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### Research questions and model purpose
- Headline question: How would economic dynamics be affected when introducing a central bank digital currency (CBDC)?
- Five specific questions:
  1. What CBDC-in-total-money shares would emerge?
  2. To what extent would bank deposit rate-to-policy rate spreads fall?
  3. How much would bank reserve borrowing needs rise in lieu of falling deposit balances?
  4. How would bank and central bank interest income and expenses (and net seigniorage) be affected via volume and price effects?
  5. To what extent would monetary policy pass-through become stronger?
- Model purpose: provide quantitative, structural, agent-level counterfactuals for CBDC introduction where no historical precedent exists.

### Model architecture and theoretical foundations
- Agents: banks, a central bank, and the private sector (“non-bank” agents).
- Key theoretical elements combined:
  - Choice theory (utility maximization; Luce and Suppes 1965)
  - Game theory (Nash equilibria and stochastic games; Nash 1951; Shapley 1953)
  - Industrial organization (pass-through pricing and competition; Drechsler, Savov, and Schnabl 2017)
  - Reinforcement learning (Sutton and Barto 1998)
  - Stock-flow consistency (Godley and Lavoie 2006)
- Defining model features:
  1. Stock-flow consistent.
  2. Dynamic, agent-interaction over time with predefined sector-dependent actions.
  3. Reinforcement learning algorithm for banks to converge to a Nash equilibrium regarding rates, market shares, and money shares.
- CBDC modeled as a central bank liability; monetary aggregates: total money = non-bank deposits + cash + CBDC (reserves excluded).

### Sequence of agent interactions (five steps per time increment; focus on Steps 1–5)
- Step 2 — Non-bank choice of money holdings (Random Utility / Nested Logit)
  - Utility: u_{n,b} = α_b + i_b β + ε_{n,b}
  - Deterministic: v_b = α_b + i_b β
  - Interest rates: non-negative for banks, zero for cash, zero or positive for CBDC.
  - Nested logit structure: P_b = P_{b|m} · P_m with Ξ_m and nesting parameters μ_m (μ_m ≥ 1; μ_m = 1 → conditional logit).
  - Behavioral extremes:
    - CBDC identical to cash (cash-like): crowds out previous cash share substantially when CBDC interest is positive.
    - CBDC like deposits (deposit-like): subtracts from deposit shares, increasing total central bank money share.
  - Money-holding moves generate reserve flows and bank reserve asset/borrowing needs.
- Step 3 — Economic activity (spending process)
  - Each non-bank picks one spending partner from N−1 and spends fraction γ of its money holding toward that partner; net spending flows create structural reserve needs.
- Step 4 — Interest and dividend payments (seven components)
  - Loan interest, deposit interest, CBDC interest (if i_{CBDC} > 0), interest on reserve holdings, interest on reserve borrowing, central bank profit distributions (seigniorage), bank dividends.
  - Central bank distributing seigniorage to non-banks expands bank balance sheets via credited reserves and deposits.
- Steps 1 and 5 — Banks set deposit rates and learn Nash equilibrium via Thompson Sampling-based reinforcement learning
  - Banks choose i_{b,t}^{d} to maximize pre-dividend profits π_{b,t} (referenced as equation (6)).
  - Profit drivers: lending rate i_L (set to i_{rb} in base model), reserve remuneration i_{rr} and i_{er}, deposit rate i_{b,t}^{d}, reserve borrowing rate i_{rb}.
  - Deposit rate grid on [0, i_{rb}] with grid length G; discrete up/down moves.
  - Two action success probabilities θ_k for k ∈ {up, down} follow beta(η_k, κ_k); η_k and κ_k initialized to 1.
  - Reward r_t = 1 if π_{t} > π_{t−1}, else 0.
  - Pseudo code (Thompson Sampling):
    - sample θ̂_up ~ beta(η_up, κ_up)
    - sample θ̂_down ~ beta(η_down, κ_down)
    - x_t ← argmax_k θ̂_k; apply x_t
    - observe reward r_t ∈ {1,0}; update (η_k, κ_k) ← (η_k + r_t, κ_k + 1 − r_t)
  - Environment ranges: monopoly (B = 1), duopoly (B = 2), oligopoly, up to many banks; banks are noncooperative.

### Stock-flow consistent grounding: balances and flows (exact account items)
- Assets/liabilities items: Loans (−L / +L), Deposits (+D / −D), Cash (+C / −C), CBDC (+DC / −DC), Reserve holding (+R / −R), Reserve borrowing (−B / +B).
- Financial flows: loan interest (−IL / +IL), deposit interest (+ID / −ID), CBDC interest (+IDC / −IDC), interest on reserve holdings (+IRH / −IRH), interest for reserve borrowing (−IRB / +IRB), dividends (+DIVB/−DIVB and +DIVCB/−DIVCB).
- Reserve acquisition channels: (1) borrowing from the central bank (“pulling”), (2) central bank open market operations (“pushing”), (3) attracting deposit liabilities from other banks (redistributive), (4) interbank borrowing (redistributive). Model currently reflects options (1) and (3); open market operations and an interbank money market are not included yet.

### Key formal expressions preserved
- u_{n,b} = α_b + i_b β + ε_{n,b}  (equation (1) structure)
- P_b = P_{b|m} · P_m  (equation (2))
- Conditional and nest probabilities: expressions in equations (3)–(5)
- Profit definition: equation (6)
- Beta density for action success probabilities: q(θ_k) (equation (7))
- Reward rule: r_t = 1 if π_{t} > π_{t−1}, else 0 (equation (8))
- Loan-transfer probability δ ∈ [0,1]; initialization η_k = 1, κ_k = 1.

### Calibration and required inputs
- Five required inputs: total money stock, currency in circulation, bank deposit rates, policy rates (reserve borrowing and remuneration rates, reserve requirements), nominal GDP.
- Some parameters aligned with empirical counterparts; others estimated to match cash-in-total-money ratios and deposit-to-policy rate spreads.
- Level of competition calibrated between monopoly and maximal competition.

### Annex III — Model parameters and calibration exact values
- Number of banks (B)
  - U.S.: 8
  - Euro area: 12
- Base utility of deposits, cash, CBDC (α)
  - Normalized to zero for all b = 1, ... , B. Estimated value in calibration: 0.52
- Price sensitivity (β)
  - U.S.: 50
  - Euro area: 150
- Velocity of money (γ)
  - U.S.: 110% (rounded, as of 2022Q2)
  - Euro area: 85% (rounded, as of 2022Q2)
- Loan transfer probability (δ)
  - U.S.: 0.75
  - Euro area: 0.75
- Reserve requirement (λ)
  - U.S./FED: 0%
  - Euro area/ECB: 1%
- Reserve borrowing rate (i_{rb})
  - U.S.: 0.035
  - Euro area: 0.02
- Interest on reserve holdings (i_{rr}, i_{er})
  - U.S.: [-, 0.035]
  - Euro area: [0.015, 0.01]
- Interest on CBDC (i_{CBDC})
  - Counterfactuals: CBDC interest rate ranges between 0 and 100 percent of the policy rate (negative rates can be considered).
- Money stock scaling (interpretability only)
  - U.S. money stock set to USD 22 trillion (rounded, end-2022Q2)
  - Euro area money stock set to EUR 15 trillion (rounded, end-2022Q2)

### Calibration procedure (Steps A, B, C)
- Step A: Estimate β via conditional logit on aggregate cash ratios and deposit rates. Point estimates: U.S. = 50; euro area = 150 (rounded).
- Step B: Calibrate γ, δ, and reserve-related parameters using GDP/M2 and recent reference values:
  - Velocity aligned to annual nominal GDP/M2 (U.S. 110%; euro area 85%).
  - δ set judgmentally to 0.75.
  - Reserve requirement as of September 2022: U.S. 0%, euro area 1%.
  - Reserve rates: U.S. i_{rb} = 0.035; i_{er} = 0.035. Euro area: [i_{rb}, i_{rr}, i_{er}] = [0.02, 0.015, 0.01].
- Step C: Estimate B and α_cash to match target cash ratio (rounded 10% for both U.S. and euro area, end-2022Q2) and deposit rate targets (1% U.S., 1.2% euro area). Money stock scaling does not affect Nash outcomes.

### Simulation design and runtimes (counterfactuals)
- Each simulation: 2,000 iterations.
- Burn-in: 500 iterations discarded.
- Analysis window: remaining 1,500 iterations per counterfactual.
- Counterfactual settings:
  1. Baseline: base utility of CBDC at −Inf (no CBDC demand).
  2. CBDC “Cash-Like”: α_CBDC = α_cash; nesting parameter μ→∞ (numerically μ = 100); CBDC interest range 0–100% of policy rate.
  3. CBDC “Deposit-Like”: α_CBDC = α_banks = 0; μ = 1; CBDC interest range 0–100% of policy rate.

### Key quantitative findings from CBDC counterfactuals (exact reported ranges and magnitudes)
- CBDC adoption (CBDC share of total money)
  - U.S.: estimated CBDC adoption ranges between 5-25 percent.
  - Euro area: estimated CBDC adoption ranges between 1-20 percent.
  - Lower/upper bounds correspond to “cash-like” and “deposit-like” extremes.
- Bank deposit rate spreads to policy rates
  - U.S.: average bank deposit rate would rise by up to a marginal 20 bps when introducing a CBDC.
  - Euro area: effect not materially different from zero.
  - Sensitivity of deposit-rate response rises when: price sensitivity is smaller; base utility for cash is lower; number of banks is smaller; nesting approaches “no nesting.”
- Reserve borrowing and deposit stock changes
  - U.S.: reserve borrowing needs increase by between USD 1.1-4.8 trillion due to non-bank CBDC adoption.
  - Euro area: deposit balances fall between EUR 0.2-2.8 trillion.
  - Reserve borrowing needs grow with CBDC interest rate and when CBDC crowds out deposits; central bank balance sheet expands accordingly.
- Bank and central bank interest income/expense and net seigniorage; private sector net financial income
  - Banks: volume effects dominate in U.S. and euro area counterfactuals → deposit expenses fall; reserve borrowing expenses rise and can outweigh deposit expense declines → bank net income and dividend distributions fall.
    - Upper bound estimates:
      - U.S. bank dividend income falls by about USD 141 billion (−30%) per annum.
      - Euro area bank dividend income falls by about EUR 22 billion (−20%) per annum.
    - U.S. banking system RoA shift at upper bound corresponds to about 70 basis points (reference RoA ≈ 1.25 percent by mid-2022). RoA remains positive in upper bound estimates.
  - Central banks: interest income from reserve lending rises, interest expense for surplus reserve balances falls, CBDC interest expense rises when paid.
    - Central bank net income is nonlinear in i_{CBDC}: first rises when i_{CBDC} is zero or small, then falls as i_{CBDC} increases.
  - Non-bank private sector net financial income remains approximately unchanged: deposit interest income falls; bank dividend income drops; CBDC interest income increases; seigniorage redistribution scales down — net effect approximately zero.
- Surplus reserves and timing
  - Surplus reserve example: U.S. surplus reserve position at end-2022Q2 amounted to USD 2.1 trillion.
  - CBDC may draw down surplus reserves before implying outright reserve borrowing; draw-down reduces reserve interest income for banks (remunerated roughly at the Fed funds rate in U.S. example).
- Monetary policy pass-through (estimates assume i_{CBDC} = policy rate; upper bound cases)
  - U.S.:
    - Baseline: 0.55
    - CBDC-cash-like: 0.85
    - CBDC-deposit-like: beyond 0.95
  - Euro area:
    - Baseline: 0.86
    - CBDC-cash-like: 0.89
    - CBDC-deposit-like: 0.97

### Scenario sensitivities and caveats (exact parameter sensitivities and uncertainties)
- Behavioral assumption (cash-like vs. deposit-like) crucial for quantitative outcomes.
- Price sensitivity β has sizable estimation uncertainty, particularly for euro area; influences deposit-rate responses materially.
- Loan transfer probability δ set to 0.75 judgmentally; shown to have no notable impact on CBDC counterfactuals presented.
- Model does not endogenize lending rates; effects on credit supply are not inferred.
- Recommendation: obtain price sensitivity estimates with maximal precision before applying model to other countries; possible refinements: micro account-level data, randomized trials.

### Discussion and policy-relevant extensions (listed options)
- Suggested model extensions (eight examples):
  - Make bank reserve borrowing collateralized to address collateral scarcity and central bank collateral eligibility needs.
  - Introduce a sovereign agent and sovereign bond markets to model open market operations and account for sizeable surplus reserves.
  - Introduce heterogeneous bank market power to align model banks with specific banks.
  - Refine bank liability and asset structures to analyze behavioral responses (e.g., incentivizing term deposits) and introduce wholesale funding endogenously.
  - Endogenize bank lending rates to complete pass-through analysis and assess impact on bank lending (would create a “two-instruments-one-target” problem requiring learning algorithm augmentation).
  - Model caps on CBDC flows and/or balances to assess policy options constraining CBDC demand.
  - Introduce multiple currency use and dollarization to extend money share buckets.
  - Integrate with larger-scale macroeconomic models to endogenize CBDC take-up and policy rates.
- Terminology recommendation: do not label falling deposit balances and rising central bank reserve borrowing driven by CBDC demand as “disintermediation.” Rationale: banks are modeled as money creators; with positive CBDC demand banks continue creating money liabilities via lending and repayments.

### Main conclusions (summary)
- CBDC-in-total-money shares depend on whether CBDC is perceived as “cash-like” or “deposit-like.”
  - Estimated ranges: U.S. 5-25 percent; euro area 1-20 percent (lower/upper bounds correspond to behavioral extremes).
- Deposit-policy rate spreads fall and monetary policy pass-through may strengthen (U.S. modest deposit-rate rise up to 20 bps; euro area negligible).
- Bank profitability declines via volume effects and higher reserve borrowing expenses; central bank seigniorage is nonlinear in i_{CBDC}.
- Net effect on private sector financial income is approximately zero due to offsetting changes (deposit income, dividends, CBDC interest, seigniorage).
- Results are conditional on calibration choices and behavioral assumptions; model codes are made available with the paper for replication and application to other jurisdictions.

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### 1.    Introduction   ...................................................................................................

### 1. Introduction

### Research questions and model purpose
- Headline question: How would economic dynamics be affected when introducing a central bank digital currency (CBDC)?
- Five specific questions the paper addresses:
  1. What CBDC-in-total-money shares would emerge?
  2. To what extent would bank deposit rate-to-policy rate spreads fall?
  3. How much would bank reserve borrowing needs rise in lieu of falling deposit balances?
  4. How would bank and central bank interest income and expenses (and net seigniorage) be affected via volume and price effects?
  5. To what extent would monetary policy pass-through become stronger?
- Model purpose: provide quantitative, structural, agent-level counterfactuals for CBDC introduction where no historical precedent exists.

### Model architecture and theoretical foundations
- Agents: banks, a central bank, and the private sector (“non-bank” agents).
- Key theoretical elements combined in the model:
  - Choice theory (utility maximization; Luce and Suppes 1965)
  - Game theory (Nash equilibria and stochastic games; Nash 1951; Shapley 1953)
  - Industrial organization (pass-through pricing and competition; Drechsler, Savov, and Schnabl 2017)
  - Reinforcement learning (Sutton and Barto 1998)
  - Stock-flow consistency (Godley and Lavoie 2006)
- Agent-level setup chosen to let the need for reserves arise structurally and to capture settlement across banks and between cash and CBDC.

### Calibration and data inputs
- The model is parsimoniously parameterized and strongly rooted in macro-financial data.
- Five required inputs for calibration and estimation purposes:
  1. the total money stock in an economy,
  2. its subset of currency in circulation,
  3. bank deposit rates,
  4. policy rates,
  5. nominal GDP.
- Some model parameters aligned directly with empirical counterparts; others estimated to match cash-in-total-money ratios and deposit-to-policy rate spreads to capture country-specific initial conditions (competition and preference for central bank money pre-CBDC).
- Level of competition is calibrated and can lie anywhere between monopoly and maximal competition.

### Primary conclusions (model and simulation-based)
- Conclusion (1): CBDC-in-total-money shares depend on behavioral assumptions about whether CBDC is perceived as more “cash-like” or “deposit-like.”
  - Estimated ranges: between 5-25 percent for the U.S. and 1-20 percent for the euro area. These represent likely lower and upper bounds.
  - Upper bounds correspond to CBDC perceived as “deposit-like” and CBDC interest paid at the policy rate.
  - Moving outside these ranges is possible if CBDC design implies notably lower or higher “base utility” beyond interest relative to cash or deposits.
- Conclusion (2): Bank deposit-policy rate spreads fall and monetary policy pass-through may strengthen.
  - Deposit rates rise, though only marginally for the U.S. and euro area calibrations (reasons discussed in Section 5).
  - Monetary policy pass-through estimates (likely upper bounds):
    - U.S.: rises from an estimated 0.55 under the no-CBDC baseline to beyond 0.9 when assuming deposit-like CBDC with interest at the policy rate.
    - Euro area: rises from a baseline at 0.85 to beyond 0.95.
- Conclusion (3): Bank profitability declines.
  - Deposit expenses are influenced by two opposing forces: deposit rates rise (assuming policy rates do not drop) while deposit balances fall.
  - For the U.S. and euro area counterfactuals, volume effects dominate and make deposit expenses fall.
  - Bank reserve borrowing expenses rise (assuming policy rates do not drop) and can outweigh declining deposit expense, making bank net income and dividend distribution fall on net.
  - For other countries, the balance between these effects might differ.
- Conclusion (4): Central bank seigniorage is a nonlinear function of CBDC interest rates.
  - With zero or low CBDC interest rates, seigniorage first rises due to higher interest income from reserve lending.
  - As CBDC interest rates continue to rise, CBDC interest expense starts dominating and seigniorage falls again.
  - The model does not answer whether interest-bearing CBDC would be net beneficial; it provides quantitative inputs for such design choices.

### Structure and availability
- Paper structure (sections): Literature (Section 2), Model (Section 3), Estimation method (Section 4), Simulation results (Section 5), Conclusions (Section 6).
- Model codes are made available along with the paper to allow replication and application to other countries.

### Literature and methodological context
- Historical emergence of CBDC debate: central bankers’ speeches between 2015-17 (examples cited).
- Six dimensions motivating CBDC in central bank reports: (1) strengthening competition, (2) expanding payment system resilience, (3) strengthening monetary policy pass-through, (4) fostering financial inclusion, (5) combating illicit use, (6) maintaining monetary sovereignty.
- Prior model classes evaluating CBDC: New Monetarist (NM), Overlapping Generations (OLG), DSGE, and other models; limitations noted regarding treatment of banks as intermediaries versus money creators.
- Distinctive methodological features of this paper:
  - Treats banks as money creators.
  - Embeds a Random Utility Model (RUM) structure for non-banks’ choice of money holdings following McFadden (1973/78/81).
  - Uses a reinforcement learning method (Thompson Sampling) for bank agents to collectively find a Nash equilibrium for deposit rates; the developed algorithm is an additional methodological contribution.

### The model’s defining features (summary)
- Three defining features:
  1. Stock-flow consistent.
  2. Dynamic, agent-interaction over time with predefined sector-dependent actions.
  3. Involves a reinforcement learning algorithm for bank agents to converge to a Nash equilibrium regarding rates, bank deposit market shares, and deposit- vs. central bank money shares.
- Time steps in the model have no calendar interpretation; only equilibria are of interest.

### Stock-flow consistent grounding: agents, balances, and flows
- Agent types: non-banks (1, ... , N), banks (1, ... , B), and a central bank.
- Integrated balance sheets: each agent’s asset is another’s liability; sectoral stocks refer to sums of individual agents (e.g., loans −L for non-banks is sum of N non-bank agents’ loans).
- Financial flows: every agent’s financial expense is another’s income; all financial flows sum to zero at the system level.
- Key account items and identities (as presented in Tables):
  - Assets/liabilities across sectors include Loans (−L / +L), Deposits (+D / −D), Cash (+C / −C), CBDC (+DC / −DC), Reserve holding (+R / −R), Reserve borrowing (−B / +B).
  - Financial flows include loan interest (−IL / +IL), deposit interest (+ID / −ID), CBDC interest (+IDC / −IDC), interest on reserve holdings (+IRH / −IRH), interest for reserve borrowing (−IRB / +IRB), dividends (+DIVB/−DIVB and +DIVCB/−DIVCB).
  - System-wide sums of financial net worth changes sum to zero.
- CBDC definition in the model: broad-based money for all non-bank agents including retail CBDC and wholesale CBDC; CBDC is modeled as a central bank liability.
- Monetary aggregates: total money = non-bank deposits + cash + CBDC; banks’ reserve assets are not money and are excluded from aggregates such as M2.
- Role of reserves:
  - Reserves reside in a segregated payment system layer and cannot be lent out to the private sector.
  - Banks obtain reserve assets through four channels, two of which change total net reserves:
    1. Borrowing from the central bank (“pulling”).
    2. Central bank “pushing” via open market operations (bond purchases).
    3. Attracting deposit liabilities from other banks (redistributive).
    4. Borrowing in the interbank money market (redistributive).
  - Only options (1) and (2) change total net reserves. The model reflects options (1) and (3); open market operations and a money market are not included yet.
  - Banks seek deposit liabilities because they bring reserve assets to the attracting bank; obtaining reserves via attracting deposits is cheaper than borrowing from the central bank. The deposit-policy rate spread reflects the cost gap and is an endogenous function of competition in the model.

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### 3.2  Sequence of Agent (Inter-) Actions in the Model

### 3.2  Sequence of Agent (Inter-) Actions in the Model

### Overview — Five Steps per Time Increment
- The model entails five steps per time increment (Figure 1). Steps 2–4 are described first; Steps 1 and 5 together implement the learning mechanism for banks.
- Net money-holding migration of non-banks between bank deposit accounts/cash/CBDC, combined with a net reserve settlement system, drives system dynamics and aligns the model with empirical data.

### Step 2 — Non-Bank Agents’ Choice of How to Hold Money (Random Utility / Nested Logit)
- Non-bank agent n’s utility for option b (out of 퐵+2 options: B banks, cash, CBDC) is:
  - u_{n,b} = α_b + i_b β + ε_{n,b}  (equation (1) structure in text)
- Deterministic component v_b = base utility α_b + interest income i_b times price sensitivity β.
  - Interest rates i_b: non-negative for banks, zero for cash, zero or positive for CBDC.
  - Base utility α_b (“market power”) is exogenous and may be heterogeneous across banks and different for cash and CBDC.
- Model uses a nested logit structure to address IIA and reflect behavioral assumptions about substitution patterns between cash, CBDC, and deposits.
  - Marginal choice probability: P_b = P_{b|m} · P_m  (equation (2))
  - Within-nest conditional probability: P_{b|m} given by expression in equation (3).
  - Nest choice probability: P_m given by equation (4), with Ξ_m defined in equation (5).
  - Parameters μ_m must be ≥ 1; μ_m = 1 collapses the model to conditional logit (Decision Layer 1 disappears).
- Behavioral implications for CBDC introduction:
  - One behavioral end: CBDC identical to cash (equal base utility, no interest) crowds out previous cash share by half or more when CBDC interest is positive.
  - Other end: CBDC subtracts from deposit shares, increasing total central bank money share.
  - The model considers different assumptions across this behavioral spectrum for CBDC counterfactuals.
- When a non-bank’s money-holding choice changes between periods (b̂_{n,t} ≠ b̂_{n,t-1}), balance-sheet transactions occur:
  - Transfers between banks create reserve flows.
  - Moves between bank deposits and cash/CBDC generate reserve asset/borrowing needs for banks.

### Step 3 — Economic Activity (Spending Process)
- “Economic activity” = spending flows between randomly drawn pairs of non-bank agents.
  - Each non-bank picks one spending partner from the other N−1 non-bank agents and spends fraction γ of its money holding toward that partner.
  - Net spending flows per non-bank can be positive or negative.
  - The random partner choice implies a stationary distribution of individual money holdings over time.
- Spending flows represent wages and household spending; the rudimentary spending process creates a structural need for reserves and dependence on the central bank policy rate and Nash-equilibrating deposit rates.

### Step 4 — Interest and Dividend Payments (Seven Components)
- Interest/dividend flows include:
  - Banks settling deposit and loan interest with non-banks.
  - Reserve holding and reserve borrowing interest flows between banks and the central bank.
  - Central bank paying interest on CBDC if i_{CBDC} > 0.
  - Central bank distributing positive net profit (seigniorage) directly to non-banks.
  - Banks distributing net profit as dividends to non-banks.
- For cash holders, bank reserves are unaffected; only central bank liabilities (cash) expand. Central bank net worth stays at nil.
- Resulting balance-sheet effects: when central bank distributes seigniorage to non-banks, bank balance sheets expand through credited reserve accounts and non-bank deposit accounts.

### Steps 1 and 5 — Banks Set Deposit Rates and Learn Nash Equilibrium via Reinforcement Learning
- Banks set bank-specific endogenous deposit rate i_{b,t}^{d} in Step 1 to maximize pre-dividend profits π_{b,t} (measured in Step 5).
- Pre-dividend profit π_{b,t} defined in equation (6) uses five interest rates:
  - lending rate i_L (in base model set to reserve borrowing rate i_{rb});
  - remuneration rates on required and excess reserves i_{rr} and i_{er};
  - bank-specific endogenous deposit rate i_{b,t}^{d};
  - reserve borrowing rate i_{rb}.
- Interest calculations use within-period averages (bars) computed from balances just before Step 3 spending and interest calculation steps in Step 4.
- A model-free reinforcement learning algorithm (Thompson Sampling-based) lets banks learn Nash-equilibrating deposit rates and implied market shares dynamically.
  - Discrete grid for deposit rates defined on [0, i_{rb}] with grid length G; “up/down” refers to 1-step grid moves.
  - Two action success probabilities θ_k for k ∈ {up, down} follow beta distributions with shape parameters (η_k, κ_k).
  - Beta density q(θ_k) given by the beta formula in equation (7). Shape parameters initialized to 1 (uniform).
  - Explorer bank chosen each iteration either increases or decreases deposit rate probabilistically (Step 1).
  - Reward r_t = 1 if net income rose relative to previous period, else 0 (equation (8)).
  - Beta parameters updated: (η_k, κ_k) ← (η_k + r_t, κ_k + 1 − r_t).
- Pseudo code (Thompson Sampling logic) summary (Table 3):
  - for t = 1,2,...
    - sample θ̂_up ~ beta(η_up, κ_up)
    - sample θ̂_down ~ beta(η_down, κ_down)
    - x_t ← argmax_k (θ̂_k) and apply x_t
    - observe reward r_t ∈ {1,0}
    - (η_k, κ_k) ← (η_k + r_t, κ_k + 1 − r_t)
- Environment ranges: monopoly (B = 1), duopoly (B = 2), oligopoly, up to many banks. Banks are noncooperative (no collusion) when B ≥ 1.
- The implementation assumes identical market power across banks and uses one pair of beta distributions; algorithm can be extended for bank-specific Nash rates with heterogeneous market power.
- Annex I demonstrates the algorithm successfully identifies Nash equilibrium; benchmark Nash can be computed analytically in simple environments.

### Parameter Dependencies and Model Calibration Notes
- Model requires five inputs: total money stock, currency in circulation, bank deposit rates, policy-related parameters (reserve borrowing and remuneration rates, reserve requirements), and nominal GDP.
- Some additional macro time series (e.g., inflation) may be needed to estimate private sector price sensitivity.
- The model comprises nine parameters (summarized in Table 4 in the source).
- Comparative statics (stated results):
  - When the number of banks B or the price sensitivity β rises (separately, ceteris paribus), then Nash equilibrium deposit rates rise and Nash cash ratios fall.
  - When the base utility of cash α_cash rises, ceteris paribus, the opposite occurs (Nash deposit rates fall and Nash cash ratios rise).
- Other dependencies (velocity of money, loan transfer probability δ, reserve remuneration rates, reserve requirement) are presented elsewhere in the source (Annex I and subsequent sections).

### Key Model Elements and Exact Numeric/Formal Expressions Preserved from Source
- Utility decomposition: u_{n,b} = α_b + i_b β + ε_{n,b}  (equation (1) structure)
- Marginal probability decomposition: P_b = P_{b|m} · P_m  (equation (2))
- Conditional probability within nest: P_{b|m} given by expression in equation (3).
- Nest probability and inclusive value: P_m given by equation (4); Ξ_m (inclusive value) defined by equation (5).
- Profit definition referenced as equation (6).
- Beta density for action success probabilities: q(θ_k) formula given in equation (7).
- Reward rule: r_t = 1 if π_{t} > π_{t−1}, else 0 (equation (8)).
- Grid for deposit rates on [0, i_{rb}] with grid length G.
- Loan-transfer follow probability between deposit and loan relationships: δ ∈ [0,1].
- Initialization of beta shape parameters: η_k = 1, κ_k = 1 (uniform prior).

*Source: Excerpt from "3.2 Sequence of Agent (Inter-) Actions in the Model" (wpiea2023009-print-pdf).*

### Annex III.

### Annex III. Table 4. Model Parameters

### Model parameterization (selected parameters and values)
- Number of banks (퐵)
  - U.S.: 8
  - Euro area: 12
- Base utility of deposits, cash, CBDC (훼)
  - Normalized to zero for all 푏ൌ1, ... ,퐵 banks. 훼௖௔௦௛ used in conjunction with 퐵 to estimate the model to match empirical deposit rate markdowns and cash ratios.
  - Estimated value in calibration: 0.52
- Price sensitivity (훽)
  - Non-bank propensity to switch banks/cash/CBDC in response to changing interest rates (deposit rates and CBDC rates).
  - U.S.: 50
  - Euro area: 150
- Velocity of money (훾)
  - Non-bank share of beginning-of-period money holdings that they spend toward other non-banks during the period.
  - U.S.: 110% (rounded, as of 2022Q2)
  - Euro area: 85% (rounded, as of 2022Q2)
- Loan transfer probability (훿)
  - Probability from [0,1] for a non-bank’s deposit account move to imply loan transfer.
  - U.S.: 0.75
  - Euro area: 0.75
- Reserve requirement (휆)
  - Required reserve ratio, required reserves over deposits.
  - U.S./FED: 0%
  - Euro area/ECB: 1%
- Reserve borrowing rate (푖ோ஻)
  - Rate that banks pay for period average reserve borrowing (RB) balances from the central bank.
  - U.S.: 0.035
  - Euro area: 0.02
- Interest on reserve holdings (푖ோோு, 푖ாோு)
  - Interest on required and excess reserves, respectively.
  - U.S.: [-, 0.035]
  - Euro area: [0.015, 0.01]
- Interest on CBDC (푖஽஼)
  - Rate that the central bank pays on non-bank period average holdings of CBDC.
  - In counterfactuals CBDC interest rate ranges between 0 and 100 percent of the policy rate (negative interest rates can be considered).

- Money stock scaling (for interpretability only)
  - U.S. money stock set to USD 22 trillion (rounded, end-2022Q2)
  - Euro area money stock set to EUR 15 trillion (rounded, end-2022Q2)

### Calibration procedure (Steps A, B, C)
- Step A: Estimate non-banks’ price sensitivity (휷)
  - Method: conditional logit model separate from the ABM using economy-aggregate historical cash ratios and aggregate deposit rates.
  - Point estimates: U.S. = 50; euro area = 150.
  - Noted estimation uncertainty, particularly for the euro area; suggested refinements include using individual bank deposit rates, micro account-level data, or country-level euro area estimation.
- Step B: Calibrate velocity (휸), loan transfer probability (휹), and reserve-related parameters (흀, 풊푹푩, 풊푹푹푯, 풊푬푹푯)
  - Velocity: aligned with annual nominal GDP over M2 → U.S. 110%, euro area 85% (as of 2022Q2).
  - Loan transfer probability 훿 set judgmentally to 0.75 for both U.S. and euro area.
  - Reserve requirement set to 0% (U.S.) and 1% (euro area) as of September 2022.
  - Reserve borrowing and remuneration rates set to recent historical reference values:
    - U.S.: 푖ோ஻ = 0.035; excess reserve remuneration 푖ாோு = 0.035 (rate on required reserves irrelevant due to 휆 = 0%).
    - Euro area: [푖ோ஻, 푖ோோு, 푖ாோு] = [0.02, 0.015, 0.01].
- Step C: Estimate number of banks (퐵) and base utility of cash (휶cash)
  - Targets: match observed cash ratio and deposit rate.
  - Cash ratio targets: rounded 10% for both U.S. and euro area (end-2022Q2).
  - Deposit rate targets: set consistent with central bank rate assumptions — 1% (U.S.) and 1.2% (euro area).
  - Money stock scaling set to observed M2 for interpretability (USD 22 trillion; EUR 15 trillion). This scaling does not affect Nash outcomes.

### Counterfactual simulation design
- Simulation runtime and equilibrium assessment
  - Each simulation: 2,000 iterations.
  - Burn-in period: 500 iterations discarded.
  - Analysis window: remaining 1,500 iterations per counterfactual.
- Counterfactual settings
  1. Baseline
     - Model estimated as above with base utility of CBDC at −퐼푛푓 (no CBDC demand).
  2. CBDC “Cash-Like”
     - Base utility for CBDC equated to that of cash (훼஼஻஽஼ = 훼௖௔௦௛).
     - Nesting parameter set to a high value to mimic “full nesting” (휇→∞). Numerically, 휇 = 100 was used.
     - CBDC interest rate allowed to range between 0 and 100 percent of the policy rate.
  3. CBDC “Deposit-Like”
     - CBDC base utility set to that of banks (훼஼஻஽஼ = 훼௕௔௡௞௦ = 0).
     - Nesting parameter set at 휇 = 1 (no nesting), implying CBDC is perceived as “deposit-like.”
     - CBDC interest rate allowed to range between 0 and 100 percent of the policy rate.

### Key quantitative findings from CBDC counterfactuals
- CBDC adoption (CBDC share of total money)
  - U.S.: estimated CBDC adoption ranges between 5-25 percent depending on behavioral assumptions and CBDC interest rate.
  - Euro area: estimated CBDC adoption ranges between 1-20 percent.
  - Lower/upper bounds correspond to “cash-like” and “deposit-like” extremes.
  - High CBDC interest rates crowd out cash almost entirely when CBDC is cash-like; when deposit-like, CBDC crowds out both deposits and cash.
- Bank deposit rate spreads to policy rates
  - U.S.: average bank deposit rate would rise by up to a marginal 20 bps when introducing a CBDC.
  - Euro area: effect not materially different from zero.
  - Sensitivity of deposit-rate response rises when: price sensitivity is smaller; base utility for cash is lower; number of banks is smaller; and nesting approaches “no nesting.”
- Reserve borrowing and deposit stock changes
  - U.S.: non-banks adopting CBDC increases bank reserve borrowing needs by between USD 1.1-4.8 trillion (to counterbalance falling deposit balances).
  - Euro area: deposit balances would fall between EUR 0.2-2.8 trillion.
  - Reserve borrowing needs grow with the CBDC interest rate and when CBDC crowds out deposits.
  - Large expansion of the central bank balance sheet is a by-product of increased reserve borrowing.
- Bank and central bank interest income/expense (net seigniorage) and private sector net financial income
  - Banks: combined effect of marginally higher deposit rates and lower deposit volumes is negative for deposit expenses; banks’ net income and dividend payouts may fall.
    - Upper bound estimates: U.S. bank dividend income falls by about USD 141 billion (−30%) per annum; euro area falls by about EUR 22 billion (−20%) per annum.
    - U.S. banking system RoA shift at upper bound corresponds to about 70 basis points (reference RoA ≈ 1.25 percent by mid-2022).
    - RoA remains positive according to upper bound estimates, implying no outright potential for capital ratios to fall.
  - Central banks: interest income from reserve lending rises, interest expense for bank surplus reserve balances falls, and interest expense for CBDC rises once CBDC interest is paid.
    - Central bank net income shows nonlinearity: it first rises when CBDC interest rate is zero or small, then falls as CBDC interest expense dominates at higher CBDC interest rates.
  - Non-bank private sector net financial income remains unaffected across scenarios:
    - Non-banks’ interest income from deposits falls; bank dividend income drops; interest income from CBDC increases; seigniorage payment flows from central bank decline in scale — net effect approximately zero.
- Considerations on surplus reserves and timing
  - At current juncture some banking systems (e.g., U.S.) have sizeable surplus reserves (surplus reserve position of the U.S. banking system at end-2022Q2 amounted to USD 2.1 trillion).
  - A CBDC introduction may draw down surplus reserves before implying outright additional reserve borrowing; draw-down implies less reserve interest income for banks (remunerated at about the Fed funds rate in the U.S.).
- Monetary policy pass-through
  - Pass-through estimated by moving the policy rate (reserve borrowing rate) by 1 percentage point and regressing Nash deposit rates on the policy rate, using a nonparametric residual bootstrap for coefficient distributions.
  - Estimates assume CBDC interest rate equals the policy rate (upper bound cases).
  - U.S. pass-through:
    - Baseline: 0.55
    - CBDC-cash-like: 0.85
    - CBDC-deposit-like: beyond 0.95
  - Euro area pass-through:
    - Baseline: 0.86
    - CBDC-cash-like: 0.89
    - CBDC-deposit-like: 0.97
  - Interpretation: pass-through increases under CBDC (upper bound estimates), with the euro area baseline pass-through exceeding the U.S. baseline consistent with larger initial competition and a larger estimated number of banks in the euro area.

### Scenario sensitivities and caveats highlighted
- The behavioral assumption on how non-banks perceive CBDC (cash-like vs. deposit-like) is crucial for all quantitative outcomes.
- The price sensitivity 휷 estimate has sizable uncertainty, particularly for the euro area, and influences equilibrium deposit-rate responses materially.
- The loan transfer probability 훿 is judgmentally set to 0.75; it affects Nash equilibrium outcomes but has no notable impact on the CBDC counterfactuals presented.
- The model does not endogenize lending rates; thus, effects on credit supply are not inferred. Further work is suggested to endogenize lending rates and refine 휷 estimation using micro data or randomized trials.

*Source: Annex III, wpiea2023009-print-pdf — Annex III.*

### 6.         Discussion and Conclusions

### 6.         Discussion and Conclusions

### Model design and purpose
- The model simulates integrated balance sheets of banks, non-bank agents, and the central bank, combining elements of choice theory, game theory, industrial organization, reinforcement learning, and stock-flow consistency.
- A subset of parameters is aligned with empirical counterparts; another subset is estimated to match endogenous cash ratios and deposit rate–policy rate spreads to empirical targets.
- This calibration captures initial competition in a banking system and people’s pre-CBDC preference for (utility derived from) central bank money.
- Counterfactuals (e.g., introduction of a CBDC) are conducted by perturbing these calibrated/estimated settings.

### Analytical variant versus agent-based model (ABM)
- An analytical model variant shows close quantitative correspondence with the simulation-based ABM for several questions, and it was useful to:
  - Prove the learning algorithm identifies Nash equilibria.
  - Confirm correspondence of counterfactual results for the first two of the five research questions.
- The analytical variant cannot address questions 3–5; the ABM is required to analyze:
  - Bank and central bank profits and their drivers.
  - Reserve dynamics.
  - Monetary policy pass-through.

### CBDC counterfactuals: take-up and substitution
- Under the assumption that interest on CBDC is equated to the central bank’s reserve borrowing rate and people perceive CBDC as close substitutes to bank deposits:
  - CBDC-in-total-money shares may approach 25% for the U.S. (upper bound).
  - CBDC-in-total-money shares may approach 20% for the euro area (upper bound).

### Deposit rates, spreads, and monetary policy pass-through
- Equilibrium deposit rate–policy rate spreads fall, though quantitatively modest for the U.S. and the euro area.
  - Deposit rate spreads are estimated to fall by up to 20 basis points for the U.S.
  - No significant variation is observed for the euro area.
- Monetary policy pass-through has the potential to become notably stronger; conditions under which spreads are more reactive are discussed in the paper.

### Volume effects on deposits and reserves
- Upper bound volume effects are economically significant:
  - Deposits may fall by USD 4.8 trillion in the U.S. (-25%) (upper bound).
  - Deposits may fall by EUR 2.8 trillion in the euro area (-21%) (upper bound).
- Reserve borrowing expenses rise more than deposit expenses fall, implying:
  - Bank net profits and dividend payouts drop.
  - Central banks may structurally lower their policy rates to counter banks’ rising reserve borrowing expense (assessment of macroeconomic consequences would require an extended model with a real economy layer).

### Net effects on bank profits and central bank seigniorage
- Net income for banking systems (upper bounds):
  - U.S. bank net income may fall by about USD 141 billion (-30%) per annum.
  - Euro area bank net income may fall by about EUR 22 billion (-20%) per annum.
- Central bank interest income through higher reserve lending would increase (or reserve remuneration expenses drop through falling reserve holding surpluses).
- Central bank net income (seigniorage) behaves nonlinearly with respect to the CBDC interest rate:
  - With zero or low CBDC interest, central bank net income would first rise.
  - With further rising CBDC interest rates, central bank net income tends to fall.
- Falling net income for banks is accompanied by central banks’ rising net income, interest payouts, and seigniorage redistribution, which let private sector financial income remain unaffected on net (a result shown by the system-wide stock-flow consistent structure of the model).

### Sources of uncertainty and key behavioral parameters
- Counterfactual estimates are surrounded by uncertainty from:
  - Behavioral assumptions: whether people perceive CBDC as closer substitutes to cash or to deposits (captured by the “base utility” and “nesting” parameters in the conditional logit model).
  - Price sensitivity parameter: plays a decisive role and is surrounded by estimation uncertainty, particularly for the euro area.
- Recommendation: any application to other countries should be preceded by obtaining price sensitivity estimates with maximal precision; the paper hints at options for doing so.

### Determinants of CBDC comparative utility
- Factors increasing base utility of CBDC relative to bank deposits:
  - People’s understanding that a central bank cannot default.
  - Deposit insurance’s role in making bank deposits closer to central bank money (deposit insurance would flank discussion).
- Factors lowering base utility relative to cash:
  - Policy-induced holding or transactions flow limits.
  - Concerns over less anonymity.
  - Cyber risk–implied leakages or operational disruptions.
- These factors map to specific parameters in the model and can be explored directionally via sensitivity analysis.

### Policy-relevant model extensions (eight examples)
- Making bank reserve borrowing collateralized: to address collateral scarcity and the need to relax central bank collateral eligibility criteria if CBDC demand rises strongly.
- Introducing a sovereign agent and sovereign bond markets: to model central bank open market operations and account for sizeable surplus reserves in the banking system post-GFC and post-Covid.
- Introducing heterogeneous bank market power: to align model banks with specific banks in a country (current model has multiple banks but they are not heterogeneous).
- Refining bank liability and asset structures: to analyze additional behavioral responses, e.g., banks incentivizing depositors to move from sight to term deposits by raising term deposit rates, and introducing wholesale funding endogenously.
- Endogenizing bank lending rates: to complete pass-through analysis and assess impact on bank lending; would create a “two-instruments-one-target” problem and require augmenting the learning algorithm.
- Caps on CBDC flows and/or balances: to reflect policy options for constraining CBDC demand; effects can be assessed by comparing metrics with and without caps.
- Multiple currency use and dollarization: extend model to include additional currencies (e.g., the U.S. dollar) so money shares include four buckets instead of three.
- Integration with larger-scale macroeconomic models: connect to macro shells (e.g., Barrdear and Kumhof (2021)) to endogenize CBDC take-up and policy rates (e.g., via a Taylor rule).

### Money creation, terminology, and interpretation
- The model implements banks as money creators in a two-layer monetary system featuring a segregated reserve-settlement structure.
- Recommendation on terminology: do not refer to falling deposit balances and rising central bank reserve borrowing driven by CBDC demand as “disintermediation.” Rationale:
  - Banks are not intermediaries in the sense implied by that term; they are money creators.
  - Intermediation is defined here as any non-banks’ on-lending of their pre-existing money asset (which leaves total money stock unchanged).
  - With positive CBDC demand, banks would still create new money liabilities and hence continue to influence M2 through lending and principal repayment.
  - Banks would increasingly generate another agent’s money liabilities—the central bank’s—yet remain money creators rather than becoming intermediaries.

### Replication and availability
- The model codes are available along with the paper to allow replication of results and application to other jurisdictions. Cross-references to all equations in the paper are placed in the model codes.

*Source: wpiea2023009-print-pdf — 6. Discussion and Conclusions*

### 1.   A rising number of banks, all else equal, means higher deposit rates (smaller markdowns).

### wpiea2023009-print-pdf - 1.   A rising number of banks, all else equal, means higher deposit rates (smaller markdowns).

### Theoretical comparative-statics and core implications
- A rising number of banks, all else equal, means higher deposit rates (smaller markdowns).
- The number of banks falling to one makes the deposit rate fall to a minimum (maximal markdown), assuming that the base utility of outside options is sufficiently small.
- A rising price sensitivity of depositors, all else equal, renders deposit rates higher (markdowns smaller). In a heterogeneous market power case, this holds for all individual banks’ Nash rates as well as their deposit volume weighted average.
- Price sensitivity extremes:
  - A price sensitivity of 훽 = ∞ implies that deposit rates approach the upper bound, the policy rate.
  - A price sensitivity of 훽 = 0 implies that deposit rates converge to a minimum.
- The theoretical results above are illustrated using the homogeneous market power model variant with no outside options and policy rate set to 푟 = 0.1 (Figures A1 and A2 referenced).

### Learning algorithm identification of Nash equilibrium
- A TS (Thompson Sampling)-based learning algorithm successfully identifies the theoretical Nash equilibrium deposit rates and Nash cash ratios.
- Exemplary parameterization (Figure A2) used:
  - 푟 = 0.1, 퐵 = 7, 훼_b = 0 ∀ b = 1, ... , 퐵, and 훼_cash = 9.
  - Deposit rate grid length on interval (0, 푟) set to 50.
  - Resulting analytical Nash equilibrium deposit rate = 8.9% and Nash cash ratio = 14.1%.
  - Algorithm converges closely to equilibrium after about 150 periods in a T = 300 simulation; time series of deposit rates and cash ratios and the sum of additions to beta distribution shape parameters confirm identification.

### Sampling choice signals from (nested) conditional logit models (Annex II)
- Without nesting (휇_m = 1 ∀ m = 1,2) and utility parameterized with 훼_b’s and 훽:
  - Draw uniform p ∈ [0,1]; compute GEV inverse G^{-1}(p) = -ln(-ln(p)); sum with deterministic utility v_{n,b} to obtain u^*_{n,b}.
  - Choice signal b̂_{n,t} = argmax_b {u^*_{n,b}}.
  - Formal expressions: u^*_{n,b} = v_{n,b} + G^{-1}(p), ∀ b = 1,...,B+2, ∀ n = 1,...,N (eq. (14)); b̂_{n,t} = argmax (eq. (15)).
- With nesting (some 휇_m ≠ 1):
  - Layer 1 (nest choice): compute scaled inclusive values and draws u^*_{n,m,t,Layer 1} = scaled inclusive value + G^{-1}(p) (eq. (16)); nest choice b̂_{n,t}^{Layer 1} = argmax over nests (eq. (17)).
  - Layer 2 (within-nest choice): u^*_{n,b,t,Layer 2} = v_{n,b} + scaled error term dependent on relevant 휇_m (eq. (18)); lower-level choice b̂_{n,t}^{Layer 2} = argmax (eq. (19)).
- Each non-bank agent makes two sequential decisions: nest selection (eq. (17)) and underlying instrument/bank choice (eq. (19)).

### Model dynamics sensitivity analyses (Annex III)
Primary model outcomes examined: equilibrium deposit rates, cash ratios, and ratios of bank end-of-period reserve holdings over bank deposit balances. Base parameterization for sensitivity analyses:
- 퐵 = 10, 훼_b=0 for b=1...B, 훼_cash = 50, 훼_CBDC = ∞, 훽 = 60, 훾 = 0.5, 훿 = 1, 휆 = 0, 푖_policy = 0.1, 푖_CBDC = 0.
- Note: rate on required reserves irrelevant when 휆 = 0; base utility of CBDC set to ∞ so CBDC share stays at zero.

(1) Velocity of Money (훾)
- Five grid points: 훾 ∈ {0.2,0.3,0.5,0.7,0.8}.
- Findings:
  - With rising money velocity, equilibrium deposit rates drop.
  - Cash ratio and reserve-to-deposit ratio grow.
  - Reserve needs increase with velocity → higher interest expense for reserve borrowing → drag on bank profits (eq. (6) in main text).
  - Falling deposit rates imply a rising cash share because relative utility of deposits decreases.
- Analytical benchmarks (no velocity dependence): Nash deposit rate = 8.2%; analytical cash ratio = 1.5%; these are approached when money velocity falls toward zero.

(2) Loan Transfer Probability (훿)
- Grid points: 훿 ∈ {0,0.25,0.5,0.75,1}.
- Findings:
  - Nash deposit rates are increasing in 훿.
  - Cash ratio is decreasing in 훿.
  - Reserve-to-deposit ratio is first increasing then decreasing in 훿.
  - Analytical Nash deposit rate and cash ratio (8.2% and 1.5%) approximated at right end of grid (훿 = 1) but still short of analytical equilibrium because velocity remains 훾 = 0.5 in simulations.

(3) Reserve Requirement (휆) and Reserve Remuneration Rates (i_RB, i_RR)
- Grid for 휆: {0,0.25,0.5,0.75,1}.
- Reserve remuneration rates treated symmetrically: i_RR = i_ER ∈ {0.1,0.05,0}.
- Policy (reserve borrowing) rate set at i_policy = 0.1.
- Findings:
  - If reserve remuneration rates equal policy rate, varying 휆 does not influence outcomes.
  - If remuneration rates are below policy rate, rising 휆 → Nash deposit rates fall and cash ratios rise; slopes steeper the lower the remuneration rates.
  - If remuneration rates sufficiently low and 휆 sufficiently high, cash ratios can reach 100%; deposit balances fall to zero and deposit rates become inconsequential.
- Simulated reserve-to-deposit ratio with 휆 = 0% equals 2.3% (reference to Figure A4/third chart where 훿 = 1 and Figure A3 where 훾 = 0.5).

### Empirical estimation and model calibration (Annex IV)
Step A. Estimate non-banks’ price sensitivity (훽)
- Multinomial logit structure embedded in ABM (eq. (1)). Aggregate-level utilities for deposits and cash at upper decision layer.
- Model specification (eq. (20)) includes cash utility 훼, controls Z_t with loadings Γ, and Type-I GEV errors.
- Cash ratio implied by model (eq. (21)): CR̂_t = τ̂_t(α̂, Γ̂) / [τ̂_t(α̂, Γ̂) + e^{...}] (as in equation).
- Estimation minimizes sum of squared residuals of observed and implied cash ratio quarter-on-quarter differences (eq. (22)); empirical performance superior using ∆(quarter-on-quarter) cash ratios.
- Required data inputs: cash-in-total-money ratio (cash/M2) CR_t, banking system deposit rate i_t^{dep}, controls Z_t (year-on-year real GDP growth, year-on-year CPI inflation, quarterly velocity of money = quarterly nominal GDP / M2). Figure A6 shows U.S. and euro area data series (data sources: St. Louis FRED and FDIC for U.S.; ECB Statistical Data Warehouse for ECB).

Price sensitivity estimates (Table A2):
- U.S.:
  - 훽 point estimate q=0.5 = 47.5; q=0.2 = 2.7; q=0.8 = 57.5.
  - R-square (level) = 0.941; R-square (diff.) = 0.438.
  - DW = 2.01.
  - obs. = 88.
- Euro Area:
  - 훽 point estimate q=0.5 = 153.6; q=0.2 = 49.8; q=0.8 = 961.1.
  - R-square (level) = 0.995; R-square (diff.) = 0.416.
  - DW = 2.03.
  - obs. = 75.
- Note: q=0.2 and q=0.8 are 20th and 80th percentiles of bootstrap-based coefficient distributions.
- Calibration choices informed by estimates:
  - Set 훽 = 50 for the U.S. (rounded).
  - Set 훽 = 150 for the euro area (rounded).
- The estimates of Γ̂ and α̂ from Step A are discarded for ABM calibration; Γ̂ not needed and α̂ (cash base utility) will be inferred in Step C consistent with ABM.

Step B. Calibrate other model parameters (훾, 훿, 휆, i_RB, i_RR, i_ER)
- Velocity of money 훾 aligned with desired annual GDP/M2 ratio:
  - U.S. 2022Q2: rounded 110%.
  - Euro area 2022Q2: rounded 85%.
- Loan transfer probability 훿 set judgmentally to 0.75; sensitivity analysis showed CBDC counterfactuals not notably sensitive to 훿.
- Reserve requirement 휆 as of September 2022:
  - U.S./FED: 0%.
  - Euro area/ECB: 1%.
- Reserve borrowing rate i_policy and remuneration rates set to recent reference values:
  - U.S.: i_policy and i_excess_reserve remuneration set to 3.5%; rate on required reserves irrelevant since 휆 = 0.
  - Euro area: [i_policy, i_required_reserve, i_excess_reserve] set to [2%, 1.5%, 1%]. i_policy assumed relates to ECB’s marginal lending facility.

Step C. Estimate number of banks (B) and base utility of cash (훼_cash) to match target cash ratio and deposit rate
- Three sub-steps:
  1. Simulate ABM over grids for 퐵 and 훼_cash (example: 10x10 grids → 100 simulations), T = 3000 periods each; record median of last 1,000 periods for Nash deposit rates and cash ratios. Figure A7 shows exemplary matrices.
  2. Fit two polynomial equations:
     - For log Nash deposit rate: f_rate(B, 훼_cash) = polynomial up to order 3 in B and 훼_cash (eq. (23)).
     - For logit cash ratio: f_CR(B, 훼_cash) = polynomial up to order 3 (eq. (24)).
     - Impose linear inequality constraints: derivative wrt B negative for cash ratio and positive for Nash deposit rate; derivative wrt 훼_cash positive in both polynomials.
  3. Solve polynomials for (B̂, α̂_cash) to match target deposit rate ĩ_dep and target cash ratio CR̃ using trust-region-dogleg algorithm (Nocedal and Wright 2006) minimizing equal-weight squared deviations (eq. (25)).
- Polynomial order (PN) set to 3. Figures A7 and A8 visualize matrices and polynomial surfaces used in optimization.
- Note on identification: cannot calibrate B first empirically because non-bank population size N has no structural meaning in model; N must be set sufficiently large for numerical stability. Analytical model in Annex I shows no structural dependence on N.

*Italic: Source — wpiea2023009-print-pdf, Annexes I–IV content excerpted from the supplied PDF content.*

### Annex V: Detailed Counterfactual Simulation Results for the U.S. and the Euro Area Figure A9. Bank Balance Sheets (USD b

### Annex V: Detailed Counterfactual Simulation Results — Figures A9–A20 (U.S. and Euro Area)

### Simulation design and general notes
- Simulation windows:
  - The first of the six simulation windows corresponds to a base simulation without CBDC.
  - The five counterfactual windows labeled A–E correspond to CBDC interest rates set to [0,25,50,75,100]% of the policy rate.
- CBDC design variants:
  - Yellow streams correspond to “full nesting” (cash-like CBDC).
  - Blue streams correspond to “no nesting” (deposit-like CBDC).
- Chart conventions:
  - For bank figures, charts in the first row show the evolution of individual bank metrics; charts in the second row show the sum across banks.
  - For non-bank agents, charts show the sum of the respective flow metrics across the underlying individual non-bank agents.
  - Central bank balance sheet figures report end-of-period levels; central bank P&L figures report flows in USD or EUR billions, per annum (as indicated).

### Figures and metrics covered (U.S.)
- Figure A9. Bank Balance Sheets (USD billions)—Counterfactual Results for the U.S.
  - Bank-level and aggregate balance-sheet metrics under the six simulation windows and two CBDC nesting designs.
- Figure A10. Bank P&L Flows (USD billions, per annum)—Counterfactual Results for the U.S.
  - Bank profit-and-loss flows per annum, displayed by individual banks and as the aggregate across banks.
- Figure A11. Non-Bank Agents’ P&L Flows (USD billions, per annum)—Counterfactual Results for the U.S.
  - Aggregate P&L flow metrics summed across non-bank agents.
- Figure A12. Central Bank Balance Sheet (USD billions, end of period)—Counterfactual Results for the U.S.
  - End-of-period central bank balance-sheet positions across the six windows and two CBDC designs.
- Figure A13. Central Bank P&L Flows (USD billions, per annum)—Counterfactual Results for the U.S.
  - Central bank annual P&L flows under each scenario.
- Figure A14. Banks’ and Central Bank Net Reserve Position (USD billions, end of period)—Counterfactual Results for the U.S.
  - End-of-period net reserve positions for banks and the central bank across scenarios.

### Figures and metrics covered (Euro Area)
- Figure A15. Bank Balance Sheets (EUR billions)—Counterfactual Results for the Euro Area
  - Bank-level and aggregate balance-sheet metrics under the six simulation windows and two CBDC nesting designs.
- Figure A16. Banks’ P&L Flows (EUR billions, per annum)—Counterfactual Results for the Euro Area
  - Bank profit-and-loss flows per annum, shown by individual banks and in aggregate.
- Figure A17. Non-Bank Agents’ P&L Flows (EUR billions, per annum)—Counterfactual Results for the Euro Area
  - Aggregate P&L flows summed across non-bank agents.
- Figure A18. Central Bank Balance Sheet (EUR billions, end of period)—Counterfactual Results for the Euro Area
  - End-of-period central bank balance-sheet positions across scenarios.
- Figure A19. Central Bank P&L Flows (EUR billions, per annum)—Counterfactual Results for the Euro Area
  - Central bank annual P&L flows under each scenario.
- Figure A20. Banks’ and Central Bank’s Net Reserve Position (EUR billions, end of period)—Counterfactual Results for the Euro Area
  - End-of-period net reserve positions for banks and the central bank across scenarios.

### Key simulation features and interpretation guidance
- Counterfactual set:
  - Scenarios A–E systematically vary CBDC interest at [0,25,50,75,100]% of the policy rate to isolate effects of CBDC remuneration on financial-sector and central-bank metrics.
- Nesting alternatives:
  - “Full nesting” (cash-like CBDC) and “no nesting” (deposit-like CBDC) produce distinct dynamic paths; visualized by yellow and blue streams respectively.
- Aggregation conventions:
  - For bank figures, comparisons are available both at the individual bank metric level and at the aggregate (sum across banks) level.
  - For non-bank agents, flows are presented as aggregates summing across all non-bank agents.
- Timing conventions:
  - Central bank balance-sheet panels report end-of-period levels; P&L panels report annualized flows (USD or EUR billions, per annum) as labeled.

*To Demand or Not to Demand: On Quantifying the Future Appetite for CBDC — Working Paper No. WP/23/9 (Annex V: Figures A9–A20).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023009-print-pdf.pdf_