Monetary policy transmission, central bank digital currency, and bank market power

Chen, Hanfeng; Hänsel, Matthias; Nguyen, Hiep Article Monetary policy transmission, central bank digital currency, and bank market power Journal of Economics and Statistics Provided in Cooperation with: De Gruyter Brill Suggested Citation: Chen, Hanfeng; Hänsel, Matthias; Nguyen, Hiep (2025) : Monetary policy transmission, central bank digital currency, and bank market power, Journal of Economics and Statistics, ISSN 2366-049X, De Gruyter Oldenbourg, Berlin, Vol. 245, Iss. 4/5, pp. 527-576, https://doi.org/10.1515/jbnst-2024-0008 This Version is available at: https://hdl.handle.net/10419/333321 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Hanfeng Chen*, Matthias Hänsel and Hiep Nguyen Monetary Policy Transmission, Central Bank Digital Currency, and Bank Market Power https://doi.org/10.1515/jbnst-2024-0008 Received January 7, 2024; accepted December 10, 2024 Abstract: Interest rates on new central bank digital currencies (CBDCs) can be expected to enter the monetary policy toolkit soon. Using an extended Sidrauski (1967) model featuring an oligopsonistic banking sector, we study the complex transmission of interest rates on CBDC, which generally involve both direct and indirect effects. This is because a CBDC rate cut does not only affect the rate on the CBDC itself, but also induces the non-competitive deposit providers to adjust their spreads, as the new substitute for their products becomes relatively less attractive. A calibration exercise suggests that the indirect effects depend strongly on the sources of deposit market power: If driven by high concentration, they substantially amplify the aggregate effects of the CBDC policy rate, both in response to transitory shocks as well as regarding its long-run welfare effects. This contrasts them with policies directed at the banking sector which are weakened by a less competitive deposit market. Keywords: CBDC; digital currency; bank market power; monetary transmission JEL Classification: E42; E43; E52 Article Note: This article is part of the special issue “Central Bank Digital Currency”published in the Journal of Economics and Statistics. Access to further articles of this special issue can be obtained at www.degruyter.com/jbnst. We are grateful to Mikael Bask, Christoph Bertsch, Mikael Carlsson, Daria Finocchiaro, Paul Klein, Per Krusell, Kurt Mitman, Lars Ljungqvist, and Yimei Zou for their useful feedback and comments. Financial support by the Jan Wallanders och Tom Hedelius Foundation, Sweden, is gratefully acknowledged by Hänsel. The opinions expressed in this article are the sole responsibility of the authors and do not necessarily reflect the views of the National Institute of Economic Research. *Corresponding author: Hanfeng Chen, Uppsala University, Uppsala, Sweden; Center for Monetary Policy and Financial Stability, Stockholm University, Stockholm, Sweden; and National Institute of Economic Research, Stockholm, Sweden, E-mail: [email protected] Matthias Hänsel, Stockholm School of Economics, Stockholm, Sweden; and Center for Monetary Policy and Financial Stability, Stockholm University, Stockholm, Sweden Hiep Nguyen, Uppsala University, Uppsala, Sweden Journal of Economics and Statistics 2025; 245(4–5): 527–576 Open Access. © 2025 the author(s), published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 1 Introduction Monetary authorities around the world are exploring the possibility of issuing a new digital payment instrument widely accessible to the public. As of May 2024, 134 countries and currency unions, which account for 98 % of the global GDP, are considering a central bank digital currency (CBDC) (Atlantic Council 2024). Motivations for such a new payment instrument include ensuring adequate public money, reducing systemic risk and improving financial stability, increasing competition in payments, and promoting financial inclusion (Engert and Ben Siu-Cheong 2017). One of the less-discussed aspects of CBDC is its potential to enable a direct implementation of monetary policy (e.g., Auer et al. 2022; Bank for International Settlement 2020). Interest on CBDC could become a new policy instrument, providing policy-makers with greater flexibility to influence the real returns of money assets and sidestep financial intermediaries. However, banks will not idly stand by if the central bank makes it more attractive to hold an asset providing similar services as their deposits. In turn, the actual equilibrium impact of CBDC rates should depend both on households’liquidity preferences and the response of financial sector agents with substantial market power. A key contribution of our work compared to the existing literature is that regarding the latter channel, we explicitly distinguish between different sources of bank market power, market concentration and differentiation: Current theories tend to focus either on one or the other (see below for further discussion), but clearly, both aspects are relevant in reality: The market for bank deposits does not only display substantial concentration (see e.g. Corbae and D’Erasmo 2020) but the respective deposits also differ in practice due to regionally differing branch networks, bundling with different payment cards, etc. Indeed, we find the actual source of bank market power to have important implications for the effect of a CBDC rate as a policy tool, which suggests assumptions on the form of imperfect bank competition to be crucial for the outcomes of quantitatively modeling CBDC policy. In particular, we conduct our analysis using an extended Sidrauski (1967) model building on the framework proposed by Niepelt (2024): In the model, households gain utility from holding different forms of liquid assets which allows us to capture various related aspects such as CBDC design and the privatesector’s need for deposits of different banks in a parsimonious way close to textbook theories. However, it should be noted that such a model has the implication that CBDC is not “special” compared to alternate forms of government-provided liquidity (in the sense of explicitly modeled design features). In turn, our theoretical analysis would equally apply to other liquidity types as long as central banks can affect their returns flexibly enough. 528 H. Chen et al. Crucially, in our framework, we allow for a banking sector in which bank market power is derived both from market concentration and households’imperfect ability to substitute between banks. We assume a common deposit market in which a set of non-competitive banks compete by offering differentiated deposits, but do not restrict it to be either monopsonistic (as e.g. in Niepelt 2024) or monopsonistically competitive (as e.g. in Bacchetta and Perazzi 2022). Rather, such settings are nested as limit cases, allowing us to vary the degree of deposit market concentration in order to demonstrate its importance for the transmission of CBDC rates. We also do not restrict different banks’deposits to be fundamentally the same to households as in Chiu et al. (2023), so that oligopsonistic settings remain characterized by different sources of market power. Additionally, we assume limited substitutability between CBDC and bank deposits, with perfect substitutability asa limit case. We believe there are good reasons toexpect that in practice, CBDC would not be almost perfectly substitutable with bank deposits. For example, bank deposits are typically bundled with other financial services such as credit lines (e.g. overdraft facilities, credit cards), while CBDC may be perceived as offering more privacy and security. Additionally, other features such as the interoperability between CBDC and deposits and the ability to conduct international transactions might also limit practical substitutability (Bacchetta and Perazzi 2022). For the purpose of this paper, we consider the interest rates on CBDC as the main policy instrument of interest but also discuss implications for reserve rates. In our model, both can be shown to affect the real allocation through the average cost of liquidity, but the influence of the CBDC spread consists of both a direct and an indirect effect. Clearly, an increase in the CBDC spread (relative to a risk-free rate) directly increases the households’cost of liquidity. At the same time, the rising spread also enables banks to widen the spreads on the deposits they offer, as the alternative source of liquidity becomes comparatively less attractive. This introduces the indirect effect, reminiscent of the deposit channel of monetary policy proposed by Drechsler et al. (2017). While the quantitative magnitude of the direct effect depends simply on the amount of CBDC households will choose to hold given its design, the strength of the indirect effect is more nuanced, depending crucially on the source of market power in the deposit market. Intuitively, if deposit market concentration is low, individual banks are small and cannot affect the amount of CBDC households will choose to hold. In turn, changes in the CBDC spread have little impact on the equilibrium deposit spread and the indirect effect is small. On the other hand, if the deposit market is highly concentrated, banks can practically compete with CBDC and adjust their spreads more, making the indirect effect relatively large. Our calibration exercises suggest the indirect effect to substantially amplify the aggregate response to CBDC rate changes in settings with high market concentration but less so if CBDC Policy Transmission 529 concentration is lower and differentiation constitutes a relatively more important source of deposit market power. Such considerations do not only apply to temporary changes in interest rates, but also to how central bank policy can affect long-run welfare through inducing a more efficient liquidity mix. Indeed, we not only demonstrate that in a setting with more deposit market concentration, the effects of the CBDC rate on deposit spreads and welfare become more pronounced, but also through a decomposition that the former is indeed the cause of the latter. This also implies that the monetary policy chosen by a Ramsey planner with limited instruments will depend on deposit market concentration. In contrast to CBDC policy, the impact of reserve rates, which we briefly analyze as stand-in for monetary policy instruments directed at the banking sector, decreases with higher deposit market concentration. Under the assumption that the shock makes it more expensive for banks to provide deposits, it causes banks to increase their deposit spreads and households’cost of liquidity. This effect is smaller if market concentration is high, as this makes the spreads charged by banks relatively more dependent on their demand schedule, which is otherwise not directly affected by the policy. Our work relates to the growing and recent literature on CBDC, which has studied these potential new payment instruments from a variety of perspectives. For example, Agur et al. (2022) analyze the trade-offs associated with CBDC design given heterogeneous household preferences over payment instruments and network effects regarding their use. They conclude that central banks should indeed issue interest-bearing CBDCs and choose their rate so that other payment instruments remain in use. Similarly, Keister and Sanches (2022) highlight trade-offs associated with CBDC design choices. In particular, they argue that a CBDC with a deposit-like design would have positive effects by increasing paymentand exchange efficiency, but may also decrease investment by inducing higher funding costs for banks. Piazzesi and Schneider (2022), in turn, warn that CBDC crowding out bank deposits may decrease efficiency in financial intermediation due to a complementarity between offering both deposits and credit lines. Other work has studied the impact of CBDC adoption on financial stability with differing findings, i.e. that CBDC may either improve financial stability (Fernández-Villaverde et al. 2021) or encourage banking panics (Williamson 2022). Given that we study CBDC in a set-up with non-competitive banks, our work is particularly related to Andolfatto (2021); Bacchetta and Perazzi (2022) as well as Chiu et al. (2023), which all study the impact of CBDC introduction in the presence of a noncompetitive banking sector. Andolfatto (2021) focuses on the impact of CBDC introduction on bank lending and economic activity and finds that a CBDC may not impede either. In fact, non-competitive banks forced to increase their deposit rates 530 H. Chen et al. will be subject to an additional inflow of deposits due to the more attractive rates and, in turn, convert this additional funding into lending. Chiu et al. (2023) obtain similar results in a different set-up allowing for differing degrees of bank market power. In contrast to our work, these papers only consider concentration as a source of deposit bank market power and focus on the long-run effects of CBDC on bank lending and general economic activity and consider, while we also analyze the transmission of short-term shocks. While Bacchetta and Perazzi (2022) also share the long-run focus, they consider a monopolistic competitive deposit market on which deposit market power is only driven by differentiation. Jiang and Zhu (2021) and Garratt et al. (2022) share our focus by studying monetary pass-through in settings with imperfectly competitive or heterogeneous banks, respectively. Jiang and Zhu (2021) study the pass-through of both reserve and CBDC rates in a framework similar to Chiu et al. (2023). In the presence of a non-competitive banking sector, the introduction of CBDC is shown to potentially weaken the reserve pass-through, as perfect substitutability forces banks to match the CBDC rate on the deposit market. CBDC can essentially “dictate”the economy. The CBDC rate, in turn, may have a particularly strong pass-through to deposit rates, while its effects on loan rates depend on the reserve rate in an ambiguous way. Major differences between the work of Jiang and Zhu (2021) and ours are that they also only consider one margin of banking competition, and, due to the assumption of perfect substitutability between bank deposits and CBDC, rule out the presence of the indirect effects discussed above, as the CBDC rate will either determine the deposit rate completely or not affect it all. Garratt et al. (2022) consider a framework with differing bank types (“large”and “small”) competing for deposits from workers having heterogeneous preferences over the non-monetary benefits (e.g. extensive branch networks) they offer. They find that the pass-through of the CBDC rate to the deposit rate is stronger if the CBDC rate is high compared to the reserve rate, which, however, hurts the “small” bank. In contrast to our work, their focus is on bank heterogeneity, from which we abstract. Also, in their setup, no one actually ends up holding CBDC (the digital currency can again perfectly substitute for bank deposits and is out-competed by banks), so their model cannot provide for indirect effects of the CBDC rate on households’liquidity costs either. Furthermore, our research is related to several studies analyzing the interaction of bank market power and monetary policy transmission. In particular, Drechsler et al. (2017) propose a deposit channel of monetary policy. As interest rate increases raise the opportunity costs of holding cash, non-competitive banks are able to increase the deposit spread in response to tighter monetary policy, consequently reducing the overall amount of deposits. This, in turn, can affect both the liquidity premium and bank lending. Choi and Rocheteau (2023) study this channel CBDC Policy Transmission 531 theoretically in a search-theoretic model of deposit markets. Additionally, estimating a structural model of the banking sector, Wang et al. (2022) similarly find bank market power to have important effects on the transmission of rate changes to deposit rates. In addition to the Drechsler et al. (2017) mechanism, their model also explicitly considers an oligopolistic lending market, where banks additionally respond by adjusting their lending rate markups. The rest of the paper is organized as follows: Section 2 describes the elements of the model economy and characterizes its equilibrium. Section 3 analyzes the transmission mechanisms of the interest rates on CBDC and reserves by qualitatively characterizing the channels through which the interest rates affect real allocation. Section 4 calibrates the model to conduct numerical exercises. Then, Section 5 quantitatively demonstrates the extent to which deposit market power affects policy transmissions under short-run monetary policy shocks. Section 6 investigates the implications of deposit market power for the efficacy of CBDC policy in the long run. Next, we conduct robustness tests in Section 7. Finally, Section 8 summarizes the results and concludes. 2 Model We study an extended Sidrauski (1967) model, building on Niepelt (2024), in which both the government and banks provide liquidity to households that gain utility from holding it. Households substitute imperfectly between a government-issued form of liquid asset that we interpret as CBDC and commercial bank deposits. Banks fund themselves by borrowing deposits from the households and invest in capital and reserves which are used to “back up”deposit issuance. We follow Drechsler et al. (2017) and assume that banks are non-competitive in the deposit market. Banks have market power due to both market concentration and imperfect substitutability between banks’deposit services. Neoclassical firms produce a common consumption good using capital and labor, and a consolidated government/central bank issues CBDC and reserves. 2.1 Households We consider an economy consisting of many identical and infinitely-lived households, with the measure normalized to one. The representative household values consumption, c t and liquidity services, z t+1 , according to a period utility function of the form 532 H. Chen et al. u(ct,zt+1)= (1−v)c1−ψ t+vz1−ψ t+1 () 1−σ 1−ψ 1−σ, where v ∈(0,1)is the relative weight of liquidity services in utility, ψ∈(0, 1) is the inverse intratemporal elasticity of substitution between consumption and liquidity, and σ> 0 is the inverse intertemporal elasticity of substitution between consumptionliquidity bundles across time. We assume that CBDC and deposits are imperfect substitutes: Liquidity services are derived from real holdings of CBDC, m t+1 , and deposits, n t+1 , according to a constant elasticity of substitution (CES) aggregator zt+1=(1−γ)m1−ϵ t+1+γn1−ϵ t+1 () 1 1−ϵ, where γ∈(0, 1) is the relative liquidity weight of bank deposits, and ϵ∈(0, 1) is the inverse elasticity of substitution between CBDC and deposits. The liquidity weight parameter, γ, captures how useful deposits are for the purpose of holding liquidity relative to the same quantity of CBDC. We follow Drechsler et al. (2017) in assuming that deposits are themselves a composite good issued by a set of Nnon-competitive banks. Each bank ihas mass 1/Nand produces deposits of a quantity ni t+1/N. The household values deposits at different banks such that nt+1=1 N∑ N i=1 ni t+1 () 1−η () 1 1−η ,(1) where ηdenotes the inverse elasticity of substitution between banks. The representative household can be thought of as an aggregation of many individual households who may have diverse preferences for holding deposits at different banks. Therefore, the representative household substitutes deposits imperfectly across banks, which implies that 0 < η<1. In our framework, in addition to deposits and CBDC, households can invest directly in capital. This is necessary for the model to feature realistic amounts of capital and liquid assets, as the aggregate amount of the former typically far exceeds the amount of the latter in modern economies. 1 Since we found it to be not crucial for our results, we abstract from a labor supply choice to simplify the analysis and instead assume the representative agent to inelastically supply a constant amount of labor l. The household’s budget constraint is then given by 1Note that our assumption is isomorphic to alternatively assuming that households do not hold capital directly but also provide funding to banks through a competitive asset market not providing liquidity services. If banks were not only the only agents able to hold capital and only obtain funding through deposits, the model would either feature way too much deposits, way too little capital, or way too low bank leverage ratios. CBDC Policy Transmission 533 ct+kh t+1+mt+1+1 N∑ N i=1 ni t+1+τt=wtl+πt+kh tRk t+mtRm t+1 N∑ N i=1 ni tRn,i t,(2) where kh t+1are direct holdings of capital, τ t is the lump-sum tax net of government transfer, w t is the wage rate, π t is the dividends from firms and banks, Rk tis the return on capital, Rm t+1is the real gross interest rate on CBDC, and Rn,i t+1is the real gross interest rate on deposits at bank i. We assume that the returns on CBDC and deposits are risk-free, i.e. Rm t+1and Rn,i t+1are known at time t. The household, taking prices, profits, and taxes as given, solves max ct,kh t+1,mt+1,ni t+1 {} ∞ t=0 E0∑ ∞ t=0 βtu(ct,zt+1) s.t.ct+kh t+1+mt+1+1 N∑ N i=1 ni t+1+τt=wtl+πt+kh tRk t+mtRm t+1 N∑ N i=1 ni tRn,i t, kh t+1,mt+1,ni t+1≥0. We now turn to the first-order optimality conditions of the household program. Detailed derivations are provided in the Appendix A.1. First, the household optimally allocates resources between deposits at individual banks according to ni t+1=nt+1 χn,i t+1 χn t+1 () −1 η ,(3) which closely resembles demand equations for differentiated consumption goods commonly derived in New Keynesian models. The relative share of deposits at bank i, ni t+1/nt+1, must relate negatively to its corresponding relative cost, χn,i t+1/χn t+1. Here, χn,i t+1 is the interest-rate differential between the risk-free rate, Rf t+1, and the deposit rate offered by bank i χn,i t+1=1−Rn,i t+1 Rf t+1 , which represents the opportunity cost of holding deposits at bank iand which we hereafter refer to as deposit spread. The risk-free rate is defined in the standard way as the inverse of the expected value of the household’s stochastic discount factor, Λ t+1 , Rf t+1=1 Et[Λt+1],(4) where the stochastic discount factor is defined as follows: 534 H. Chen et al. ct+kt+1−kt(1−δ)=atkα tl1−α−Qt, where Qt=mt+1μ+nt+1νt+ζt+1ρ () . The resource constraint has the standard interpretation that available output in the economy is split between consumption, c t , and investment, k t+1 −k t (1 −δ). However, there are resource costs associated with the provision of liquidity to the household, summarized by the term Q:μper unit of CBDC and ν t +ζ t+1 ρper unit of deposit. The resource cost of deposits has two terms because the banking sector incurs a cost of deposit issuance, ν t , and the government incurs a cost of issuing reserves used by the banking sector, ζ t+1 ρ,to“back up”deposit issuance. As the household demands liquidity services in proportion to consumption, we can combine the terms c t and Q t , and rewrite the resource constraint as ctΩrc t+kt+1−kt(1−δ)=atkα tl1−α,(25) where Ωrc t≥1 is given by Ωrc t=1+v 1−v 1 χz t+1 () 1 ψ (1−γ)χz t+1 χm t+1 () 1 ϵ μ+γχz t+1 χn t+1 () 1 ϵ ω+ϕζ1−φ t+1+ζt+1ρ () ⎛ ⎝⎞ ⎠.(26) 2.6 Policy and Equilibrium The consolidated government sets the interest rates on CBDC and reserves and elastically supplies these assets to households and banks to meet demand. A policy consists of Rm t+1,Rr t+1,τt {} t≥0and an equilibrium conditional on policy consist of –a set of positive prices, wt,Rk t+1,Rf t+1,χm t+1,χn t+1,χz t+1,χr t+1 {} t≥0; –a positive allocation, {ct,kt+1}t≥0; –and positive CBDC, deposits and reserves holdings, {mt+1,nt+1,zt+1,rt+1}t≥0, such that (4), (6)–(11), (14)–(16), (22), (23) and (25) are satisfied. 3 Monetary Policy Transmission In this section, we elaborate on the transmission mechanisms of the interest rates on CBDC and analytically characterize the channels through which they affect the allocation. We also briefly discuss the transmission of reserve rates, given that we will contrast their effects with CBDC below. Overall, this analysis builds the CBDC Policy Transmission 541 foundation for the quantitative exercise in the next sections where we study how monetary policy affects the real economy. 3.1 Real Effects of Monetary Policy The two key conditions that characterize the equilibrium allocation, the Euler equation (11) and the resource constraint (25), all closely parallel the conditions of a textbook RBC model. The differences relative to an RBC model are the quantities Ωc t+1 and Ωrc t+1. Importantly, the direct impact of liquidity on the household’s consumption/ savings decision, captured by Ωc t+1, depends solely on the average cost of liquidity, χz t+1. So we will mostly focus on the effects of policy on χz t+1when studying transmission below. For this purpose, it is instructive to first lay down how the average cost of liquidity works through our model economy. The Euler equation (11) shows that the household’s consumption/savings choice depends on liquidity through the marginal utility of consumption, which changes with the average cost of liquidity according to ∂uc,t ∂χz t+1 =c−σ t ∂Ωc t ∂χz t+1 ,where ∂Ωc t ∂χz t+1 ∝σ−ψ ψ. We see that the sign of the impact on the marginal utility of consumption depends on the relative magnitudes of ψand σ. If the household’s intratemporal elasticity of substitution between consumption and liquidity is smaller than the intertemporal elasticity of substitution, i.e. ψ>σ, an increase in the cost of liquidity leads to a decrease in the marginal utility of consumption. This is driven by the fact that an increase in the cost of liquidity, according to (6), reduces the household’s demand for it. A decrease in the level of liquidity then decreases the marginal utility of consumption, and hence there is consumption–liquidity complementarity. On the other hand, when ψ<σ, an increase in the cost of liquidity leads to an increase in the marginal utility of consumption. In the case where ψ=σ, the household’s utility is separable in consumption and liquidity and the cost of liquidity has no direct impact on consumption/savings choices. Moreover, the spreads on CBDC and deposits also show up in the aggregate resource constraint (25) through the term Ωrc t. This reflects the resource costs associated with liquidity provision, incurred by the government and the banking sector. In the special case where the household does not value liquidity services, i.e. v →0, both Ωc t+1and Ωrc t+1converge to one. At this “cashless limit”, the cost of liquidity has no impact on the household’s consumption/savings since no liquid assets are held. Therefore, there are also no resource costs associated with liquidity provision. Then, the model collapses into a standard RBC model. 542 H. Chen et al. To conclude, we have seen that the household’s consumption/savings decision only depends on the average cost of liquidity, which in turn is a function of the spreads on CBDC and deposits. As these are also the sole endogenous determinants of the liquidity cost term Ωrc t+1, the government can affect the allocation only insofar as it affects these spreads. While the government controls the CBDC spread directly through the CBDC rate, the deposit spread is determined by the banking sector. But as we will see below, the government can influence its behavior through the interest rates on both reserves and CBDC. 3.2 Interest on CBDC We now explain the channels through which the household’s average cost of liquidity can be influenced by the CBDC rate. Suppose the government lowers the CBDC rate so that the CBDC spread widens. 2 Differentiating the average cost of liquidity, given by (7), with respect to the CBDC spread yields ∂χz t+1 ∂χm t+1 =(1−γ)1 ϵχm t+1 χz t+1 () −1 ϵ ⏟⏞⏞⏟ direct effect +γ1 ϵχn t+1 χz t+1 () −1 ϵ∂χn t+1 ∂χm t+1 ⏟⏞⏞⏟ indirect effect .(27) This expression shows that the CBDC spread works through two channels: Firstly, it directly increases the cost of liquidity by the first term. The strength of this direct effect is increasing in the relative liquidity weight of CBDC, 1 −γ, and decreasing in how costly CBDC is relative to the average cost of liquidity, χm t+1/χz t+1. Comparing the direct effect with the household’s demand for CBDC (9), we see that it is just the share of CBDC in the total stock of liquidity, m t+1 /z t+1 . Intuitively, the more important CBDC is as a source of liquidity for the household, the larger the impact of its cost on liquidity’s average cost. Secondly, the CBDC spread affects the cost of liquidity through the deposit side, given by the second term. The strength of this indirect effect is increasing in the relative liquidity weight of deposits, γ, and decreasing in how costly deposits are relative to the average, χn t+1/χz t+1. Comparing the indirect effect with the household’s demand for deposits (10), we see that it is equal to the product of the share of deposits in the total stock of liquid, n t+1 /z t+1 , and the change in the deposit spread caused by a change in the CBDC spread, ∂χn t+1/∂χm t+1. Analogous to the direct effect, the more important deposits are as a source of liquidity the larger is this indirect effect. However, the sign and the magnitude of the second effect also depend on how the banking sector responds to an increasing CBDC spread, captured by ∂χn t+1/∂χm t+1. 2For simplicity, we assume here that the reserve spread is constant. CBDC Policy Transmission 543 In this regard, the optimality condition (16) shows that the CBDC spread can influence the deposit spread through the bank’s marginal benefit of deposit issuance (left-hand side). Specifically, CBDC spread affects the elasticity of demand for deposits that the bank faces, given by (18). As we discussed previously, the demand elasticity depends on a weighted average of the household’s elasticities of substitution to consumption, 1/ψ, and CBDC, 1/ϵ. The CBDC spread determines this average through the relative weight s t , given by (19). Taking the partial derivative of the demand elasticity (18) with respect to the CBDC spread, we get 1 N ∂st ∂χm t+1 () 1 ψ−1 ϵ ()with ∂st ∂χm t+1 =−1−ϵ ϵ st(1−st) χm t+1 <0.(28) The partial derivative shows that the marginal impact of CBDC spread is non-zero only if ψ≠ϵ. Intuitively, banks collectively face competition from CBDC and consumption for the household’s resources. Therefore, any outflow from deposits depends on the household’s elasticities of substitution to CBDC and consumption. The CBDC spread only influences the relative importance of these two sources of deposit outflow, indicated by s t . If the household finds it as easy to substitute from deposits to consumption as it does to CBDC, i.e. ψ=ϵ, then the two sources of competition for the banks are equally important and the CBDC spread does not influence the banks’ deposit spread. In such a case, the equilibrium spread is set equal to the marginal cost of deposit issuance plus a constant markup, similar to the case where banks are monopsonistically competitive. In general, it seems reasonable to expect that deposits will be more substitutable with CBDC than with consumption, i.e. ψ>ϵ. Then, an increase in the CBDC spread makes the demand elasticity for deposits (18) less negative in value and, in turn, decreases the marginal benefit of deposit issuance. The intuition is that when its spread widens, CBDC becomes a comparatively expensive source of liquidity and a larger fraction of potential substitution out of deposits will go to consumption (indicated by a decrease in s t and more weight being put on 1/ψ). The elasticity of demand moves closer to 1/ψ, which is smaller than 1/ϵ, and thus decreases in absolute value. Therefore, an increase in the CBDC spread makes the household’s demand for deposits less elastic. For banks with market power, a less elastic demand means that in order to attract additional deposits from the household, the spread needs to be lowered by more than before. That is, the marginal benefit of deposit issuance decreases. Given a fixed marginal cost, this implies that the equilibrium deposit spread increases. In other words, an increase in the CBDC spread is akin to giving banks more market power. Banks take advantage of this and charge a higher spread on deposits in equilibrium. As we alluded to previously, market conditions in the deposit market also play a central role. If deposits at different banks are perfect substitutes or the deposit 544 H. Chen et al. market is perfectly dispersed, the equilibrium deposit spread is determined without the influence of the CBDC spread. If the household does not differentiate between banks, each individual bank’s choice of how much deposits to issue does not matter for the equilibrium spread, which will equal the marginal cost of deposit issuance (20): the market is competitive. Similarly, if the deposit market is perfectly dispersed and the only source of market power is differentiation, the impact of each individual bank’s spread on the aggregate deposit spread goes to zero. The deposit market becomes monopsonistically competitive with a constant markup over marginal cost solely depending on the substitutability between banks, given by (21). In both cases, the government cannot use the CBDC spread to influence the banking sector. To sum up, when the government decreases the CBDC rate and widens the CBDC spread, it directly increases the household’s average cost of liquidity and affects allocation. Moreover, a higher CBDC spread increases the spread on bank deposits, provided that banks have sufficient market power, which raises the household’s cost of liquidity further. The transmission of the CBDC rate through the banking sector is similar to the deposit channel of monetary policy proposed by Drechsler et al. (2017). The authors describe a situation in which the household holds cash issued by the government and deposits issued by banks with market power. Policy-makers can induce an increase in the deposit spread by increasing the household’s opportunity cost of holding cash, captured by the nominal interest rate on risk-free bonds. In our model, instead, the alternative to bank deposits is CBDC. The government can similarly affect banks’deposit spread by changing the household’s opportunity cost of holding this alternative, i.e. its spread. 3.3 Interest on Reserves While interest rates on central bank reserves have traditionally not been a particularly salient policy tool (for example, the Fed only started to remunerate reserves in 2008), we briefly analyse them as a stand-in for monetary policy instruments directed at the banking sector: Since the reserve rate shock effectively increases the banks’cost of deposit provision, we expect that other (unmodeled) monetary policy shocks that affect consumers only through the banking system to be affected by deposit market power in a qualitatively similar way. In particular, the interest on reserves affects the household’s average cost of liquidity only through its impact on the deposit spread. Suppose the government decreases the reserve rate so that the reserve spread increases. 3 Taking the first derivative of the average cost of liquidity with respect to the reserve spread, we get 3For simplicity, we assume here that the CBDC spread is constant. CBDC Policy Transmission 545 ∂χz t+1 ∂χr t+1 =γ1 ϵχn t+1 χz t+1 () −1 ϵ∂χn t+1 ∂χr t+1 . Notice that the marginal impact of the reserve spread is very similar to the indirect effect of the CBDC spread in (27). This is not surprising since both effects work through the banking sector. The impact of the reserve spread is the product of the share of deposits in the total stock of liquid, n t+1 /z t+1 , and the change in the deposit spread caused by the change in the reserve spread, ∂χn t+1/∂χr t+1. Again, the more important deposits are as a source of liquidity, the larger is this effect. But, its sign and the magnitude also depend on how the banking sector responds to an increasing reserve spread, ∂χn t+1/∂χr t+1. This key term is in turn determined by how banks react to the higher cost of deposit provision induced by χrand depend to what extent potential outflows to CBDC are considered in banks’deposit rate setting. 4 Calibration To gauge the importance of deposit market concentration for the efficacy of a CBDC rate as a policy instrument, we calibrate our model to conduct various numerical exercises below. A detailed description of our procedure is provided in Appendix A.5. A period is interpreted as a quarter. Following Niepelt (2024), we adopt the case with a monopsonist bank as a benchmark but will compare it with the case N=3 below: Given the symmetric banks, this value implies a Hirschman-Herfindahl-Index (HHI) of 1/3, close to the average county-level HHI of 0.35 estimated by Drechsler et al. (2017) for the U.S. over the period from 1994 to 2013. We start with exogenously setting several parameters to values from the literature: In line with standard convention, we choose the household’s risk aversion parameter σto be 2, the capital share αto be 1/3 and the depreciation rate to be 2.5 %. We normalize l=1/3. Additionally, we follow Bacchetta and Perazzi (2022) and assume for our benchmark exercises that CBDC is designed so that its elasticity of substitution with respect to the deposit aggregate is ϵ= 1/6. Given the uncertainty about whether this will be the practically relevant magnitude for any actual CBDC, we also consider different values of ϵin the robustness exercises in Section 7. We also exogenously set the initial steady state’s policy: Firstly, we assume that the central bank chooses to pay a nominal interest rate of 0 on the CBDC, i.e. it has the same return as cash. This is in line with many central seeming reluctant to remunerate a potential CBDC, and, assuming 2 % trend inflation, implies a real annual gross return of 0.98 and Rm= 0.981/4. We furthermore assume the annual gross return on reserves to amount to Rr= 0.991/4, implying a nominal reserve rate of 1 % annually: 546 H. Chen et al. The Fed started to pay nominal interest rates on reserves only in 2008 and they averaged roughly 1 % in the time since. Now, to be able to clearly identify the effect of deposit market concentration in our model, we restrict the model versions with N= 1 and N= 3 to be identical in all other dimensions: In particular, the CBDC is designed so that aggregate CBDC holdings amount to just 12 % of deposit holdings while we induce the consumption velocity c/zto be equal to 1.2. The former target reflects that in many jurisdictions considering the implementation of a CBDC, policymakers seem reluctant to induce substantial disintermediation of the banking sector. 4 It also implies that the total amount of CBDC held is in line with just physical currency being replaced, which typically amounted to approx. 12 % of aggregate deposit holdings in the post-war US. 5 In contrast, the c/ztarget ensures that even after the introduction of a CBDC, the overall liquidity velocity is similar to current levels. 6 The targets are achieved by setting γ= 0.5938 and v =0.0252. Finally, as in Niepelt (2024), we aim to induce a deposit markdown of 1.5, which we also induce by choosing ψaccordingly in the N=1 version. To achieve the same in the model version with N= 3, we additionally use the parameter ηgoverning the substitutability between different banks’deposits. This results in ψ= 0.3774 and η= 0.33, respectively. 7 Note that here ψ<σ, which implies consumption and liquidity services to be substitutes. This calibration result is in line with Niepelt (2024), who calibrates a very similar value for ψ. While substitutability between liquidity services and consumption may seem counterintuitive if the former are taken to solely represent transaction services, it is perhaps less so if one considers additional benefits of liquidity. For example, liquid assets may also be useful for insuring against idiosyncratic risk, as for example in Huggett (1993). 8 4For example, Federal Reserve Governor Michelle Bowman voiced related concerns, stating that ”a CBDC, if not properly designed, could disrupt the banking system and lead to disintermediation, potentially harming consumers and businesses, and could present broader financial stability risks” (Bowman 2023). 5This statement is based on the series MBCURRCIR and DPSACBW027SBOG from FRED. 6According to FRED (Series: M2V), M2 velocity in the US is typically between 1.4 and 1.8, while aggregate consumption is usually around 60–70 % of output. Thus, in the current situation without CBDC, a c/zaround 1.2 seems reasonable. 7Note that with N= 1, the parameter ηplays no role. 8The calibration of ψis limited by the parameter restrictions on the class of utility functions used by us and Niepelt (2024). For the model to feature a well-defined deposit spread in the monopsonistic case (N= 1), it is necessary that ψ< 1 (see e.g., equation (9) in Niepelt (2024)). Although complementarity could still be achieved by setting 1 > ψ>σ, we chose to prioritize having a value for the intertemporal elasticity of substitution that is in line with the standard convention σ≥1 in the literature. Besides, substitutability is also necessary for the model to conform with the conventional wisdom that higher policy rates reduce aggregate consumption demand. CBDC Policy Transmission 547 Regarding the costs of providing different forms of liquidity, Niepelt (2024) discusses various evidence suggesting that the annual cost of depositand reserve provision may amount to up to 1.2 % and 0.05 %, respectively. We thus choose ω= 0.003 and ρ= 1.3 ×10−4for our quarterly calibration. We furthermore restrict μ=ω+ρ, implying that the costs of providing CBDC are equivalent to the government operating a narrow bank. Again following Niepelt (2024), we set φ= 1.5: Note that this parameter will effectively only matter for exercises with changing reserve rates, as we always choose ϕto induce a reserve-to-deposit ratio of 0.1945, the midpoint of the range considered in said paper. This is achieved by ϕ= 0.0021. Table 1 presents the full list of calibrated parameters. 5 Short-Run Analysis Armed with the calibrated model, we can now assess the implications of deposit market power for the CBDC rate as a policy tool, and, in particular, its role in shaping Table :Baseline calibration. Parameter Value Source/Target Household β.–/Annual Rf=% σStandard l  /Normalization ν. c/z=. ψ. See text ϵ/Bacchetta and Perazzi () η. See text γ. m/n=. Banks ω. Niepelt () φ.Niepelt () ϕ. ζ=. (Niepelt ) Firms α/Standard δ. Standard Government ρ.×−Niepelt () μ. μ=ω+ρ Rr./% nom. return Rm./% nom. return 548 H. Chen et al. the transmission of policy through the direct and indirect effects outlined above. We start by analyzing to what extent it matters if a central bank aims to use the CBDC rate as a tool to influence business cycle fluctuations, for which we compute linearized impulse response functions (IRFs) of the economy to shocks to the CBDC and reserve rates under the different assumptions on deposit market power (i.e. the cases with N= 1 and N= 3). Note that according to the Blanchard–Kahn criterion, all model versions feature locally unique equilibria. 5.1 Policy Shocks When analyzing the impact of a CBDC rate shock numerically below, we assume it to follow a log AR (1) process log Rm t+1 () =(1−ρm)log(Rm)+ρmlog Rm t () +em t, where ρmis the persistence parameter, Rmis the steady state CBDC rate, and em tis the exogenous shock. The exogenous shock is non-zero in the first period of the simulation and returns to zero afterward. In order to properly isolate the effect, when analyzing the CBDC rate, we assume that the reserve rate is set so that the reserve spread is constant at its steady state level, i.e. it fulfills Rr t+1=Rf t+1(βRr), where Rris the steady state reserve rate. When we discuss the case of the reserve rates, equivalent assumptions are made, effectively interchanging the processes for Rrand Rm. 5.2 Impulse Responses 5.2.1 Response to a CBDC Rate Shock Figure 1 shows the IRFs, as deviations from the non-stochastic steady state, to a negative 10 basis points shock to the quarterly CBDC rate. Naturally, the decrease in the CBDC rate immediately widens the CBDC spread by essentially the same magnitude as the aggregate capital stock and the risk-free rate changes little. The increasing CBDC spread raises the household’s average cost of liquidity in both the baseline N= 1 and the alternative N= 3 cases so that households choose to enjoy less liquidity services. This decreases the household’s demand for liquidity services but increases the household’s current marginal utility of consumption, reflected in a higher Ωc t+1. This is due to our calibration featuring ψ<σ. In other words, the CBDC Policy Transmission 549 household’s opportunity cost of saving, in utility terms, goes up. The household is incentivized to save less and increase current consumption. 9 Overall, the effect of the cut is not overly strong in either case, reflecting its size and the assumed scenario of limited CBDC adoption. Nevertheless, there is a substantial difference in the relative magnitudes of the responses, with e.g. the overall impact of the shock on aggregate consumption being almost twice as large in the baseline case with N= 1. As outlined above, in our model, the real effects of a CBDC rate are transmitted via its effect on the liquidity cost χzand the impact of the CBDC spread on this term can be decomposed into a direct effect and an indirect effect, shown in (27). Now, Figure 2 displays this decomposition of the responses of the liquidity cost: The green dashed lines show the direct effects and the red solid lines show the indirect effects. The sum of the lines equals the original impulse responses of the cost of liquidity in Figure 1. Figure 1: Impulse responses to 10 basis points decrease in CBDC rate. Figure shows the impulse responses of key economic variables to a 10 basis points decrease in the CBDC rate under two different market structures (monopsony N= 1 and oligopsony N= 3). 9Note that in the absence of a labor supply margin, the response of output is determined by the response of the capital shock, implying that the rate shocks do not induce the positive comovement of consumption and output typical for e.g. New Keynesian models. 550 H. Chen et al. that banks’deposit spreads are kept fixed at the level calibrated in the baseline steady state with Rm= 0.98: The difference between the true and counterfactual CE gains then isolates the contribution of the indirect effects to welfare. 15 Figure 6 illustrates these CE welfare effects for the by now familiar cases of a monopsonistic bank (N= 1) and the oligopsonistic environment with three banks (N= 3). Clearly, the model suggests that paying a higher real return on CBDC has welfare benefits, which reflects the results by Niepelt (2024) who finds CBDC to be a very efficient means of liquidity provision: As the higher rate on CBDC induces the representative agent to hold relatively more of it, the aggregate costs of liquidity provision are reduced. Overall, the contribution of the indirect effect to aggregate welfare is substantial and can account for almost a third of the overall CE gains with N= 1. With N= 3, the share is smaller but still noticeable. Of course, this is again because with differentiation being the more important source of deposit market power, the smaller banks react less to the decreasing CBDC spreads. Figure 5: Deposit spread against different CBDC rate changes. Figure shows how the deposit spread responds to different CBDC rates. The responses are shown for two market structures (monopsony N=1 and oligopsony N= 3). 15 Note that this decomposition differs from the one used in the previous Section 5, which applied to the cost of liquidity χz. CBDC Policy Transmission 557 Considering the results above, it seems that indirect effects through the banking sector can be quantitatively important for the welfare gains a central bank can achieve by paying positive interest on CBDC. This is particularly the case ifdepositmarketpowerisdeterminedbyconcentration and reiterates the point that the source of bank market power is an important determinant of the model effects of CBDC policy. Interestingly, given that welfare gains are higher for N=1 than for N= 3, the exercise above also suggests that for given policy rates, moving from low to high banking concentration can potentially increase aggregate efficiency as the differing deposit rates change the composition of the aggregate liquidity mix. 7 Robustness Tests Whilewedisciplinemostparametersofourmodelinlinewiththeliterature,akey uncertainty is the substitutability between CBDC and deposits, captured by ϵ.Since Figure 6: CE welfare gain versus Counterfactual welfare gain. Figure compares the consumptionequivalent (CE) welfare gains and counterfactual welfare gains given different CBDC rates. The comparisons are shown for two market structures (monopsony N= 1 and oligopsony N= 3). 558 H. Chen et al. for most countries, CBDC still largely remains a theoretical possibility, we are left to speculate regarding some aspects of the relationship. Therefore, we test the robustness of the main insights in our paper by changing the substitutability between CBDC and deposits. The main specifications assume a “medium”degree of substitutability between CBDC and deposits, i.e. ϵ= 1/6, following Bacchetta and Perazzi (2022). The test is then changing the degree of substitutability to ϵ= 1/4 and ϵ= 1/10 and repeating the exercises above: Note that this entails re-calibrating parameters such as ψand γto achieve the same steady state targets as in the baseline model, ensuring that the resulting differences reflect only the differing choice of ϵ. 7.1 Short-Run Analysis Figures A1 and A2 in Appendix C show the impulse responses to a 10 basis points decrease in the CBDC rate with the alternative specifications described above. We see that the main takeaways from Section 5 still stand: An increase in the CBDC spread increases consumption and lowers capital investment. The extent to which deposit market power is shaped by concentration still has a noticeable impact: Higher market concentration noticeably amplifies the impulse responses, although both the aggregate impact of the shock as well as the relative contribution of concentration decrease (increase) for higher (lower) ϵ. Naturally, if the representative agent finds it harder to substitute deposits with CBDC, banks need to be less concerned with the impact of its return Rmon deposit demand, which dampens the indirect effect and in turn the strength of the overall response. If there is thus less scope overall for indirect effects due to lower substitutability, the impact of deposit market concentration on the effects of the shock is lower. We also consider how the response to the reserve shock is shaped by η, for which Appendix Figures A3 and A4 display the IRFs for again the cases ϵ= 1/4 and ϵ= 1/10, respectively. In contrast to CBDC, lower (higher) substitutability now increases (decreases) the aggregate effects of the shock but also dampens (strengthens) the impact of concentration N. With less scope for substitution between CBDC and deposits, banks have more scope to react to their increases in costs due to the lower reserve rates, amplifying their aggregate effects. Overall, the above results imply that the attention central banks will need to pay to deposit market power and its sources is going to depend on how they design their potential CBDC: In case it becomes rather easy for consumers to switch between deposits and the digital currency, it is going to matter more for the effects of monetary policy. CBDC Policy Transmission 559 7.2 Long-Run Analysis Similar considerations as for the short-run analysis above apply to the long-run effects of the CBDC rate: Figures A5 and A6 illustrate the CE welfare gains for different levels of the elasticity of substitution between CBDC and bank deposits for the cases of a monopsonistic bank (N= 1) and the oligopsonistic setting (N= 3), respectively. In either case, a higher elasticity of substitution (lower value of ϵ) corresponds to overall stronger welfare effects of the CBDC rate in the long run, as it ends up affecting deposit spreads and hence the aggregate liquidity mix more. 8 Conclusions This paper has studied the transmission and effects of interest rates on CBDC, a potential new policy lever at central bankers’disposal. The analysis focused on bank market power and highlighted that if the deposit market is concentrated, a CBDC rate will in general affect the aggregate economy through both direct and indirect effects: A higher return for holding the digital currency not only affects households’savingand portfolio decisions by itself, but also incentivizes the non-competitive banks to adjust their deposit spreads, which may in turn strengthen the aggregate effect of the policy innovation. Our simple theoretical model implies that the latter indirect channel has the potential to substantially amplify the general equilibrium consequences of CBDC policy, but only if concentration is the key determinant of deposit market power. If deposit markdowns are instead due to banks providing differentiated liquidity services and concentration is lower, the direct channel typically dominates. We establish these insights both for the short run, by studying policy shocks, as well as the long run, by comparing steady states. Regarding the latter, we also extend the insights of Niepelt (2024) on optimal policy, which depends on deposit market concentration and the potential scope for indirect effects particularly if the government cannot subsidize banks directly. Moreover, deposit market concentration is also relevant for other monetary policy tools directed at the banking sector (such as reserve rates) but tends to affect them in the opposite way, weakening their efficacy. We view our findings as relevant for both practice and theory: Regarding the former, it means that using interest rates on CBDC as a policy tool necessitates a detailed understanding of the deposit market and competition thereon. For modeling, in turn, they reveal that specific assumptions on bank competition can importantly affect results for models of CBDCand monetary policy, even if consistency with the same aggregate moments is ensured. It thus seems interesting for future research to 560 H. Chen et al. explore this further by incorporating oligopsonistic banks into quantitative models of CBDC allowing for richer frictions and shocks. Appendix A Derivations A.1 Households The household, taking prices, profits and taxes as given, solves max ct,kh t+1,mt+1,ni t+1 {} ∞ t=0 E0∑ ∞ t=0 βtu(ct,zt+1) s.t.ct+kh t+1+mt+1+1 N∑ N i=1 ni t+1+τt=wtl+πt+kh tRk t+mtRm t+1 N∑ N i=1 ni tRn,i t, kh t+1,mt+1,ni t+1≥0. Focusing on the interior solution, the first-order conditions with respect to capital, CBDC and deposits are kh t+1:1=EtΛt+1Rk t+1 [] (A.1) mt+1: uz,tzm,t+1 uc,t =χm t+1(A.2) ni t+1: uz,tzni,t+1 uc,t =1 Nχn,i t+1(A.3) where fa,tdenotes the partial derivative of function f with respect to its argument a, Λ t+1 is the household’s stochastic discount factor Λt+1=βuc,t+1 uc,t , χm t+1and χn,i t+1are the CBDC spread and deposit spread at bank i, respectively, χm t+1=1−Rm t+1 Rf t+1 ,χn,i t+1=1−Rn,i t+1 Rf t+1 , and the risk-free rate is defined as Rf t+1=1 Et[Λt+1]. CBDC Policy Transmission 561 A.1.1 Demand for Individual Bank Deposits Household’sfirst-order condition (A.3) with respect to deposits at any bank ican be written as uz,tzn,t+1 uc,t nt+1 ni t+1 () η =χn,i t+1.(A.4) Since the last expression holds for any bank, it means that for any two banks i and j χn,i t+1 ni t+1 nt+1 () η =χn,j t+1 nj t+1 nt+1 () η , from which we find the demand for bank deposits j nj t+1=χn,i t+1 χn,j t+1 () 1 η ni t+1.(A.5) Let Tdenote the sum of deposit spreads that the household incurs, and insert (A.5) into the expression, T=1 N∑ N i=1 ni t+1χn,i t+1=1 N∑ N j=1 χn,i t+1 χn,j t+1 () 1 η ni t+1χn,j t+1, to find an expression for ni t+1 ni t+1=NT χn,i t+1 () −1 η ∑N j=1χn,j t+1 () η−1 η .(A.6) We plug equation (A.6) into the definition of aggregate deposit, given by (1), nt+1=Nη η−1T∑ N i=1 χn,i t+1 () η−1 η () η 1−η .(A.7) Let χn t+1be the spread associated with one unit of aggregate deposit, n t+1 .By setting n t+1 = 1, we see from equation (A.7) that χn t+1=1 N∑ N i=1 χn,i t+1 () η−1 η () η η−1 .(A.8) Given equation (A.8), we see that equation (A.6) can be written as 562 H. Chen et al. ni t+1=T χn t+1 χn,i t+1 χn t+1 () −1 η ,(A.9) and inserting the resulting expression into (1), we get nt+1=1 N∑ N i=1 T χn t+1 χn,i t+1 χn t+1 () −1 η ⎛ ⎝⎞ ⎠1−η ⎛ ⎜ ⎜ ⎝⎞ ⎟ ⎟ ⎠ 1 1−η =T χn t+1 .(A.10) Combining the expression for Tand (A.10) we see that T=1 N∑ N i=1 ni t+1χn,i t+1=nt+1χn t+1.(A.11) Lastly, inserting equation (A.11) into (A.9), we get the household’s demand for deposits at bank i ni t+1=nt+1 χn,i t+1 χn t+1 () −1 η .(A.12) Combining the household’s demand schedule with the first-order condition (A.4), we see that (A.4) can be expressed as uz,tzn,t+1 uc,t =χn,i t+1 nt+1 ni t+1 () −η =χn t+1.(A.13) A.1.2 Optimality Conditions Given the functional form assumptions, the household’sfirst-order conditions (A.2) and (A.13) become mt+1: vz−ψ t+1 (1−v)c−ψ t (1−γ)zt+1 mt+1 () ϵ =χm t+1(A.14) ni t+1: vz−ψ t+1 (1−v)c−ψ t γzt+1 nt+1 () ϵ =χn t+1.(A.15) We combine (A.14) and (A.15) to get the ratio mt+1 nt+1 =(1−γ)χn t+1 γχm t+1 () 1 ϵ .(A.16) We plug equation (A.16) into CES function for z t+1 and solve for the ratio of z t+1 to n t+1 CBDC Policy Transmission 563 zt+1 nt+1 = (1−γ)χn t+1 () 1−ϵ () 1 ϵ+γχ m t+1 () 1−ϵ () 1 ϵ () ϵ 1−ϵ γχm t+1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 1 ϵ .(A.17) Inserting (A.17) into equation (A.15) and solve for z t+1 , we get the household’s optimal demand for liquidity zt+1=ct v 1−v 1 χz t+1 () 1 ψ ,(A.18) where χz t+1is the average cost of liquidity faced by the household χz t+1=χm t+1χn t+1 (1−γ)1 ϵχn t+1 () 1−ϵ ϵ+γ1 ϵχm t+1 () 1−ϵ ϵ () ϵ 1−ϵ . Given household’s optimal demand for z t+1 ,wefind the household’s demand for m t+1 and n t+1 mt+1=zt+1(1−γ)χz t+1 χm t+1 () 1 ϵ nt+1=zt+1γχz t+1 χn t+1 () 1 ϵ. (A.19) Plugging optimal z t+1 , given by (A.18), into the first-order condition (A.1), we find the household’s Euler equation c−σ tΩc t=βEtRk t+1c−σ t+1Ωc t+1 [] (A.20) where Ωc tis given by Ωc t=(1−v)1−σ 1−ψ1+v 1−v () 1 ψχz t+1 () 1−1 ψ () ψ−σ 1−ψ . A.2 Banks The date-tprogram of a typical bank is max ri t+1,Rn,i t+1 −ni t+1νi t+EtΛt+1ki t+1Rk t+1+ri t+1Rr t+1−ni t+1Rn,i t+1 ()[] s.t.ni t+1=nt+1 χn,i t+1 χn t+1 () −1 η ki t+1=ni t+1−ri t+1, where 564 H. Chen et al. νi tζi t+1 () =ω+ϕζ i t+1 () 1−φ,ζi t+1=ri t+1 ni t+1 . The first-order conditions for bank iwith respect to its deposit rate and reserve holdings are, respectively, Rn,i t+1:χn,i t+1+χn,i t+1 en,i t+1 =νi t−νi ζ,tζi t+1(A.21) ri t+1:−νi ζ,t=χr t+1,(A.22) where χr t+1=1−Rr t+1/Rf t+1and en,i t+1denotes the elasticity of demand for deposits at bank iwith respect to its deposit spread, χn,i t+1, which in a symmetric industry equilibrium can be shown to be en,i t+1=∂ni t+1 ∂χn,i t+1 χn,i t+1 ni t+1 . Given functional form assumptions, the first-order condition (A.21) becomes χn,i t+11+1 en,i t+1 () =ω+φϕ ζi t+1 () 1−φ, where bank i’s optimal reserves-to-deposits ratio is given by the first-order condition (A.22) ζi t+1=χr t+1 ϕ(φ−1) () −1 φ . To find the demand elasticity, en,i t+1,wedifferentiate the household’s demand for deposit at bank i, equation (A.12) with respect to χn,i t+1and multiply it with the ratio χn,i t+1/ni t+1 en,i t+1=− 1 η nt+1 χn,i t+1 χn t+1 χn,i t+1 () 1 η +1 η nt+1 χn t+1 χn t+1 χn,i t+1 () 1 η∂χn t+1 ∂χn,i t+1 +χn t+1 χn,i t+1 () 1 η∂nt+1 ∂χn t+1 ∂χn t+1 ∂χn,i t+1 ⎛ ⎝⎞ ⎠χn,i t+1 ni t+1 =−1 η+1 η χn,i t+1 χn t+1 ∂χn t+1 ∂χn,i t+1 +χn,i t+1 χn t+1 () 1 ηχn,i t+1 nn,i t+1 ∂nt+1 ∂χn t+1 ∂χn t+1 ∂χn,i t+1 (A.23) In a symmetric industry equilibrium, where χn,i t+1=χn,j t+1and ni t+1=nj t+1for any bank iand j, CBDC Policy Transmission 565 χn t+1=1 N∑ N i=1 χn,i t+1 () η−1 η () η η−1 =χn,i t+1 nt+1=1 N∑ N i=1 ni t+1 () 1−η () 1 1−η =ni t+1 Then, equation (A.23) reduces to en,i t+1=1 N ∂nt+1 ∂χn t+1 χn t+1 nt+1 () −1−1 N () 1 η. To find the aggregate demand elasticity, we differentiate household’soptimal deposit demand, equation (A.19), with respect to the liquidity premium on deposits, χn t+1, ∂nt+1 ∂χn t+1 =∂zt+1 ∂χz t+1 ∂χz t+1 ∂χn t+1 γχz t+1 χn t+1 () 1 ϵ +zt+1γ ϵχn t+1 ∂χz t+1 ∂χn t+1 γχz t+1 χn t+1 () 1−ϵ ϵ −zt+1γχz t+1 ϵχ n t+1 () 2 γχz t+1 χn t+1 () 1−ϵ ϵ and multiply the last expression with the ratio χn t+1/nt+1 ∂nt+1 ∂χn t+1 χn t+1 nt+1 =−1 ψγ1 ϵχz t+1 χn t+1 () 1−ϵ ϵ −1 ϵ(1−γ)1 ϵχz t+1 χm t+1 () 1−ϵ ϵ . Lastly, we write the optimality condition as it applies to a representative bank (and dropping the individual superscript i) χn t+1+χn t+1 1 N−1−st ψ−st ϵ () −1−1 N () 1 η () −1 =ω+φϕζ1−φ t+1,(A.24) where ζt+1=χr t+1 ϕ(φ−1) () −1 φ (A.25) and s t ∈[0, 1] is st=(1−γ)1 ϵχz t+1 χm t+1 () 1−ϵ ϵ . A.3 Aggregate Resource Constraint To find the aggregate resource constraint, we start by inserting total profit, π t , into the household’s budget constraint, imposing market clearing for labor and capital and rearranging 566 H. Chen et al. Figure A2: Higher CBDC-deposits substitutability ϵ=1 10. Figure shows the impulse responses of key economic variables to a 10 basis points decrease in the CBDC rate when ϵ= 1/10. The responses are shown for two different market structures (monopsony N= 1 and oligopsony N= 3). Figure A3: Lower CBDC-deposits substitutability ϵ=1 4. Figure shows the impulse responses of key economic variables to a 10 basis points decrease in the reserve rate when ϵ= 1/4. The responses are shown for two different market structures (monopsony N= 1 and oligopsony N= 3). CBDC Policy Transmission 573 Figure A4: Higher CBDC-deposits substitutability ϵ=1 10. 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