CBDC as Imperfect Substitute to Bank Deposits: A Macroeconomic Perspective
What this paper finds — and why it matters
Layer 1: Overview
Research question and motivation: As central banks worldwide explore retail central bank digital currency (CBDC), the macroeconomic consequences depend heavily on how CBDC interacts with bank deposits. Prior work spans a wide range of conclusions — from “no effect” (Brunnermeier and Niepelt 2019) to disintermediation that reduces lending and output (Keister and Sanches 2022; Chiu et al. 2022) to large output gains (Barrdear and Kumhof 2021, +3% GDP). Bacchetta and Perazzi argue these differences hinge on (i) how substitutable CBDC is with checking deposits, (ii) how easily banks replace lost deposits with other funding, (iii) the interest rate on CBDC, and (iv) the competitive structure of banking. The paper provides quantitative welfare estimates in a model where CBDC and deposits are imperfect substitutes and banks are in monopolistic competition.
Model setup: A closed-economy steady-state model (akin to Gali 2015 and Del Negro-Sims 2015) with households, “bank owners,” firms, banks, government, and central bank. Money reduces a transaction cost on consumption (Schmitt-Grohe-Uribe 2004 style). Deposits and CBDC combine via a CES composite liquid asset characterized by three CBDC design dimensions: its interest rate (rc), its relative liquidity (alpha_c/alpha_b, the CES weight), and its substitutability with deposits (elasticity epsilon_cb). Crucially, with monopolistic competition each bank takes the average deposit rate as given, so the equilibrium deposit rate is unaffected by CBDC (Lemma 1); and because firms can fund at the risk-free rate, bank credit extension and loan rates are also unaffected by CBDC in steady state. Calibration (US-based): risk-free rate 4%, deposit spread 2%, loan spread 1%, reserve ratio 5%, deposit management cost 25 bps, interest semi-elasticity of money demand -0.05, inverse Frisch elasticity gamma=1, wealth/consumption=4. The two extreme ownership cases are zeta=1 (“case a,” households fully own banks) and zeta=0 (“case b,” a zero-measure set of bankers receives all profits).
Main findings (welfare in consumption-equivalent basis points): Welfare can improve via three channels — (1) seigniorage allowing lower distortionary labor taxes, (2) a lower opportunity cost of holding money (raising money holdings, cutting transaction costs, stimulating labor and consumption), and (3) redistribution of bank deposit rents from bankers to the general population. The optimal CBDC rate trades off seigniorage versus opportunity-cost reduction and is decreasing in the labor tax rate and decreasing in the share of banks owned by households (Proposition 3). The first two channels alone yield only modest gains: +9 bps at a 25% labor tax and +20 bps at 45%. Adding the redistribution channel (“case b”) raises non-bankers’ welfare to +54 bps (25% tax) and +59 bps (45% tax); the headline maximum is about 60 bps. From Table 2 (epsilon_cb=20, equal liquidity): consumption rises +27 bps (case a) / +54 bps (case b) at 25% tax, and +41 / +62 bps at 45% tax. All benefits require historically normal interest rates (baseline 4%); near the zero lower bound seigniorage, money’s opportunity cost, and deposit rents all vanish, so the welfare gain falls roughly linearly to zero with the deposit spread.
Policy/theoretical implications: CBDC is a tool to mitigate two distortions — distortionary taxation and the gap between the opportunity cost and the (low) production cost of money — plus a redistributive lever against the concentration of bank rents. The pure efficiency gains are modest; the larger gains come from redistribution and are larger where labor taxes (e.g., EU-14 averaging >40% vs. US ~25%), the Frisch elasticity, or the interest semi-elasticity of money demand are higher.
Layer 2: Deep Dive
What is the model’s identification/derivation strategy, since this is a theoretical paper rather than an empirical one?
There is no econometric identification; results come from a calibrated closed-economy steady-state general equilibrium model. The ‘identification’ of the welfare channels is analytical: three propositions (proved in an online appendix) characterize how seigniorage and the optimal CBDC rate depend on CBDC liquidity (alpha_c), substitutability (epsilon_cb), and the labor tax rate, and numerical experiments on a US-calibrated economy quantify the welfare changes. The key structural assumption enabling the results is monopolistic competition in banking plus a financial-market funding alternative for banks at the risk-free rate.
Why does the introduction of CBDC leave the deposit rate and bank lending unchanged in this model?
Lemma 1: under monopolistic competition each individual bank takes the aggregate deposit rate as given and does not internalize how aggregate deposit demand shifts with CBDC, so its optimal deposit rate (eq. 30) is invariant to CBDC’s interest rate or liquidity. CBDC lowers aggregate deposit demand, so banks simply rely more on other liabilities (bonds/equity). Lending is unaffected because the marginal cost of bank funding remains the risk-free rate (banks can borrow from the market), so the loan rate (eq. 32) and quantity of loans do not change. This contrasts with monopoly/Cournot banking (Andolfatto 2021; Chiu et al. 2022) where CBDC moves the deposit rate.
What are the three welfare channels and how is each maximized?
(1) Seigniorage: higher central-bank seigniorage finances lower distortionary labor taxes; maximized by setting rc to raise seigniorage revenue (peak occurs at rc < rb in the cases analyzed). (2) Opportunity cost of money: paying high interest on CBDC raises money holdings and cuts the transaction cost, stimulating labor and consumption; maximized by setting rc equal to the risk-free rate so households drop deposits entirely and drive the transaction cost toward zero. (3) Redistribution: CBDC lets non-bankers capture deposit rents previously held by bankers (via tax cuts or interest on CBDC), maximal when zeta=0 and rc near the risk-free rate. Channels (1) and (2) conflict, generating the optimal-rate tradeoff.
What does seigniorage look like as a function of the CBDC rate, and what do Propositions 1-2 say?
Seigniorage is non-monotonic in rc: a higher rc lowers seigniorage per unit of CBDC but raises CBDC demand. Proposition 1 (under alpha_b^{epsilon_cb}*epsilon_cb > 1 and negligible CBDC management cost): the seigniorage-maximizing rc exceeds the deposit rate rb; if epsilon_cb>1.5 the optimal rc decreases in CBDC liquidity alpha_c; and the peak seigniorage rises with both alpha_c and epsilon_cb. Proposition 2: within that parameter region, maximum seigniorage is achieved as epsilon_cb to infinity (perfect substitutes) with rc set infinitesimally above rb — i.e., outcompete deposits. In the numerical cases shown, the seigniorage peak occurs at rc < rb, moving closer to rb as CBDC liquidity rises.
What heterogeneity / cross-country variation does the paper document?
Two dimensions. (i) Labor tax level: US ~25% vs EU-14 averaging >40% (Trabandt-Uhlig 2011). Higher taxes raise the value of the seigniorage/tax-cut channel, lower the optimal CBDC rate, and raise welfare gains (efficiency gains +9 bps at 25% to +20 bps at 45%). (ii) Bank ownership (zeta): ‘case a’ (households own banks) gives small gains (7-8 bps at 20% tax to 18-20 bps at 45%); ‘case b’ (bankers own banks) gives large gains (52-53 bps at 20% to 58-60 bps at 45%) via redistribution. The optimal CBDC rate is higher in case b than case a and rises with the tax rate (Proposition 3 / Figure 3).
What robustness / alternative-parameter checks are run (Table 3)?
Frisch elasticity (gamma=0.25 i.e. Frisch=4, and gamma=4 i.e. Frisch=0.25): higher Frisch raises case-a gains (e.g., +28 bps at 25% tax) but case-b gains are roughly independent of Frisch. Interest semi-elasticity of money demand set to -0.12 (Benati et al. 2021 for Switzerland): with 45% taxes, gains reach +35 bps (case a) and +85 bps (case b) — this parameter has the biggest impact. Other variations with small effects: deposit/loan management costs, reserve ratio (0% vs 10%), bank-profit tax tau_b (15% vs 35%; lower tau_b means more inequality and larger CBDC gain), loan elasticity epsilon_l, working-capital share phi, wealth/consumption ratio (2 vs 4). Loan-side parameters and household wealth essentially do not matter because lending is unaffected by CBDC. With lump-sum (non-distortionary) taxes, case-a gains shrink (the seigniorage-tax channel is inactive) while case-b gains are essentially unchanged. At the zero lower bound the welfare gain is approximately linear in the deposit spread and zero when the spread (net of management cost) is zero.
How does this paper relate to and differ from the closest prior work?
Versus Barrdear and Kumhof (2021): shares the transaction-cost money-demand approach but estimates a much smaller welfare benefit; their large +3% GDP gain comes mainly from the central bank buying public debt and lowering the government bond rate — a channel absent here. Versus Brunnermeier-Niepelt (2019): they get equivalence (no effect) under specific funding conditions; here CBDC does affect outcomes through seigniorage, opportunity cost, and redistribution. Versus Andolfatto (2021, monopoly bank) and Chiu et al. (2022, Cournot): in those the CBDC rate moves the deposit rate, whereas monopolistic competition here insulates the deposit rate (Lemma 1). Versus Chiu-Davoodalhosseini (2021): the opportunity-cost channel is shared. The paper abstracts from cyclical issues (cf. Burlon et al. 2022 DSGE; Piazzesi et al. 2022 monetary-policy use of rc) by focusing on steady state.
What are the main caveats and scope conditions on the welfare results?
(1) Steady-state only — no transitional or cyclical analysis. (2) Requires historically normal interest rates; near the ZLB all three channels are inert. (3) Liquidity and substitutability are treated as fixed design constraints in the welfare optimization, with only rc as the policy lever, because they may be technologically hard to set. (4) The headline ~60 bps gain relies on the extreme ‘case b’ (zero-measure bankers own all banks) and on the welfare function ignoring bankers — i.e., it is largely a redistribution result, not a pure efficiency result. (5) The model deliberately shuts down CBDC effects on bank lending (banks fund at the risk-free rate), so disintermediation-of-credit channels stressed elsewhere are absent by construction. (6) Bank profits in the model equal net interest income (~1.5-2% of consumption), comparable to US bank NII but higher than actual bank profits.
Is cash incorporated, and does it change the conclusions?
The baseline model excludes cash, but an appendix adds cash as a third zero-interest money in a nested CES (cash and CBDC combine, then that composite substitutes for deposits). The paper shows that if the ‘composite interest’ of cash-plus-CBDC equals the rc of the two-instrument baseline, economic outcomes are unchanged: households rebalance across the three instruments so the equilibrium transaction cost and total cost of holding money are the same.