Large excess reserves, central bank digital currency, and monetary policy

Abstract

In this paper I examine the effect of introducing an account-based central bank digital currency (CBDC) on liquidity insurance and monetary policy implementation. An asset-exchange model is constructed with idiosyncratic liquidity risk, in which one type of agents require currency and/or CBDC to consume while the other type of agents can use any assets to trade. There arises a liquidity insurance to distribute assets efficiently by type. Since central bank reserve accounts are accessible by the public directly, the large excess reserves (LER) in a floor system can make it difficult to separate the types under private information. Therefore, raising the interest on reserves in the floor system can reduce the aggregate liquidity excessively, and the equilibrium allocation with the LER can be suboptimal.

1. Introduction

Recent development in blockchain technology have led not only a number of private digital currencies such as Bitcoin and Ethereum to be widely spread in the real world, but also the central bankers to be interested in issuing its own digital currency to the public. Technically, central banks have already issued account-based digital currency in a form of central bank reserves.¹ However, only depository institutions can access to reserve accounts and use it for interbank transactions through a real-time gross settlement system. In this respect, introducing central bank digital currency (CBDC) amounts to provide consumers an opportunity of holding a reserve account with the central bank directly.² This “central bank digital money for all” idea is proposed and promoted explicitly in the recent literature.³ Barrdear and Kumhof (2022) refer to a retail CBDC by meaning that “the central bank grants universal, electronic, 24 x 7, national-currency-denominated and interest-bearing access to its balance sheet.” Bordo and Levin (2017) provides a way of implementing CBDC as “such accounts could be held directly at the central bank itself or made available via public-private partnerships with commercial bank.” Berentsen and Schar (2018) discuss the advantages of introducing CBDC for all, which includes that the monetary policy becomes more transparent.⁴

Nevertheless, there are still many questions that need to be carefully addressed before this proposal is implemented. Specifically, this proposal needs to be examined in a perspective of the monetary policy framework such as channel or floor system, because the effect of monetary policy may vary by the amount of excess reserves if the reserves is allowed to be held by the public. In response to the last Great Recession, the Federal Reserve System has implemented various types of unconventional monetary policy with asset-purchase programs. As a result the excess reserves held in the banking sector had grown substantially, and the Federal Reserve began paying interest on reserve balances and shifted the monetary policy framework from the channel system to the floor system in October 2008. Afterwards, as a policy normalization plan the Federal Reserve announced to put a priority on a liftoff before unwinding in October 2019, so the level of reserves remains ample.⁵ Therefore, introducing CBDC, especially in this floor system, provide a challenge to policy makers, because we have limited experience with the large excess reserves (LER) and have not considered seriously the central bank “open to all” yet in our monetary history.

In this paper I pay attention to the liquidity insurance of the banking sector, which is one of their primary functions. When the consumers are exposed to an idiosyncratic liquidity shock, it would be optimal to distribute liquidity by type. For instance, if one type of consumers require cash and/or CBDC for retail transactions while the other type of consumers can use any asset holdings including reserves and government bonds for collateral transactions, then it is efficient to provide cash/CBDC to the former type and the rest of assets to the latter one. If the consumers can access to reserve accounts at any time and use CBDC, then the reserves can be used in both types of transactions. Thus, if the whole banking sector’s balance sheet is filled with the large excess reserves to the extreme, it would be nearly impossible to distinguish their types under anonymity. This difficulty of revealing types could reduce the bank’s demand for reserves, while raise the demand for less liquid assets such as government bonds that cannot be used in retail transactions. The monetary policy is implemented by changing the supplies of liquid and illiquid assets such as reserves and government bonds in channel system or by adjusting the relative price between these assets in floor system. Thus, when the banks choose their asset portfolio, this change in the demand for assets can also make an impact on the monetary policy implementation. In this respect it will be meaningful to analyze the effect of introducing CBDC in each monetary policy regime by considering the banks’ asset portfolio choice for liquidity insurance.

In order to study this issue I develop a search-theoretical monetary model a la Lagos and Wright (2005) and Rocheteau and Wright (2005) with a Diamond and Dybvig (1983) type banking contract shown in Sanches and Williamson (2010) and Williamson (2012, 2016). This model has an advantage of incorporating limited commitment and private information frictions in a simple way and is also highly tractable with insurance contracts and an array of assets. In the model agents can produce consumption goods with an elastic labor supply, but cannot consume by themselves. Under limited commitment and lack of record-keeping, agents need a medium of exchange to trade for consumption: one type of agents can use only cash and CBDC based on reserve accounts, while the other type can use the whole asset portfolio including reserves, government bonds and private assets.⁶ The lack of memory assumption not only makes assets essential for transactions, but also keeps the type information of the agents secret.⁷ An insurance contract with truth-telling incentive constraints arises to reveal the types by offering different types of assets. After announcing their type and receiving the assets, each agent has an opportunity to access their reserve accounts in the central bank.⁸ The central bank issues cash, CBDC and reserves by holding government bonds and can provide positive interests on CBDC and reserves.

There could be three equilibrium regimes according to the monetary policy variables, that is, channel system, floor system (without the LER), and floor system with the LER. In the channel system there are no excess reserves. The monetary policy adjusts the supplies of reserves and government bonds through open-market operations (OMOs) to control the liquidity premium of each asset separately and to affect the real interest rates. OMOs represents a conventional monetary policy tool to affect the real allocations by exchanging reserves with government bonds.⁹ This mechanism will be maintained and work well when account-based CBDC is introduced. Since both cash and CBDC are generated from reserve accounts, although cash is replaced by CBDC and nothing will be changed.

In the floor system, OMOs are no longer effective because there is no return dominance between reserves and government bonds with excess reserves. Therefore, the real allocations are determined solely by the level of the interest on reserves, and adjusting the interest rate on reserves can make the same effect as OMOs in the channel system. In this case the public access to the reserve accounts does not make any changes as long as the types can be separated well. The agents who will participate in retail transactions would choose an offer with reserves only to withdraw cash or to use CBDC, while the agents who will engage in collateral trades would choose the asset portfolio including government bonds.

If the excess reserves are sufficiently large, one of the truth-telling incentive constraints could bind as agents can access to reserve accounts. In this new regime, that is the floor system with the LER, the OMOs can be effective. Suppose that the bank’s balance sheet is filled with the large excess reserves. When the bank distributes the liquid assets by types, one type agents prefer cash/CBDC and the other type agents will receive a larger amount of reserves and bonds instead. Since the agents can access to the central bank and convert reserves into CBDC, the former agents have an incentive to mimic the latter agents and receive reserves to convert it into CBDC. Therefore, the quantities of cash/CBDC to the former agents must be greater than or equal to the quantity of reserves to the latter agents. Since it is difficult to separate the types with large excess reserves, government bonds are more useful than reserves. Thus, OMOs can relax or tighten the binding truth-telling constraint and affect the real interest rates.

In this situation the assets are used not only for medium of exchange, but also for revealing the types in this case. Thus, the liquidity premium of one asset can depend on the conditions of the corresponding asset market and the other asset market. For example, reserves are useful for the collateral transactions, but also can tighten the binding truth-telling constraint because it can be used as CBDC in retail transactions. Therefore, the liquidity premium of reserves reflects the frictions in both trading markets as a weighted average.

More importantly, the liquidity distribution across the types can be inefficient in the floor system with the LER. With the large excess reserves, the agents for retail transactions may pretend to be an agent who will use reserves as collateral, and use CBDC with their reserve accounts afterwards. Thus, the insurance contract has to provide less reserves to the agents for collateral trades inevitably and the efficiency of the asset transactions in this economy is lowered.

These results can provide some implications for introducing account-based CBDC in monetary policy implementations. First, the effectiveness of OMOs relies not only on the liquidity of the assets that are exchanged, but also on the convertibility of the illiquid asset to the liquid asset or vice versa. Both reserves and government bonds are used for the same type of collateral trade, but they are not perfect substitutes. If reserve accounts are accessible, then reserves can be converted to CBDC and used for retail transactions. Therefore, the liquidity premia of reserves and government bonds could be different and OMOs can be effective. Second, if the truth-telling constraint binds with the large excess reserves, the effect of raising the level of interest on reserves on the real interest rates depends on the amount of excess reserves. Therefore, the amount of excess reserves must be considered even though the level of the interest on reserves is used as a main implementation tool in the floor system. Finally, the floor system with the LER is suboptimal in a respect of aggregate liquidity and welfare, although OMOs are effective. Therefore, reducing the amount of excess reserves by using overnight reverse repurchase (ON-RRP) facility before allowing the access to reserve accounts can be helpful to improve welfare.

1.1. Related literature

There has been a growing literature on CBDC including its motivations and implications.¹⁰ Specifically, Sanches and Keister (2019) and Williamson (2019a) construct a model in which CBDC can replace either cash or deposit, and evaluate the welfare effect along with its negative impact on financial intermediation and the cost for central bank independence, respectively. Andolfatto (2021) and Chiu et al. (2023) develop models in which an interest-bearing CBDC competes with bank deposits and show that the introduction of CBDC need not disintermediate banks by reducing their lending, if the banks’ market power in the deposit market is limited by CBDC.¹¹ On the other hand, in this paper CBDC is assumed to compete only with cash, because we focus on the efficiency of distributing liquidity when the people can use CBDC with their own reserve accounts.

There is also a number of literature on monetary policy with excess reserves in floor system. Since the interest on reserves can be used as a policy tool independently, Goodfriend (2002), Ennis and Weinberg (2007), and Keister et al. (2008) point out that the central bank can have an additional degree of freedom to choose the quantity of reserves by providing the interest on reserves.¹² Recently, Ireland (2014) and Ennis (2018) investigated the decoupling between the supply of reserves and the price level given the interest on reserves: Ireland (2014) finds out that the quantity of reserves can affect the nominal variables when there exists a cost for managing reserves. Similarly, Ennis (2018) shows that the link between the supply of reserves and prices can be tightened again when the bank capital constraint binds with a large balance sheet. In my model the price variables are determined by the government budget constraint rather than the supply of reserves, which is similar to Cochrane (2014).¹³ However, the supply of reserves can be used as a policy variable because the accessibility to reserve accounts provides an opportunity to use reserves directly as CBDC.

In terms of the (in)effectiveness of the monetary policy, the results of this paper are consistent with the literature. The ineffectiveness of monetary policy in this paper is close to the liquidity trap in Wallace (1981) rather than Williamson (2012) and Rocheteau et al. (2018): Since excess reserves and government bonds can be used for the same type of transactions, open-market-operations are irrelevant at any positive interest rate on reserves. Note that the main result of this paper is different from Williamson (2012). Williamson (2012) finds out that a new type of liquidity trap equilibrium, where the open-market operations are ineffective as the rates of return on money and bonds are equal, can arise due to the scarcity of the asset and also exist with a strictly positive nominal interest rate in a floor system. This paper extends his model and finds out that another new equilibrium, where the open-market operations are effective even with the large excess reserves, can arise if the reserves are accessible and convertible to the retail CBDC.

In the model when the truth-telling incentive constraint binds with a sufficiently large excess reserves, monetary policy is less effective. This result is similar to Agenor and Aynaoui (2010) and Bech and Klee (2011), but the mechanism is different from their models. Agenor and Aynaoui (2010) show that given a precautionary demand for excess reserves, contractionary monetary policy can be less effective with prevailing excess reserves. Bech and Klee (2011) build a limited participation model with over-the-counter market to show that the federal funds rate rises less when the interest on reserves increases. In Armenter and Lester (2017) the interest rate spread in a corridor system can have a real effect because the spread between the two rates determines the trading gain and the efficiency of matching in their model.

This paper is also related to the literature that studies the bank’s optimal decision with the interest on reserves and the excess reserves. Dressler and Kersting (2015) and Martin et al. (2016) focus on the bank’s lending behavior given the excess reserves, while Dutkowsky and VanHoose (2013) and Dutkowsky and VanHoose (2017) study the bank’s decision on sweeping and the size of bank balance sheet given the interest on reserves. In this paper, banks provide liquidity insurance given idiosyncratic liquidity risk as shown in Williamson (2016, 2019b), but I focus more on how introducing CBDC with the LER can affect the bank’s decision on liquidity allocation under private information and the effectiveness of the monetary policy.¹⁴

For the welfare evaluation between channel system and floor system, Berentsen et al. (2014) show that there is a welfare improvement in channel system when lending is costly, because banks will hold more reserves ex ante, which can internalize the pecuniary externality in holding reserves. Williamson (2016) shows that the welfare can improve in the floor system by injecting more reserves given the interest on reserves, when reserves are more useful than government bonds in a liquidity aspect. In this paper, the channel system without excess reserves is preferred to the floor system in a perspective of aggregate liquidity supply, because the monetary policy is more effective when the trading markets are sufficiently segregated.

Finally, this paper is related to some papers that study the liquidity premium of illiquid assets. Herrenbrueck and Geromichalos (2017) show that the price of illiquid assets has a liquidity premium if the illiquid assets can be traded to obtain liquid assets. In this paper when the interest on reserves is implemented, the liquidity premium of the reserves decreases and the liquidity premium in the government bonds goes up, because the less liquid government bonds are more useful for revealing private information as the reserves are more easily convertible to currency. Therefore, when the truth-telling incentive constraint binds, injecting illiquid assets can be beneficial. In a respect of the social benefit of illiquid assets, the result of this paper can be interpreted along with Kocherlakota (2003) and Shi (2008). Kocherlakota (2003) shows that illiquid assets are beneficial because agents can trade liquid and illiquid assets after observing idiosyncratic shock. Shi (2008) shows that legal restriction on government bonds improves welfare when low marginal utility type cannot trade with bonds.

2. The environment

The basic model structure is based on Rocheteau and Wright (2005) in which ex ante heterogeneous agents trade in bilateral meetings and rebalance their portfolios in the centralized market. Time is discrete over infinite horizon and each period is divided into two sub-periods—the Centralized Market (CM) followed by the Decentralized Market (DM). There is a continuum of buyers, sellers and bankers, each with unit mass. An individual buyer has preferences

\begin{equation*}E_{0}\overset {\infty }{\underset {t=0}{\sum }}\beta ^{t}[\!-H_{t}+u(x_{t})]\end{equation*}

where $H_{t}\in \mathbb{R}$ is the labor supply of the buyer in the CM, $x_{t}\in \mathbb{R}_{+}$ is the consumption of the buyer in the DM, and $0\lt \beta \lt 1$ . Assume that $u(\cdot )$ is strictly increasing, strictly concave, and twice continuously differentiable with $u'(0)=\infty, \, u(0)=0$ , and $-x\frac{u^{''}(x)}{u'(x)}=\gamma \lt 1$ for all $x\gt 0$ .¹⁵ For the proofs, a specific utility function $u(x)=\frac{x^{1-\gamma }}{1-\gamma }$ is used. Each seller has preferences as

\begin{equation*}E_{0}\overset {\infty }{\underset {t=0}{\sum }}\beta ^{t}[X_{t}-h_{t}]\end{equation*}

where $X_{t}\in \mathbb{R}$ is the consumption of the seller in the CM, and $h_{t}\in \mathbb{R}_{+}$ is the labor supply of the seller in the DM. All the agents can consume and produce in the CM. But in the DM only buyers can consume and only sellers can produce. One unit of labor inputs can produce one unit of perishable consumption goods in either the CM or the DM.

In the CM all agents meet together and then production and consumption occur. Buyers receive a lump-sum transfer from the government, and the share holders of the Lucas trees receive the realized dividends. The previous debts are paid off, and assets and consumption goods are traded in a Walrasian market.

In the DM each buyer meets each seller randomly, and the terms of trade are determined by bargaining in the bilateral meeting. For simplicity, we assume that the buyer makes a take-it-or-leave-it offer to the seller in the meetings. Basically, there is no record-keeping technology, so the buyers and the sellers cannot verify the trading history of their partners in the DM. Moreover, no one can be forced to work under limited commitment. Thus, recognizable assets are essential for transactions in the DM and all of the trades must be quid pro quo. In a manner similar to Sanches and Williamson (2010) and Williamson (2012), there are two types of DM meetings. In a $\rho$ proportion of DM meetings, which represents retail transactions, buyers will meet a seller who accepts only currency and CBDC. In the rest of DM meetings buyers will meet a seller who accepts the claims for the whole asset portfolio as collateral. It is assumed that the seller would prefer to accept the central bank’s claims instead of the bank’s deposit claims due to the credit risk in the settlement process.¹⁶ In the following we refer these buyers as type 1 and type 2 buyers, respectively.

Figure 1. Time-line.

At the beginning of the CM, buyers do not know which type of match they will be in the DM. Given this idiosyncratic shock, buyers can write an insurance contract with banks. After the Walrasian market is closed in the CM, buyers learn their types and their types are private information. Once the type information is realized, each buyer can contact with his or her bank. Finally, buyers can access their reserve accounts in the central bank at the end of the CM.

There are an array of assets such as currency, CBDC, reserves and government bonds, which are the liabilities of both the fiscal authority and the central bank, in this economy. The fiscal authority can issue government bonds and provide nominal interests by collecting lump-sum taxes from buyers. One unit of government bonds trades at price $q_{t}$ in terms of currency in the period $t$ CM and pays one unit of currency in the period $t+1$ CM. The central bank can issue currency, CBDC, and reserves by purchasing government bonds. One unit of currency sells at price $\phi _{t}$ in terms of consumption goods in the period $t$ CM and provides no interest. While currency is a physical object produced by the central bank, CBDC and reserves are account balances in the central bank. We define CBDC as an amount of reserve balances that is used in the $\rho$ proportion of DM meetings. One unit of CBDC yields the CBDC holder in the next CM a total of $R^d_t$ units of CBDC where $R_t^d$ represents a gross nominal interest on CBDC. One unit of reserves, if the ownership is not changed in the DM, earns a total of $R^d_t R^m_t$ units of reserves in the next CM, where $R_t^m$ represents an additional gross nominal interest on reserves. As CBDC is also an account balance, we assume that the central bank can pay a non-negative interest on CBDC less than or equal to the interest on reserves, $1\leq R^d_t \leq R^d_t R^m_t$ to characterize the allocations in a simple way.^{17, 18} Moreover, we assume that CBDC transactions are not traceable as the same as currency, because we focus on the effect of using reserves accounts in retail market rather than replacing currency with CBDC itself.¹⁹

In the model currency and CBDC are substitutable because both assets can be used in the retail transactions and converted into the reserves immediately. Therefore, buyers would use CBDC if it provides a strictly positive nominal interest and use both cash and CBDC if the nominal interest rate is zero. However, reserves and government bonds are quite different, because buyers can use reserve balances as CBDC in the retail transactions, although both assets are useful in the $1-\rho$ proportion of transactions as collateral.²⁰

Timing is described in Figure 1. At the beginning of the CM, government debt holders receive a unit of currency by redeeming a unit of government bonds. Buyers receive lump-sum transfers (or pay lump-sum taxes). Then all buyers and sellers provide labor and trade assets in a Walrasian market. Buyers deposit consumption goods and/or assets with a bank in exchange for an insurance contract. After the liquidity shock is realized, buyers learn their types and $\rho$ proportion of buyers can withdraw currency and/or receive CBDC from the bank. Then, the buyers can access to the central bank before moving to the DM. In the DM buyers meet sellers randomly in the bilateral meeting and can trade with a take-it-or-leave-it offer.

2.1. Banks

Given the idiosyncratic liquidity risk, a banking contract arises endogenously to allocate currency and CBDC and the other assets across the types of buyers. Without this insurance arrangement, type 1 buyers could run out of currency/CBDC and holding idle government bonds while type 2 buyers could hold low-yielding currency/CBDC instead of other high-yielding assets. Therefore, the banking contract can provide liquidity insurance by allocating currency/CBDC to type 1 buyers and the rest of assets to type 2 buyers. Given the perfect competition, any agents can suggest an optimal banking contract that maximizes the expected utility of buyers and play a role as banks.

Under private information the banks cannot verify the type of individual buyers, so one type of buyers can mimic the other type of buyers. In general this private information friction does not matter because the reserves cannot be accessed by the buyers as shown in Sanches and Williamson (2010) and Williamson (2016). However, if CBDC is introduced and buyers can access to their reserve accounts additionally, then it is difficult for banks to reveal the types by providing different types of assets. For example, type 1 buyers are willing to mimic type 2 buyers for receiving a sufficient amount of reserves, because they can use their reserve balances as CBDC in the retail transactions. Therefore, given the excess reserves, the banks are required to adjust the amount of assets for both types of buyers to separate the types efficiently, and thus the effect of monetary policy could be changed.

We assume that all the claims issued by the banks and the central bank are not counterfeitable and buyers can meet only one bank after their liquidity shock is realized to prevent the banking contract from being unraveled.²¹

In order to know whether this problem is created by the insurance contract or not, we introduce a type of liquidity market from Berentsen et al. (2007) as shown in the Appendix B, and find that the main results are maintained.

2.2. Government

At $t=0$ government bonds are issued by the fiscal authority, and then currency, CBDC and reserves are injected by the central bank through open-market purchases. The initial revenue of issuing currency, CBDC, reserves, and government bonds is transferred to buyers. After $t=0$ , outstanding currency, CBDC, reserves and government bonds amounts can be supported by collecting taxes or providing transfers over time. So the consolidated government budget constraints can be written as

\begin{equation*}\phi _{0}(C_{0}+D_{0}+M_{0}+q_{0}B_{0})=\tau _{0}=V\end{equation*}

and

\begin{align*} \phi _{t}\{C_{t}-C_{t-1}+D_{t}-R^{d}_{t-1}D_{t-1}+M_{t}-R^{m}_{t-1}M_{t-1}+q_{t}B_{t}-B_{t-1}\}&=\tau _{t},\: t=1,2,3,\ldots \end{align*}

where $C_{t}$ , $D_{t}$ , $M_{t}$ and $B_{t}$ denote the nominal quantities of currency, CBDC, reserves and government bonds held by the private sector in the CM at time $t$ , respectively. In the model the central bank can buy and sell the private assets such as the Lucas trees.²² $\tau _{t}$ denotes the real value of the lump-sum transfer from the fiscal authority to each buyer in the CM at period $t$ . As described in Williamson (2012, 2016), we assume that the fiscal authority keeps the total value of the outstanding consolidated government debt as a constant, $V$ , after the fixed amount, $\tau _{0}$ , is transferred at $t=0$ . This fiscal rule plays a role as separating fiscal policy and monetary policy. The fiscal policy manages the total quantity of government debt, and the monetary policy adjust the composition of the debt by exchanging liquid assets such as currency, CBDC, reserves with government debt. Since this fiscal rule restricts the total supply of public safe assets as scarce, in the model we analyze the monetary policy effect when the fiscal policy is limited.²³

On the other hand, the amount of private liquidity in this economy can vary by the price level of the Lucas tree. Since the real rate of return on the Lucas tree is negatively related to its equilibrium price level in this model, the aggregate liquidity supply could increase as the real interest rate falls. With this setting, we can observe the effect of monetary policy on aggregate liquidity in the private sector.

In every period to maintain the real value of outstanding consolidated government debt, the real term of lump-sum transfer $\tau _{t}$ is derived passively from

\begin{align*} & \begin {array}{cccc} \tau _{t}= & \underbrace {(1-\frac {\phi _{t}}{\phi _{t-1}})V} & +\underbrace {\frac {\phi _{t}}{\phi _{t-1}}\{(1-R^d_{t-1})\phi _{t-1}D_{t-1}+(1-R^m_{t-1})\phi _{t-1}M_{t-1}+(q_{t-1}-1)\phi _{t-1}B_{t-1}}\},\: \\ & seigniorage & real\, interest\, payment \end {array}\\[5pt] &t=1,2,3,\ldots . \end{align*}

Note that the lump-sum transfer consists of seigniorage from inflation, real interest payments for government bonds and reserves, and investment earning from the Lucas tree.

3. Maximization problem

Under perfect competition a representative bank suggests an insurance contract to maximize the buyer’s ex ante expected utility. In this respect, the buyer’s problem is trivial because it is solved by the bank. Moreover, the seller’s problem is also trivial since sellers always accept the buyer’s take-it-or-leave-it offer in equilibrium. Thus, to construct an equilibrium, we focus on the bank’s maximization problem and asset market clearing conditions. A representative bank solves the following problem in the CM of period $t$ :

(1) \begin{equation} \underset{k_{t},c_{t},d_{t},m_{t},b_{t},x_{1t},x^{m}_{2t},x^{b}_{2t}}{Max}-k_{t}+\rho u(x_{1t})+(1-\rho ) u(x^{m}_{2t}+x^{b}_{2t}) \end{equation}

subject to the participation constraint of the bank,

(2) \begin{equation} \begin{array}{ll} k_{t}-c_{t}-d_{t}-m_{t}-q_{t}b_{t}+ \{\frac{\beta \phi _{t+1}}{\phi _{t}}c_{t}+\frac{\beta \phi _{t+1}}{\phi _{t}}R^{d}_{t}d_{t}-\rho x_{1t}\}\\ +\{\frac{\beta \phi _{t+1}}{\phi _{t}}R^{d}_{t}R^{m}_{t}m_{t}-(1-\rho ) x^{m}_{2t}\} +\{\frac{\beta \phi _{t+1}}{\phi _{t}}b_{t}-(1-\rho )x^{b}_{2t}\}\geq 0 \end{array} \end{equation}

and the currency/CBDC, reserves and assets constraints,

(3) \begin{equation} \frac{\beta \phi _{t+1}}{\phi _{t}}c_{t}+\frac{\beta \phi _{t+1}}{\phi _{t}}R^{d}_{t}d_{t}-\rho x_{1t} \geq 0, \end{equation}

(4) \begin{equation} \frac{\beta \phi _{t+1}}{\phi _{t}}R^{d}_{t}R^{m}_{t}m_{t}\zeta -(1-\rho )x^{m}_{2t}\geq 0, \end{equation}

(5) \begin{equation} \frac{\beta \phi _{t+1}}{\phi _{t}}b_{t}-(1-\rho )x^{b}_{2t}\geq 0, \end{equation}

and the truth-telling constraints,

(6) \begin{equation} R^m_t x_{1t}\geq x^{m}_{2t}, \end{equation}

(7) \begin{equation} x^{m}_{2t}+x^{b}_{2t}\geq R^m_t x_{1t}, \end{equation}

and non-negative constraints,

(8) \begin{equation} k_{t},c_{t},d_{t},m_{t},b_{t},x_{1t},x^{m}_{2t},x^{b}_{2t}\geq 0. \end{equation}

The problem (1) subject to constraints (2)-(8) states that a banking contract is chosen in equilibrium to maximize the expected utility of the representative buyer subject to the participation constraint for the bank (2) and the resource constraints for currency/CBDC, reserves and private assets (3)-(5) and the truth-telling incentive constraint for type 1 and 2 buyers, respectively, (6)-(7) and non-negativity constraints (8). In (1)-(8) $k_{t}$ denotes deposit of buyers, $c_{t}$ , $d_{t}$ , $m_{t}$ , and $b_{t}$ denote the quantities of currency, CBDC, reserves and government bonds in terms of the CM good in the period $t$ held by banks. $x_{1t}$ denotes the consumption of type 1 buyers in the period $t$ DM, while $x_{2t}^{m}$ and $x_{2t}^{b}$ denote the consumption of type 2 buyers in the period $t$ DM based on reserves and the rest of assets, respectively, and we define $x_{2t}$ as the sum of $x_{2t}^{m}$ and $x_{2t}^{b}$ .

The participation constraint (2) implies that the net payoff for the bank must be non-negative. In the period $t$ CM, the bank receives $k_{t}$ deposits and invest in assets, and then provides $x_{1t}$ amount of currency/CBDC to type 1 buyers at the end of the CM and $x_{2t}$ amount of consumption goods to agents who hold the deposit claims in the next period $t+1$ CM. The currency/CBDC, reserves, and assets constraints (3)-(5) represent the resources of the bank. $\zeta$ reflects a credit risk of the claims in a settlement process: The bank’s deposit claims backed by CBDC/reserves can be used with the lower pldgeability $\zeta =\hat{\zeta }\lt 1$ while CBDC/reserves can be used with $\zeta =1$ .²⁴ Truth-telling incentive constraints (6)-(7) imply that each type of buyers prefers their own offer to the offer for the other type. Since buyers can access to their reserve accounts, type 1 buyers can receive reserves and other assets by mimicking type 2 buyers, and use CBDC in the retail market. Similarly, type 2 buyers can receive currency/CBDC by mimicking type 1 buyers and deposit it to the central bank as reserves.

From now on I focus on stationary equilibrium without time scripts on variables where $\frac{\phi _{t+1}}{\phi _{t}}=\frac{1}{\mu }$ holds for all time $t$ and $\mu$ denotes the gross inflation rate. From the maximization problem, the first-order conditions can be derived as

(9) \begin{equation} \frac{\mu }{\beta R^d}=u'(x_{1})+\frac{R^m(\lambda _{1}-\lambda _{2})}{\rho }, \end{equation}

(10) \begin{equation} \frac{\mu }{\beta R^d}=u'(x^{m}_{2}+x^{b}_{2})R^m-\frac{R^m(\lambda _{1}-\lambda _{2})}{1-\rho }, \end{equation}

(11) \begin{equation} q\frac{\mu }{\beta }=u'(x^{m}_{2}+x^{b}_{2})+\frac{\lambda _{2}}{1-\rho }, \end{equation}

where $\lambda _{1}$ and $\lambda _{2}$ denote each multiplier associated with the constraints (6)-(7), respectively. In equilibrium asset markets clear in the CM with

(12) \begin{equation} \begin{array}{rl} c&=\phi C,\\ d&=\phi D,\\ m&=\phi M,\\ b&=\phi B. \end{array} \end{equation}

Since the supplies of currency, CBDC and reserves are equal to the central bank’s government bond holdings, we have

(13) \begin{equation} c+d+m = (V -qb), \end{equation}

and the proportion of currency, CBDC and reserves among the supply of the total assets can be defined as $\delta$ by

(14) \begin{equation} c+d+m = \delta V. \end{equation}

I assume that the central bank can set the interests on reserves and CBDC, $R^d R^m$ and $R^m$ , and choose the proportion of currency, CBDC and reserves in the total asset supply, $\delta$ , to implement monetary policy.

Finally, the quantity of government bonds held by the private sector must be less than or equal to the total government bonds issued by the fiscal authority as

(15) \begin{equation} 0 \leq qb \leq V. \end{equation}

Definition 1. Given parameters $(\rho, y,V)$ and the policy variables $(R^d,R^m,\delta )$ , a stationary monetary equilibrium consists of quantities $(x_{1},x^{m}_{2},x^{b}_{2})$ and prices $(\mu, q)$ and multipliers $\lambda _{1}$ and $\lambda _{2}$ which solve equations (9) - (15) .

Since a quasi-linear utility is adopted in the model, the rates of return on assets such as currency, $\frac{1}{\mu }$ , CBDC, $\frac{R^d}{\mu }$ , reserves, $\frac{R^d R^m}{\mu }$ , government bonds, $\frac{1}{q\mu }$ , and the Lucas tree, $1+\frac{y}{\psi }$ , cannot exceed the inverse of the time preference, $\frac{1}{\beta }$ . In the model if a truth-telling incentive constraint binds, the rate of return on reserves can be higher or lower than the rates of return on government bonds and the Lucas tree in equilibrium, although reserves are used for transactions as collateral the same as the government bonds and the Lucas tree. Thus, we have no-arbitrage conditions in equilibrium as

(16) \begin{equation} \begin{array}{rl} \frac{1}{\mu }\leq \frac{R^d}{\mu }\leq \frac{R^d R^m}{\mu }\leq &\frac{1}{\beta }, \\ \frac{1}{q\mu }=\frac{\psi +y}{\psi }\leq &\frac{1}{\beta }. \end{array} \end{equation}

In the following analysis we assume that the total supply of assets in this economy is sufficiently small as $V\lt x^{*}$ , where $x^{*}$ satisfies with $u'(x^{*})=1$ , in order to make the first-best allocation, that is $x_{1}=x_{2}=x^{*}$ , infeasible.²⁵

4. Monetary equilibrium

In this section, we characterize the equilibrium allocations. Equilibrium cases can be distinguished by whether excess reserves exist or not, and also by which of the incentive constraints (6)-(7) bind or not.²⁶ In the following we describe channel systems and floor systems as different equilibrium cases and will compare the effect of monetary policy and the welfare in these equilibrium allocations.

4.1. Channel systems

In a channel system the central bank can set the interests on CBDC and reserves, $R^d$ and $R^d R^m$ , lower than the nominal interest rate on government bonds, $\frac{1}{q}-1$ , which is controlled by open-market-operations, $\delta$ . So banks will not hold any excess reserves in equilibrium, $m=0$ and $x^{m}_{2}=0$ , in this case. We also know that the constraints (3)-(5) always bind when the first-best allocation is not feasible. The first step of characterization is to solve for the equilibrium conditions without the binding incentive constraints (6)-(7). If $\lambda _{1}=\lambda _{2}=0$ , then the first-order conditions (9) and (11) can be reduced into

(17) \begin{equation} \frac{\mu }{\beta R^d} =u'(x_{1}), \end{equation}

(18) \begin{equation} q\frac{\mu }{\beta }=u'(x_{2}), \end{equation}

respectively. By using these first-order conditions (18)-(19) and the binding constraints (3) and (5), the government budget constraint (13) can be transformed into a feasibility condition,

(19) \begin{equation} \rho x_{1}u'(x_{1})+(1-\rho )x_{2}u'(x_{2})=V. \end{equation}

Note that given a nominal interest rate target $\frac{1}{q}-1$ , $x_{1}$ and $x_{2}$ have a strictly positive relationship as $q R^d u'(x_{1})=u'(x_{2})$ in (17)-(18) and a strictly negative relationship in (19). So there is a unique equilibrium allocation $(x_{1},x_{2})$ , which satisfies with (17)-(19). In order to support this equilibrium allocation $(x_{1}, x_{2},q)$ , the central bank can implement open-market-operations by choosing $\delta$ as

(20) \begin{equation} \delta = \frac{\rho x_{1}u'(x_{1})}{V}, \end{equation}

which can be derived from (14) and (19). Notice that the equilibrium allocation $(x_1,x_2)$ is determined by the relative rate between the interest on CBDC, $R^d$ , and the nominal interest rate, $\frac{1}{q}$ . That means, when $R^d$ varies, only the nominal interest rate and the inflation rate, $\mu$ , will change along with it and the other real allocations are maintained. Thus, without the loss of generality, we assume that the interest on CBDC is normalized as one, $R^d=1$ , from now on. Further details on how monetary policy is implemented in the channel system are described in the Appendix A, because the mechanism is the same as shown in Williamson (2012, 2016).

Lemma 1. In channel systems the incentive constraints (6) - (7) do not bind.

Proof. See the appendix.

Lemma 1 shows that both the incentive constraints do not bind in the channel system. The insurance contract provides currency for type 1 buyers and government bonds for type 2 buyers to make the marginal rate of substitution between currency and government bonds equivalent to the given nominal interest rate. Without the excess reserves, type 1 buyers prefer currency/CBDC because it is unavailable to use government bonds in retail transactions. Type 2 buyers prefer government bonds as long as the rate of return in government bonds is greater than the rates of return on currency and CBDC. Therefore, the optimal insurance contract does not violate the truth-telling incentive constraints without excess reserves.

Therefore, if the monetary policy is implemented in a channel system, there is no effect on real allocations when an account-based CBDC is introduced to the agents.

4.2. Floor systems

In a floor system the interest on reserves must be greater than or equal to the nominal interest rate, $R^m \geq \frac{1}{q}$ , to have excess reserves. Given the interest on reserves, $R^m$ , if $\delta =\bar{\delta }$ then we have the same equilibrium as in channel systems with no excess reserves, $m=0$ and $x^{m}_{2}=0$ , so when $\delta \gt \bar{\delta }$ , banks hold strictly positive excess reserves on their balance sheets as $m\gt 0$ and $x^{m}_{2}\gt 0$ . In order to implement floor systems, the central bank can set $ R^m\geq 1$ and $\delta \gt \bar{\delta }$ where $\bar{\delta }$ satisfies with $R^m=\bigg (\frac{1-\rho }{\rho }\frac{\bar{\delta }}{1-\bar{\delta }}\bigg )^{-\frac{\gamma }{1-\gamma }}$ , given the utility function $u(x)=\frac{x^{1-\gamma }}{1-\gamma }$ .

Define $\alpha$ as a proportion of type 2 buyer’s consumption supported by excess reserves, $\alpha =\frac{x^{m}_{2}}{x^{m}_{2}+x^{b}_{2}}$ . Note that $\alpha \in [0,1]$ increases in $\delta$ and corresponds to $\delta \in [\bar{\delta },1]$ since $\alpha =\frac{\delta -\bar{\delta }}{1-\bar{\delta }}$ holds from the binding constraints (3)-(5) and (13)-(14).²⁷ Then the truth-telling constraint (6) can be rewritten as

(21) \begin{equation} R^m x_{1}\geq \alpha x_{2}. \end{equation}

Lemma 2. In the floor system, the truth-telling constraint (7) never binds, but the truth-telling constraint (6) can bind when the excess reserves are sufficiently large as $\delta \gt \hat{\delta }$ where $\hat{\delta }$ satisfies $\frac{\hat{\delta }-\bar{\delta }}{1-\bar{\delta }}=(R^m)^{1-\frac{1}{\gamma }}$ .

Proof. See the appendix.

Lemma 2 shows that the incentive constraint for type 1 buyers can bind when the excess reserves are sufficiently large. In a floor system, the interest on reserves plays the same role as the nominal interest rate in the channel system to adjust the marginal rate of substitution between currency/CBDC and government bonds. However, when CBDC is introduced, type 1 buyers can use CBDC by accessing to the reserve accounts. So, the optimal allocation cannot be obtained when the large excess reserves are provided to type 2 buyers, because type 1 buyers can mimic type 2 buyers.

Notice that only type 1 buyers deviate in this model unlike Diamond and Dybvig (1983) where type 2 buyers deviate in a similar setting, because we assume that the coefficient of relative risk aversion, $\gamma$ , is less than 1. Since the insurance contract makes type 1 buyers worse off and type 2 buyers better off with $\gamma \lt 1$ , type 1 buyers tend to deviate and the incentive constraint (6) is binding here.²⁸

Therefore, we can describe our equilibrium cases as shown in Figure 2. Given $R^m\gt 1$ , if $\tilde{\delta } \leq \delta \leq \bar{\delta }$ then we can conduct open-market operations in channel systems. Given $R^m\gt 1$ , if $\delta \gt \bar{\delta }$ we need to implement monetary policy in floor systems. In case of $\bar{\delta }\lt \delta \leq \hat{\delta }$ then the incentive constraint (21) does not bind, but it binds in case of $\hat{\delta }\lt \delta \leq 1$ .²⁹

Figure 2. Equilibrium cases.

4.3. Floor system without large excess reserves

When $\lambda _{1}=0$ , the equilibrium allocation at $R^m\gt 1$ in a floor system is the same as one in a channel system with $\frac{1}{q}=R^m\gt 1$ , because given $\lambda _{1}=0$ , $\frac{1}{q}=R^m$ holds in (10)-(11) and the equilibrium conditions (17)-(19) still hold. Therefore, introducing CBDC does not affect the real allocations in a floor system without large excess reserves. Additional explanations on how monetary policy is implemented in the floor system without large excess reserves are described in the Appendix A.

4.4. Floor system with large excess reserves

When $\lambda _{1}\gt 0$ , the feasibility condition (19) does not change, but the truth-telling constraint (21) binds with equality instead of the first-order conditions (17)-(18).³⁰ Thus, in this case given $(R^m,\delta )$ , the equilibrium allocation $(x_{1},x_{2})$ in the floor system with $\lambda _{1}\gt 0$ is determined by (19) and (21) with equality. Define $(x^{i}_{1}, x^{i}_{2})$ as the equilibrium allocations in the channel system ( $c$ ) and floor system ( $f$ ), respectively, where $i=\{c,f\}$ . As shown in Figure 3, given the same level of interest on reserves, $R^m$ , the equilibrium allocation $(x^{f}_{1},x^{f}_{2})$ in the floor system with $\lambda _{1}\gt 0$ can be described as the point $F$ while the equilibrium allocation $(x^{c}_{1},x^{c}_{2})$ in the channel system(or in the floor system with $\lambda _{1}=0$ ) can be shown as the point $C$ . Notice that $x^{c}_{1}\lt x^{f}_{1}$ and $x^{c}_{2}\gt x^{f}_{2}$ in the graph, because $(x^{c}_{1},x^{c}_{2})$ and $(x^{f}_{1},x^{f}_{2})$ still satisfy with the same feasibility condition (19) while (21) violates at $(x^{c}_{1}, x^{c}_{2})$ because of $\lambda _{1}\gt 0$ . Given the large excess reserves, the bank cannot provide plentiful reserves to type 2 buyers because type 1 buyers could mimic them. Therefore, the bank would provide more currency/CBDC to type 1 buyers and less reserves to type 2 buyers.

Therefore, given the interest on reserves, $R^m$ , the inflation rate and the real interest rate on government bonds can be changed from the ones in the channel system. When $\lambda _{1}\gt 0$ , from (9)-(11) we obtain

(22) \begin{equation} \frac{1}{\beta r^{f}_{m}} := \frac{\mu ^{f}}{\beta R^m} =\frac{\rho u'(x^{f}_{1})}{R^m}+(1-\rho )u'(x^{f}_{2}), \end{equation}

(23) \begin{equation} \frac{1}{\beta r^{f}_{b}} :=q\frac{\mu ^{f}}{\beta } =u'(x^{f}_{2}). \end{equation}

where $\mu ^{i}$ , $r_{m}^{i}$ , $r_{b}^{i}$ denote the inflation rate, the rate of return on reserves, and the rate of return on government bonds in the channel system ( $c$ ) and floor system ( $f$ ), respectively, where $i=\{c,f\}$ .³¹

Figure 3. Equilibrium in floor system.

In this case, the liquidity premia of reserves do not depend only on the collateral transactions market condition, because excess reserves can tighten the truth-telling constraint for the retail market transactions. Thus, the liquidity premium in reserves is calculated as a weighted average of the marginal utility in both types of trades as shown in (22).

Given $\lambda _{1}\gt 0$ , since the point $C$ is not achieved, the marginal rate of substitution between type 1 buyer’s consumption and type 2 buyer’s consumption, $\frac{u'(x^{f}_{1})}{u'(x^{f}_{2})}$ , is lower than the interest on reserves, $R^m$ . It is optimal for the banks to raise $x_{2}$ and reduce $x_{1}$ , but it is not feasible because they cannot reveal the types under private information. Therefore, there arises a positive liquidity premium on currency in (9) and a negative liquidity premium on reserves in (10) with $\lambda _{1}\gt 0$ . Moreover, the liquidity premium on government bonds is greater than the liquidity premium on reserves with $\lambda _{1}\gt 0$ in (11), because government bonds are more useful than reserves to reveal the types. Therefore, in equilibrium, we have a return dominance between reserves and government bonds, $r_m^f\gt r_b^f$ , as shown in (22)-(23) because $\frac{1}{q}\lt R^m$ and $R^m u'(x_{2})\gt u'(x_{1})$ holds when (21) binds. Thus, open-market operations, the exchange between reserves and government bonds, can be effective in this case. Notice that the nominal interest rate, $\frac{1}{q}$ , is lower than the interest on reserves, $R^m$ , in the model which actually happened in the U.S. federal funds market recently. In reality, the reason is based on a specific feature of the U.S. financial system, but our model shows that a similar phenomenon can occur when the reserve accounts are allowed to the public with large excess reserves.³² When the people can use the reserves as CBDC, the reserves become more liquid. Thus, it can be difficult to distribute liquidity by types efficiently because the supply of illiquid assets such as government bonds is scarce with the large excess reserves. In this respect our result also emphasizes the role of illiquid assets to distribute liquidity efficiently as shown in Kocherlakota (2003).

4.4.1. Open-market operations ( $\triangle \delta$ )

Notice that we have two policy variables, the proportion of currency/CBDC and reserves, $\delta$ (or $\alpha$ ), and the interest on reserves, $R^m$ . Given $R^m$ , if the proportion of currency and reserves, $\delta$ , is raised by open-market operations, the equilibrium allocation $(x^{f}_{1},x^{f}_{2})$ moves toward the point $C$ in Figure 3.

Proposition 1. Given $R^m$ , the inflation rate decreases in $\delta$ while the real interest rate on government bonds increases in $\delta$ in the floor system with LER.

Proof. See the appendix.

In this case, there can be two different effects on the liquidity premia in currency and government bonds. Open-market purchases increase the quantity of reserves and decrease the quantity of government bonds in the market. Since the truth-telling incentive constraint (21) is tightened further, the bank provides more currency/CBDC to type 1 buyers and less reserves to type 2 buyers.

Consequently, the liquidity premia on currency/CBDC and government bonds are reduced as the gap between type 1 and 2 buyers’ consumption becomes smaller. Thus, given that the level of the interest on reserves is maintained, the inflation rate rises. The real interest rate falls as type 2 buyers’ consumption decreases.

The opposite movement of the inflation rate and the real interest rate is similar to the Mundell-Tobin effect, in which capital investment is considered. By raising money supply, the inflation rate goes up and the real interest rate decreases, because capital investment is more demanded than currency trade. However, the reason that the retail transactions are restricted in this model is based on the scarcity of illiquid assets rather than the tradeoff between the currency demand and capital investment.

4.4.2. Interest on reserves ( $\triangle R^m$ )

Given $\delta$ , if the interest rate on reserves, $R^m$ , is raised, then the equilibrium allocation $(x^{f}_{1},x^{f}_{2})$ also moves from point $F$ toward point $C$ in Figure 3 as the $FOC$ curve (21) shifts to the left.

Proposition 2. Given $\delta$ , both the inflation rate and the real interest rate on government bonds increase in the interest of reserves, $R^m$ , in the floor system with LER.

Proof. See the appendix.

When the interest on reserves goes up, the truth-telling constraint (21) is relaxed with higher $\lambda _{1}$ . Therefore, the liquidity premium in currency goes up while the liquidity premium in government bonds decreases additionally in the floor system with LER. So, the real interest rate on government bonds increases further than it would in the floor system without LER.

Table 1. Comparative statics

The effects of monetary policy implementations on the inflation rate and the real interest rates by different cases are summarized in Table 1. In the channel system only the open-market operations are effective, while only adjusting the interest on reserves is effective in the floor system without LER. In the floor system with LER, both adjusting the interest on reserves and the open-market operations are effective. Moreover, the direction of the real interest rate is opposite in the case of open-market operations.

4.4.3. Effectiveness of the interest on reserves ( $\triangle R^m$ )

Note that raising the interest on reserves in the floor system either without LER or with LER has a positive effect on the inflation rate and the real interest rates. However, the effectiveness of adjusting the interest on reserves must be different between these two regimes whether the incentive constraint binds or not. We can compare the effects of changing $R^m$ on the inflation rate and the real interest rate, given the equilibrium allocations at the same level of $R^m$ .

Proposition 3. At the same level of $\delta$ , when the interest on reserves is raised, the inflation rate increases less and the real interest rate on the government bonds decreases more in the floor system with LER compared to the floor system without LER.

Proof. See the appendix.

Proposition 3 shows that raising the interest on reserves in the floor system with LER can increase the real interest rate more than that in the floor system without LER. If the truth-telling incentive constraint binds, the constraint is relaxed with higher $\lambda _{1}$ when the interest on reserves goes up. Thus, it raises the inflation further in (9) and reduces the real interest rate on government bonds further in (11). That means, if CBDC is introduced, raising the interest on reserves can be more contractionary with the LER.

5. Equilibrium with private assets

In this section I introduce private assets to analyze the effect of monetary policy on the aggregate liquidity in an economy. In the model it is assumed that the total real value of government debt, which includes money and bonds, is kept as constant in order to separate monetary policy from fiscal policy. For instance, when the fiscal authority issues money or government bonds, then the total supply of the assets in the market increases and the liquidity premia of the assets decrease, and thus the real interest rates on the assets would go up in the model. Therefore, in order to focus solely on the effect of open-market operations, which exchanges money with government bonds, I assume that the total supply of government debt is fixed. As a result, we can figure out the monetary policy effect on the real interest rates, but cannot tell the effect on the aggregate liquidity in the economy. In the real world, if the real interest rate falls, then the loan of the financial intermediaries increases and the prices of the real assets rise, so the aggregate liquidity in this economy increases. To capture this monetary policy effect on the aggregate liquidity, I extend the model with the private assets.³³

Suppose that a divisible Lucas tree with a fixed supply of $A$ is endowed to buyers in the initial period, $t=0$ , CM and pays off $y$ units of consumption goods as a dividend in the CM in every period and trades at the price of $\psi _{t}$ in terms of goods in the period $t$ CM.

Given the fixed supply of the Lucas tree, if the asset price goes up then aggregate liquidity can increase in this economy. to focus on the effectiveness of monetary policy without considering the effect on the aggregate liquidity supply through the price of the private assets. In the model the real quantity of public assets is assumed to be constant as $V$ to restrict the fiscal policy effect, so to know the effect on Agg. liquidity, introduce private asset. real interest rates decreases then the asset price goes up and liquidity increases. It can also be interpreted as loans, because the real

The maximization problem can be modified as

(24) \begin{equation} \begin{array}{ll} k_{t}-c_{t}-d_{t}-m_{t}-q_{t}b_{t}-\psi _{t}(1-\theta _{t})a_{t}+ \{\frac{\beta \phi _{t+1}}{\phi _{t}}c_{t}+\frac{\beta \phi _{t+1}}{\phi _{t}}R^{d}_{t}d_{t}-\rho x_{1t}\}\\ +\{\frac{\beta \phi _{t+1}}{\phi _{t}}R^{d}_{t}R^{m}_{t}m_{t}-(1-\rho ) x^{m}_{2t}\} +\{\frac{\beta \phi _{t+1}}{\phi _{t}}b_{t}+\beta (\psi _{t+1}+y)(1-\theta _{t})a_{t}-(1-\rho )x^{b}_{2t}\}\geq 0 \end{array} \end{equation}

(25) \begin{equation} \beta (\psi _{t+1}+y)(1-\theta _{t})a_{t}-(1-\rho )x^{b}_{2t}\geq 0, \end{equation}

(26) \begin{equation} \frac{\psi }{\beta (\psi +y)}=u'(x^{m}_{2}+x^{b}_{2})+\frac{\lambda _{2}}{1-\rho }, \end{equation}

(27) \begin{equation} a=A, \end{equation}

(28) \begin{equation} c+d+m = (V -qb)+\psi \theta a, \end{equation}

where $\theta _{t}$ denotes a proportion of the Lucas trees purchased by the central bank and $a_{t}$ denotes the demand for the asset holdings of the bank and $\psi _{t}$ .³⁴

Given $A\gt 0$ , instead of (19), we can obtain

(29) \begin{equation} \rho x_{1}u'(x_{1})+(1-\rho )x_{2}u'(x_{2})=V+\psi A. \end{equation}

where $\psi =\frac{\beta y u'(x_{2})}{1-\beta u'(x_{2})}$ from the binding constraints (3) and (5), government budget constraint (13) and the first-order conditions (17)-(18) in the channel system. In the floor system, even with $\lambda _{1}\gt 0$ , we still have (24) from (3)-(5), (13) and the first-order condition (21). Therefore, the equilibrium allocation $(x_{1},x_{2})$ is uniquely determined from (17)-(18) and (24) in the channel system and the floor system without LER, while determined from (21) and (24) in the floor system with LER.

Figure 4. Open-market operations ( $\triangle \delta$ ).

Notice that the proportion of private assets held by the central bank, $\theta \in [0,1]$ , is irrelevant to determine the equilibrium allocation as long as (15) does not violate, because both government bonds and private assets are traded as identical securities by type 2 agents in the model.^{35, 36}

Unlike the previous case of $A=0$ , the feasibility equilibrium condition (29) can be shifted by the level of the asset price, $\psi$ , in (24). For example, if the real interest rates in the channel system, $r^{c}_{b}$ , in (18) and the real interest rates in the floor system with LER, $r^{f}_{b}$ , in (23) increase, then the asset price falls in (11), so the feasibility condition (24) shrinks toward the origin. If the central bank implement monetary policy in the channel system with open-market operations, $\delta \in [\tilde{\delta },\rho ]$ , at zero interest on reserves, $R^m=1$ , then the feasibility condition (24) can be described as $IC(V+\psi )$ curve in Figure 4 a. Note that the slope of $IC(V+\psi )$ curve (24) is less steep than the slope of $IC(V)$ curve (19), because when $\delta$ increases, the real interest rate goes down.

It is not simple in the case of floor system with LER, because the central bank can use either the interest on reserves ( $R^m$ ) or open-market operation ( $\delta$ ) and also the effect on the real interest rate can vary by these policy tools. Therefore, we consider the shape of the feasibility curve (24) by considering the effect of each policy variable, $R^m$ or $\delta$ , on the real interest rates.

5.1. Open-market operations( $\triangle \delta$ )

In Figure 2, given $R^m\in (1,\frac{1}{\tilde{q}})$ , as the central bank raises $\delta$ from $\tilde{\delta }$ to 1, the equilibrium allocation starts with the channel system regime and passes the floor system without LER regime and reaches floor system with LER regime. Since the open-market operations are ineffective in the floor without LER regime, in Figure 4 a once the equilibrium allocation arrives at the point $H$ with $\delta =\tilde{\delta }$ , it remains when $\delta$ increases further. When $\delta$ reaches $\hat{\delta }$ in the floor system with LER regime, raising $\delta$ tightens the binding incentive constraint (21), so the real interest rate goes up and the asset price falls. Therefore, we have a kink at the point $H$ and the feasibility of equilibrium allocations in the floor system with LER becomes restricted more than that in the channel system.

Corollary 1. When $\delta$ is raised, the feasibility equilibrium condition (24) moves towards the origin in the floor system with LER.

Proof. See the appendix.

As summarized in Table 1, when $\delta$ increases, both the inflation rate and the real interest rate decrease in the channel system whereas the inflation rate falls and the real interest rate goes up in the floor system with LER, so we can describe the inflation rate and the rate of return on government bonds as shown in Figure 4 b. The dotted lines in Figure 4 b show the paths of the rates of return on currency/CBDC and government bonds when $\delta$ increases only in the channel system.

5.2. Interest on reserves( $\triangle R^m$ )

Similarly, given $\delta \in (\rho, 1)$ when the central bank raises $R^m$ from 1 to $\frac{1}{\tilde{q}}$ , the equilibrium allocation starts with the floor system without LER and reaches the floor system with LER as shown in Figure 2. Since changing the interest on reserves in the floor system without LER has the same effect as in the channel system, when $R^m$ is raised, the equilibrium allocation moves along the curve (24) from point $I$ to point $J$ in Figure 5 a. However, when $R^m$ increases, the real interest rate in the floor system with LER goes up further than one in the floor system without LER as shown in Proposition 3. Thus, the asset prices go down further and the feasibility equilibrium condition (24) moves towards the origin as described in Figure 5 a.

Corollary 2. When the interest on reserves, $R^m$ , is raised, the feasibility equilibrium condition (24) moves towards the origin in the floor system with LER.

Proof. See the appendix.

Given a sufficiently large excess reserve, $\delta$ , if $R^m$ is raised then the allocation moves from the floor system without LER into the floor system with LER in Figure 2. In this case both the inflation rate and the real interest rate increase either in the floor system without LER or in the floor system with LER. However, both the inflation rate and the real interest rate rise further in the floor system with LER as described in Figure 5 b.

Figure 5. Interest on reserves ( $\triangle R^m$ ).

5.3. Welfare

Since the utility and production functions are linear in the CM, all the surplus is generated from trades in the DM. So the welfare function is

(30) \begin{equation} W(x_{1},x_{2})=\rho \{u(x_{1})-x_{1}\}+(1-\rho )\{u(x_{2})-x_{2}\}+yA, \end{equation}

which consists of the trading gains in the DM and the dividend from the Lucas tree. However, it is important to consider the costs of operating a currency/CBDC system when we evaluate the effects of monetary policy. It includes not only the direct cost of managing the stock of currency and data in the central bank server, but also the indirect social costs to inhibit the illegal activities such as theft, counterfeiting, hacking, etc. One way to reflect this operating cost is to assume that a fraction $\omega$ of currency/CBDC trade is socially useless as suggested in Williamson (2012, 2016). Then, our welfare measure becomes

(31) \begin{equation} \hat{W}(x_{1},x_{2})=\rho \{(1-\omega )u(x_{1})-x_{1}\}+(1-\rho )\{u(x_{2})-x_{2}\}+yA. \end{equation}

As shown in Figures 4 a and 5 a, raising $\delta$ and $R^m$ in the floor system with LER regime lead the feasibility equilibrium condition (24) to move towards the origin. So, if given $(\delta, R^m)$ an equilibrium allocation is located at the floor system with LER in Figure 2, there is a possibility to improve welfare by expanding the feasibility set. For example, by reducing $\delta$ to $\bar{\delta }$ , and then reducing $R^m$ in the floor system without LER regime, the allocation can move from point $G$ to point $H$ , and then to point $G'$ in Figure 4 a, where $\hat{W}$ represents the welfare curve. Similarly, by reducing $R^m$ to reach the floor system without LER regime, and then reducing $\delta$ to the channel system in Figure 2, the allocation can move from point $J$ to point $I$ , and then to point $J'$ in Figure 5 a. If there are no restrictions on the policy variables $(\delta, R^m)$ , the channel system with $\delta \in (\tilde{\delta },\rho ]$ and $R^m=1$ is desirable to maximize the feasibility of the equilibrium allocations.

Proposition 4. If $\omega$ is sufficiently large, then the equilibrium allocations in the floor system with LER are suboptimal.

Proof. See the appendix.

Proposition 4 shows that reducing the amount of excess reserves can be beneficial when the cost of operating a currency/CBDC system is sufficiently large. If the cost of operating a currency/CBDC system is large, then it is optimal to raise the nominal interest rate high enough because it can discourage the currency trade. When the nominal interest rate goes up, the aggregate liquidity supply decreases further with the large excess reserves, thus a policy mix that raises the interest on reserves and reduces the excess reserves simultaneously can improve welfare.

5.4. Discussion

When private assets are introduced, the monetary policy effect on the inflation rate and the real interest rates is maintained in the channel and floor systems and also with the large excess reserves. However, with the large excess reserves, the aggregate liquidity shrinks further due to the general equilibrium effect. When the real interest rate rises by monetary policy, the price of the private asset falls and the total supply of assets becomes more scarce. Thus, the real interest rate becomes higher in equilibrium and the effectiveness of the monetary policy increases when the private assets are additionally introduced.

This result is not desirable from the welfare perspective, because it can reduce the aggregate liquidity supply excessively in an economy. It is difficult to distribute the liquidity efficiently by types with the large excess reserves, because reserves are accessible and convertible into CBDC. Thus, when the incentive constraint binds with the large excess reserves, the more liquidity is required to separate the types and it leads to a welfare loss.

Therefore, if the large amount of excess reserves is inevitable, there could be an alternative solution: Instead of liquid excess reserves, the central bank can issue illiquid bonds such as the central bank’s bill or ON-RRP.³⁷ Issuing the illiquid debt can prevent the banks from holding liquid assets excessively in their portfolio, which can be used as CBDC at any time.

6. Conclusion

In this paper we study the effect of introducing CBDC on monetary policy tools in the floor system with the large excess reserves. In the model the banks provide a liquidity insurance to the agents who are exposed to the idiosyncratic liquidity risk. Since the large excess reserves can inhibit the banks from separating the types by liquidity needs, there could exist a new regime in the floor system where open-market operations can be effective with a return dominance. When we consider the aggregate liquidity effect with private assets, the feasibility equilibrium condition can move towards the origin in this regime. In this case we can rewind it by reducing either the interest on reserves or the proportion of excess reserves in the bank’s balance sheet or both.

This paper takes a step forward to understand monetary policy with the large excess reserves when CBDC is introduced. It provides a theoretical model in which we can implement both the interest on reserves and open-market operations in the floor system. However, it also leaves some questions unanswered. For example, this model uses an insurance contract with truth-telling constraints instead of secondary financial markets under private information. In a respect of convertibility we may ask how the secondary market friction can also influence the effectiveness of these monetary policy tools. Moreover, since the truth-telling constraints can prevent bank runs in the model, we may further study how the fragility of banks can be associated with the introduction of CBDC. Finally, there are other respects of excess reserves which could be considered in future research. For example, the excess reserves can be very helpful for banks when they are exposed to an aggregate liquidity shock.