bank can offer a better contract with π < π^C thus undercutting the central bank. Through Bertrand competition between the commercial banks, profits will be driven to 0 and the socially optimal contract will prevail, attracting all deposits, namely f = 0. If however π^C < 0, this implies that the contract offered by the CBDC is better than the domestic socially optimal contract. As banks are not able to replicate this contract, they will lose depositors, up to the capital constraint of f = k. ■

2.6 Domestic Commercial Bank Runs

So far we have considered a situation in which depositors behaved according to their types, and found that in this situation, the socially optimal contract is offered by commercial banks. When the foreign CBDCs mimics the payoffs of the optimal contract, there is competition for deposits with domestic banks, which may give rise to capital outflows (trivially, this also occurs if the foreign CBDC deposit payoffs dominate). However, when consumers do not behave according to their types, a different equilibrium is also possible, in which their payoffs may deviate from those promised in the deposit contract.

In this section, we consider the possibility of panics in the domestic commercial bank system. These occur when patient consumers, instead of consuming in period 2, instead rush to the banks and demand their deposits early, under the apprehension that other agents may do the same. Since an agent’s type is unobservable, it is possible for the agent to pretend to be impatient, with the consequence that a self-fulfilling run occurs.

When patient depositors withdraw early, e.g. if α ∈ (λ, 1], the commercial bank needs to liquidate some portion of the long-term investment in order to service the early redemptions. The timing of events is as follows. In period 1, depositors arrive at the bank in random order. The bank satisfies a sequential-servicing constraint which requires that it attends to all withdrawal requests on a first-come first-served basis. If the bank is forced to liquidate the long-term asset, it uses the proceeds from the early liquidation $(1-\overline{y})l$ and the return on the short-term investment $\overline{y}$ to satisfy as many as possible of the withdrawals. This may not be able to fully satisfy every consumer if $\alpha {\overline{c}}_1>r\overline{y}+(1-\overline{y})l$. The expected payoff of an agent who liquidates early in case of a run is thus rationed by the number of agents in the queue who demand to be served, namely $\frac{r\overline{y}+(1-\overline{y})l}{\alpha {\overline{c}}_1}$.

Furthermore, early liquidation of the long-term asset reduces the payoffs to those depositors who roll-over at period 1. However, agents internalize the reduced payoffs to depositors who remain patient, which determines them, if they believe that others would run on the bank, to also run. If all agents attempt to withdraw their deposits in period 1, the bank will run out of resources and fail before it can meet all the claims made on it, which implies that the payoff to consumer from rolling over during a bank-run is 0.

The payoffs matrix can thus be described as follows:

Event/Action	Withdraw	Roll-over
No	$u\left({\overline{c}}_1\right)$	$u\left(\frac{R[(1-\overline{y})-(\alpha -\lambda ){\overline{c}}_1/l]}{1-\alpha }\right)$	(6)
Run	$\frac{r\overline{y}+(1-\overline{y})l}{\alpha {\overline{c}}_1}u\left({\overline{c}}_1\right)$	0

View Table

Event/Action	Withdraw	Roll-over
No	$u\left({\overline{c}}_1\right)$	$u\left(\frac{R[(1-\overline{y})-(\alpha -\lambda ){\overline{c}}_1/l]}{1-\alpha }\right)$	(6)
Run	$\frac{r\overline{y}+(1-\overline{y})l}{\alpha {\overline{c}}_1}u\left({\overline{c}}_1\right)$	0

The payout structure makes clear that this game exhibits strategic complementarity in actions: conditional on a bank run, the payoff from withdrawing exceeds the payoff form rolling over and withdrawing is optimal. However, in the event of no run (patient consumers acting according to their type), then the payoff from rolling over exceeds the payoff from withdrawing and roll-over is optimal (as ${\overline{c}}_1<{\overline{c}}_2$).

Proposition 5 The withdrawal game of commercial bank deposits has two pure equilibria. There is one “good” equilibrium, in which all patient agents roll-over their deposits in period 1 (α = λ), and the socially-optimum contract is attained. However, there is also a second “bank-run “ equilibrium, where all patient depositors panic and demand to withdraw in period 1, assuming that others would do the same (α = 1). In this “bad” equilibrium, a lower allocation than in the socially optimum contract is attained.

2.7 Equilibrium with Runs in the Foreign CBDC

Relative to the framework presented before, we further assume the presence of the foreign CBDC riskless deposit which competes with domestic commercial bank deposits. The safety features of CBDC deposits are empirically credible, as central banks can operate even with negative equity without being forced into bankruptcy (even abstracting from their ultimate fiscal backing). However, there are various ways to rationalize the risk-free central bank deposits, without imposing it exogenously (two such mechanisms are presented in Annex B).

The presence of ‘safe’ CBDC deposits leads to the following result.

Lemma 6 If the foreign central bank offers a riskless deposit contract which mimics the payoff of the socially-optimal contract, then it will attract all deposits up to the capital account constraint.

The intuition for this result is simple. Even if the domestic commercial banks offer the socially-optimal contract, they can still face the risk of runs, leading to a worse outcome. If the foreign central bank were to offer the socially-optimal allocation, it would be able to guarantee the returns, thus it would attract all deposits. The domestic economy will experience capital outflows into the foreign CBDC up to the limit imposed by capital account restriction.

Given that the CBDC deposit is less risky than domestic commercial bank deposits, the foreign central bank enjoys a type of market power which allows it to offer a “safe” deposit contract even with inferior payoffs to the socially-optimum contract, and still attract all deposits up to the capital account constraint. Essentially, the risk-free asset that the central bank provides can be offered as a discount because of its superior safety features.

Proposition 7 As consumers internalize that the central foreign bank deposit contract is perfectly safe, the foreign central bank can offer a deposit contract with lower payoffs than the socially optimal one, and still attract the highest possible amount of deposits (up to the capital account constraint).

Proof. Let’s denote by U₁ the utility the agent derives from the socially-optimal commercial bank deposit contract, which can be subject to runs, and by U₂ the utility derived from a ‘safe’ CBDC deposit contract with the same payouts. As runs can occur with commercial banks, but not with the central bank, it has to be that U₁ < U₂. In this case, the central bank can offer lower payoffs $c_1^{*}<c_{1}and{c}_2^{*}<c_2$ such that $U_1<U(c_1^{*},c_2^{*})<U_2$ and still attract all deposits up to the capital account constraint. ■

We have previously seen that the presence of a foreign CBDC as an alternative international “safe” asset may increase the risk of financial instability for the domestic banking sector even in the absence of panics, if it is at least as attractive as domestic deposits. When runs are possible, the result is strengthened. The foreign CBDC could pay lower interest rates and still, due to its superior safety features, lead agents to flee domestic assets – absent CBDC features or other frictions that would prevent such an outcome. This case is indeed the most relevant one in practice, given that in general interest rates in countries with credible central banks and/or reserve currency issuers would tend to be lower than those in most emerging and developing economies.

3.1 Conclusions

In a streamlined model of banks runs with a foreign CBDC, we find that the introduction of the CBDC leads to competition for deposits between the foreign central bank and private sector banks. The foreign CBDC can become an attractive asset to hold especially if it is issued by a credible foreign central bank and pays interest. The model allows for an equilibrium in which self-fulfilling runs on domestic banks can occur, if investors believe others would withdraw early. Given the likelihood of domestic runs, the foreign CBDC which offers superior safety ends up attracting all deposits from the commercial banking sector, up to a capital account constraint on outflows. Thus, domestic commercial banks may be drained of a significant fraction of their deposits. Absent some form of capital controls or CBDC design features which would limit non-resident inflows, large scale disintermediation of the banking system is possible, with negative implications for bank lending and financial stability more broadly.

The efficient allocation in this case will represent an equilibrium in which the domestic economy experiences higher capital outflows, as domestic agents shift part of their deposits into the foreign CBDC. The salient feature driving the results is that the foreign CBDC opens up the central bank balance sheet to foreigners, and given its high credibility (effectively offering a global safe asset), domestic agents find such deposits attractive, even when they offer lower returns than the more productive domestic investment technology. Thus, the forces driving currency substitution in countries with weak policies and fundamentals may be further reinforced.

This paper highlights one key avenue through which allowing non-residents to use a CBDC issued in another jurisdiction may give rise to financial instability due to capital outflows. The presence of elevated risks to financial stability in both issuer and recipient countries calls for international cooperation and coordination in the design of CBDCs. Regardless of the particular architectural solutions, key design principles for international CBDCs would benefit from being coordinated at the global level, so that they best benefit all countries and minimize the risk of currency substitution and financial instability.