## 2.1–3.5: Model setup and impact of CBDC introduction (wpiea2023236-print-pdf)

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### Model setup and equilibrium (agents, assets, key parameters)
- Framework: portfolio choice model with an imperfectly competitive banking sector (in the spirit of Monti (1972); Klein (1971); Drechsler et al. (2017)).
- Agents: households, banks, central bank.
- Household assets:
  - Liquid: notes (cash) N, CBDC C, deposits D.
  - Illiquid: bonds earning rate f.
- Key elasticity/parameter restrictions:
  - elasticity of substitution between wealth and liquidity: ρ < 1.
  - elasticity of substitution between liquid assets: ϵ > 1.
  - substitutability across banks: η > 1.
  - number of banks: J.
  - derived aggregate deposit-demand elasticity with respect to the spread: M (equation (12)).
- Household utility and budget (identities preserved):
  - Utility: U(W0) = max ( W^{ρ−1}_{ρ} + λ L^{ρ−1}_{ρ} )^{ρ/(ρ−1)}.
  - Liquidity services: L(N,C,D) = ( N^{(ϵ−1)/ϵ} + δ_C C^{(ϵ−1)/ϵ} + δ_D D^{(ϵ−1)/ϵ} )^{ϵ/(ϵ−1)}.
  - Budget constraint (opportunity-cost form): W = W0(1 + f) − N f − C(f − r_C) − D(f − r_D).
  - First-order conditions (selected):
    - L/W = λ ρ s_L^{−ρ} with s_L ≡ ( f^{1−ϵ} + δ_D^{ϵ} (s^*)^{1−ϵ} + δ_C^{ϵ} (f−r_C)^{1−ϵ} )^{1/(1−ϵ)}.
    - C/N = δ_C^{ϵ} ( (f − r_C)/f )^{−ϵ}.
    - C/D = ( δ_C/δ_D )^{ϵ} ( (f − r_C)/(f − r_D) )^{−ϵ}.
    - D_j/D = ( (f − r_{D,j})/(f − r_D) )^{−η}.
- Banks:
  - Aggregate deposits: D = ( (1/J) Σ_{j=1}^J D_j^{(η−1)/η} )^{η/(η−1)}.
  - Banks funded by deposits (baseline) and invest in bonds at rate f.
  - Bank profits per bank j: (f − r_{D,j}) D_j.
  - Pricing condition: ∂D_j/∂(f − r_{D,j}) · (f − r_{D,j})/D_j = −1 (equation (9)), leading to aggregate elasticity M:
    - − ∂D/∂(f − r_D) · (f − r_D)/D = 1 − (η − 1)(J − 1) = M (equation (12)).
    - M decreases with greater competition (higher J) and higher η.
- Central bank: chooses policy rate f and CBDC rate r_C; supplies bonds and CBDC with infinite elasticity.
- Baseline analytical tractability in limit λ → 0:
  - For ϵ > M > ρ, closed-form expressions exist for equilibrium spread s^* ≡ f − r_D^* (equation (13)) and equilibrium deposits D^* (equation (14)).
  - Comparative statics (bank deposit channel):
    - ∂s^*/∂f ≥ 0 (equation (15)): higher f raises equilibrium spread s^*.
    - ∂D^*/∂f ≤ 0 (equation (16)): higher f reduces total deposits D^*.

### Impact of CBDC introduction in homogeneous-household (λ → 0) model
- Definitions and baseline no-CBDC expressions:
  - Deposit spread s^* ≡ f − r_D^*.
  - Without CBDC (δ_C = 0) equilibrium:
    - ̃s∗ = δ−1D [M − ρ−M]1−1 × f (equation (17)).
    - ̃D∗ = δ(1−ρ)1−D (̃s∗)−ρ [1 + δ−D (̃s∗ f)−1]ρ−−1 (equation (18)).
- Two opposing effects of CBDC:
  - Diversification (love-of-variety): ϵ > 1 implies households diversify across liquid assets; CBDC reduces but does not eliminate demand for cash and deposits.
  - Competitive pressure on banks: CBDC induces banks to reduce deposit spread (raise deposit returns), which increases households’ demand for deposits.
- Proposition 1 (λ → 0, δ_C > 0):
  - The equilibrium spread on deposits always declines after CBDC introduction: s∗ − ̃s∗ = ∆s < 0.
  - The equilibrium level of deposits always increases after CBDC introduction: D∗ − ̃D∗ = ∆D > 0.
- Mechanism (brief):
  - s∗/̃s∗ = [1 + δC ( (f − rC)/f )1− ]1/(1−) < 1.
  - D∗/̃D∗ = [s∗/̃s∗]−ρ > 1 because 0 < ρ < 1 and s∗/̃s∗ < 1.
  - CBDC lowers the opportunity cost of liquidity and banks adjust s∗ so margin indifference holds, raising aggregate deposits.

### Heterogeneous households: fixed access costs and extensive vs. intensive margins
- Enrichments:
  - Heterogeneity in initial household wealth W0 (Pareto Type I with shape α; set W0 = 1).
  - Fixed costs of holding deposits φ_D and CBDC φ_C measured in utility; assume φ_D > φ_C.
- Fixed-cost utility specification:
  - u(W0) = max [ (Wρ−1/ρ + λLρ−1/ρ)ρ/(ρ−1) − 1(φ) ], where 1(φ) equals φ_C if C > 0 and D = 0; φ_D if D > 0 and C = 0; φ_C + φ_D if C > 0 and D > 0.
- Fixed costs generate wealth cutoffs and an extensive margin (who holds deposits) and an intensive margin (how much deposit-holders hold).
- Possible equilibrium orderings (four scenarios) depending on cutoffs ˆW11, ˆW12, ˆW13, ˆW21, ˆW22 (see Appendix B formulas).
- Preferred parametrization equilibrium (numerical solution): 
  - Low-wealth households: only cash.
  - Medium-wealth households: cash + CBDC.
  - High-wealth households: cash + CBDC + deposits.

### Baseline calibration (parameters preserved exactly)
- λ Relative importance of liquid assets = 1.5∗10−6
- ρ Complementarity b/w wealth & liquidity = 0.15
- ϵ Substitutability b/w different liquid assets = 3
- η Substitutability b/w deposits at different banks = 1.1
- J Number of banks = 8
- δD Share of deposits = 1.5
- δC Share of CBDC = 2
- f Interest on bonds = 3%
- rC Return on CBDC = 0
- φD Fixed cost of accessing deposits = 0.06 × λρ
- φC Fixed cost of accessing CBDC = 0.001 × λρ
- W Normalized lowest wealth = 1
- α Shape of wealth distribution = 1.52
- Calibration targets/choices:
  - λ chosen to generate share of non-liquid assets to total household wealth of 83 percent (U.S. Census).
  - δD chosen to match share of wealth in cash versus deposits in the U.S. when no CBDC (≈11%, Cash/M2).
  - δC set assuming CBDC more helpful than deposits for liquidity services.
  - φD set to have share of population fully banked around 80% (FDIC).
  - α set to match U.S. Gini coefficient of 0.49.
  - Baseline assumes non-interest bearing CBDC; interest-bearing CBDC considered in robustness checks.

### Numerical results (baseline heterogeneous calibration; values preserved exactly)
- Table 2 key variables before and after CBDC introduction:
  - Cash (%) before CBDC: 1.8% ; with CBDC: 0.4%
  - CBDC (%) before CBDC: 0.0% ; with CBDC: 3.5%
  - Deposits (%) before CBDC: 15.4% ; with CBDC: 11.2%
  - Non liquid wealth (%) before CBDC: 82.8% ; with CBDC: 84.9%
  - Interest on deposits before CBDC: 2.25% ; with CBDC: 2.72%
  - Banks profit before CBDC: 0.0012 ; with CBDC: 0.0003
  - Financial inclusion (%) before CBDC: 79.5% ; with CBDC: 100%
  - % Deposits for those with bank before CBDC: 16.6% ; with CBDC: 19.6%
  - % Wealth held by those with bank account before CBDC: 92.5% ; with CBDC: 56.9%
  - Intensive margin: 1.69 p.ps.
  - Extensive margin: -5.92 p.ps.
- Interpretation (baseline):
  - Banks raise deposit remuneration by about 47 basis points (from 2.25% to 2.72%) to compete with CBDC.
  - Extensive margin (depositors abandoning accounts) dominates intensive margin, yielding aggregate deposits fall from 15.4% to 11.2% of total wealth (about a 4% decline).
  - Share of non-liquid wealth increases to 84.9% as liquid-assets share falls.
  - Financial inclusion rises from 79.5% to 100% due to low φC.

### Illustration of margins (descriptive)
- Deposit-by-wealth (D/W) patterns:
  - Before CBDC: households with wealth < WA (20.5% of households, financially excluded) hold only cash; wealth > WA hold cash + deposits.
  - After CBDC: cutoff shifts from WA to WB; households between WA and WB switch from deposits to CBDC only (extensive margin).
  - Remaining depositors increase D/W (intensive margin).
  - Net baseline effect: extensive margin area larger than intensive margin area; net decline in aggregate deposits of 4.23 percentage points of initial aggregate deposits.

### Sensitivity and alternative calibrations (preserved results)
- Mechanisms determining which margin dominates:
  - Key parameters: relative fixed costs φC/φD and wealth-distribution shape α.
  - When φC << φD and the mass of poorer households is large (α high), extensive margin dominates and aggregate deposits decline.
- Alternative calibrations (Table 3 excerpts; No CBDC / CBDC):
  - High CBDC cost (c = 0.03):
    - Cash (%) : 1.8% / 0.0%
    - CBDC (%) : 1.6% / 0.0%
    - Deposits (%) : 15.4% / 15.5%
    - Non liquid wealth (%) : 82.8% / 82.7%
    - Interest on deposits : 2.25% / 2.67%
    - Banks profit : 0.0012 / 0.0005
    - Financial inclusion (%) : 79.5% / 100%
    - % Deposits for those with bank : 16.6% / 19.2%
    - % Wealth held by those with bank account : 92.5% / 80.7%
    - Intensive margin* : 2.08 p.ps.
    - Extensive margin** : -1.96 p.ps.
    - Net: deposits grow slightly (intensive margin dominates).
  - High alpha (α = 1.62):
    - Cash (%) : 1.9% / 0.5%
    - CBDC (%) : 0.0% / 3.9%
    - Deposits (%) : 15.6% / 10.3%
    - Non liquid wealth (%) : 82.5% / 85.3%
    - Interest on deposits : 2.32% / 2.74%
    - Banks profit : 0.0011 / 0.0003
    - Financial inclusion (%) : 80.4% / 100%
    - % Deposits for those with bank : 17.0% / 19.9%
    - % Wealth held by those with bank account : 92.0% / 51.6%
    - Intensive margin* : 1.52 p.ps.
    - Extensive margin** : -6.86 p.ps.
    - Net: extensive margin dominates; larger disintermediation.

### Robustness: remunerated CBDC and δC comparative statics
- Interest-bearing CBDC (Figure 4 summary):
  - Change in aggregate deposits is linear in rC.
  - Increasing rC from zero to 0.7% reduces aggregate deposits by a further 1 percentage point relative to non-remunerated CBDC and raises deposit remuneration by a further 6 basis points, further decreasing banks’ profits.
  - Decreasing rC leads to a smaller drop in aggregate deposits; model allows negative rC.
- Comparative statics on δC (Figure 5):
  - Baseline: δN = 1, δC = 2, δD = 1.5.
  - Higher δC (CBDC provides better liquidity services) leads to a larger drop in aggregate deposits because CBDC substitutes more for deposits.
- Results for remunerated CBDC (Table 4, rC = 3%; No CBDC / CBDC):
  - Cash (%) : 1.8% / 0.3%
  - CBDC (%) : 0.0% / 4.4%
  - Deposits (%) : 15.4% / 10.4%
  - Non liquid wealth (%) : 82.8% / 84.9%
  - Interest on deposits : 2.25% / 2.78%
  - Banks profit : 0.0012 / 0.0002
  - Financial inclusion (%) : 79.5% / 100%
  - % Deposits for those with bank : 16.6% / 20.4%
  - % Wealth held by those with bank account : 92.5% / 51.2%
  - Intensive margin* : 1.97 p.ps.
  - Extensive margin** : -5.88 p.ps.

### Banking extension: effects on lending and wholesale funding
- Extension: banks fund with deposits (D_i) and wholesale funding (H_i), lend L_i; lending opportunities characterized by parameters l0, l1 and wholesale funding cost h.
- Analytical insight (Equation (23)):
  - If h = 0 (riskless wholesale funding at policy rate), lending is constant after CBDC introduction.
  - If h > 0 and profitable lending opportunities (l0/l1 sufficiently high), banks use wholesale funding to offset deposit losses and protect lending.
- Quantitative calibration for lending extension (parameter choices preserved):
  - l0 = 0.001, l1 = 0.001, h = 0.00002.
- Table 5 (No CBDC / CBDC; preserved values):
  - Cash (%) : 1.7% / 0.4%
  - CBDC (%) : 0.0% / 3.4%
  - Deposits (%) : 15.6% / 11.7%
  - Non liquid wealth (%) : 82.7% / 84.5%
  - Interest on deposits : 2.3% / 2.8%
  - Banks profit : 0.0011 / 0.0003
  - Financial inclusion (%) : 80.6% / 100%
  - % Deposits for those with bank : 16.8% / 20.2%
  - % Wealth held by those with bank account : 92.9% / 57.9%
  - Intensive margin * : 1.90%
  - Extensive margin ** : -6.88%
  - Change in Lending : -0.18%
- Textual summary (Section 4.2):
  - CBDC introduction leads to a drop in aggregate deposits of 4 percentage points of total wealth, but lending drops by only 0.14% (alternative reported magnitude).
- Interpretation:
  - Even when CBDC causes deposit disintermediation, the negative effect on bank lending is quantitatively small (below 0.2%) because banks access alternative funding or exploit profitable lending opportunities.

### Comparative-statics in homogeneous-household model (Appendix A; exact signs and inequalities)
- Effect of higher CBDC rate r_C:
  - ∂|∆s|/∂r_C > 0  (Equation (25))
  - ∂∆D/∂r_C > 0  (Equation (26))
  - Interpretation: higher r_C forces banks to raise deposit rates to retain households, increasing |∆s| and ∆D.
- Effect of higher policy rate f:
  - ∂|∆s|/∂f > 0  (Equation (27))
  - ∂∆D/∂f < 0  (Equation (28))
  - Interpretation: higher f raises decline in deposit spread but reduces increase in aggregate deposits after CBDC introduction.
- Effect of greater competition (η or J):
  - ∂|∆s|/∂η < 0,  ∂|∆s|/∂J < 0  (Equation (29))
  - ∂∆D/∂η > 0,  ∂∆D/∂J > 0  (Equation (30))
  - Interpretation: more competition implies smaller decline in deposit spread and larger increase in aggregate deposits after CBDC introduction.

### Aggregate-deposit expression with heterogeneity and deposit-spread equilibrium (Appendix B; preserved expressions)
- Aggregate deposits when some households hold both CBDC and deposits:
  - D = δ^ϵ_D ( s_B^l / s )^ϵ λ ρ ( s_B^l )^(−ρ) (1 + λ ρ (s_B^l)^(1−ρ)) (1 + f) × ∫_{W0}^{ˆW2} dF(W0)  (Equation (32))
- Deposit-spread equilibrium condition (exact expression, Equation (33)) preserved in source (see Appendix B for full formula).
- Cutoff-wealth expressions (selected, preserved):
  - ˆW1 = φ_C (1+f) ( t(s_C^l) − t(s_N^l) )
  - ˆW2 = φ_D (1+f) ( t(s_B^l) − t(s_C^l) )
  - General cutoffs ˆW11, ˆW12, ˆW13, ˆW21, ˆW22 given in Appendix B.

### Policy-relevant implications emphasized in the source
- Assessment of CBDC impacts must account for:
  - household heterogeneity in wealth,
  - fixed access costs to deposit and CBDC accounts,
  - imperfect substitutability between cash, CBDC, and deposits,
  - banks’ strategic pricing responses and market structure (J and η).
- Empirical needs: quality data on household preferences over means of payment to estimate demand for liquid assets and to quantify intensive versus extensive margin effects.
- Caveats:
  - Baseline static model abstracts from dynamic channels whereby compressed bank profits and eroded capital could affect lending over time.
  - Sections 3–4 refine baseline conclusions by introducing heterogeneity and alternative funding.

*Source: wpiea2023236-print-pdf - 2.1, 2.3, 3.5 (IMF working paper content preserved exactly)*

### 2.1    Setup .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .1

### 2.1 Setup

### Model environment
- Framework: portfolio choice model with an imperfectly competitive banking sector (in the spirit of Monti (1972); Klein (1971); Drechsler et al. (2017)).
- Agents: households, banks, and a central bank.
- Liquid assets available to households: notes (cash) N, CBDC C, deposits D; illiquid asset: bonds earning rate f.
- Key elasticity/parameter restrictions explicitly used:
  - elasticity of substitution between wealth and liquidity: ρ < 1.
  - elasticity of substitution between liquid assets: ε > 1.
  - substitutability across banks: η > 1.
  - number of banks: J.
  - symmetric-bank derived elasticity of aggregate deposit demand with respect to the spread defined as M (equation (12)).

### Households (preferences and budget)
- Utility: U(W0) = max ( W^{ρ−1}_{ρ} + λ L^{ρ−1}_{ρ} )^{ρ/(ρ−1)}, with wealth and liquidity complements (ρ < 1) and parameter λ.
- Liquidity services: L(N,C,D) = ( N^{(ε−1)/ε} + δ_C C^{(ε−1)/ε} + δ_D D^{(ε−1)/ε} )^{ε/(ε−1)}, where δ_C and δ_D are relative usefulness of CBDC and deposits versus cash.
- Budget constraint (opportunity-cost form): W = W0(1 + f) − N f − C(f − r_C) − D(f − r_D). Cash earns no return; CBDC earns r_C ≥ 0; deposits earn r_D.
- First-order conditions for household portfolio choice (identities retained from source):
  - liquidity-versus-wealth condition: L/W = λ ρ s_L^{−ρ} (equation (5)), with s_L ≡ ( f^{1−ε} + δ_D^{ε} (s^*)^{1−ε} + δ_C^{ε} (f−r_C)^{1−ε} )^{1/(1−ε)}.
  - relative liquid-asset choices:
    - C/N = δ_C^{ε} ( (f − r_C)/f )^{−ε} (equation (6)).
    - C/D = ( δ_C/δ_D )^{ε} ( (f − r_C)/(f − r_D) )^{−ε} (equation (7)).
  - cross-bank deposit allocation: D_j/D = ( (f − r_{D,j})/(f − r_D) )^{−η} (equation (8)).

### Banks (technology and behavior)
- Aggregate deposits: D = ( (1/J) Σ_{j=1}^J D_j^{(η−1)/η} )^{η/(η−1)} (equation (4)).
- Banks are funded by deposits (baseline assumption) and invest in bonds at rate f (assumption relaxed in Section 4).
- Bank profits (per bank j): (f − r_{D,j}) D_j.
- First-order condition for bank pricing: ∂D_j/∂(f − r_{D,j}) · (f − r_{D,j})/D_j = −1 (equation (9)), which combined with aggregate elasticity yields equilibrium condition (11).
- Derived elasticity of aggregate deposit demand with respect to the spread:
  - − ∂D/∂(f − r_D) · (f − r_D)/D = 1 − (η − 1)(J − 1) = M (equation (12)).
  - M decreases with greater competition (higher J) and higher η.

### Central bank
- Chooses the policy rate f and the CBDC rate r_C.
- Supplies bonds and CBDC with infinite elasticity.

### Equilibrium highlights (baseline homogeneous-household case)
- Analytical tractability in the limit λ → 0: for ε > M > ρ, closed-form expressions for equilibrium spread and deposits exist:
  - equilibrium spread s^* ≡ f − r_D^* given by equation (13) (explicit functional form retained).
  - equilibrium deposits D^* given by equation (14) (explicit functional form retained).
- Parameter regime considered: focus on case ε > M > ρ so r_D^* < f (banks have market power and set deposit returns below f).
- Comparative-static channel (bank deposit channel):
  - ∂s^*/∂f ≥ 0 (equation (15)): higher f raises the equilibrium spread s^*.
  - ∂D^*/∂f ≤ 0 (equation (16)): higher f reduces total deposits D^*.
- Intuition: higher policy rate f raises the opportunity cost of holding liquid assets, allowing banks to increase deposit rates but not fully match f, increasing spreads and reducing deposit holdings as bonds become relatively more attractive.

### Key mechanisms of CBDC introduction (summary of main qualitative findings)
- Intensive margin (competition effect):
  - When liquid assets are costlessly accessible and households value variety, introducing CBDC reduces banks’ market power.
  - Banks optimally respond by increasing deposit rates; households may hold even more bank deposits overall.
  - Intensive-margin effect of CBDC introduction on aggregate deposits is always positive in the homogeneous baseline.
  - Net effect on bank profits is negative due to compressed spreads.
- Extensive margin (access-cost and heterogeneity effect, described later in Section 3):
  - When households are heterogeneous in wealth and fixed costs to access deposits or CBDC exist, an extensive margin arises: poorer households may abandon deposits if deposit access costs are high relative to CBDC.
  - If CBDC is more liquid and easier to access than deposits, poorer households can switch fully to CBDC and cash despite higher deposit rates, generating bank disintermediation.
  - The extensive margin can offset or dominate the intensive margin leading to aggregate deposit declines.

### Quantitative/numeric findings (from calibrated heterogeneous model and lending extension)
- Calibration to US data in the heterogeneous-household model yields:
  - Aggregate bank deposits can fall by 4 percent following CBDC introduction when CBDC access is easy and the mass of poorer households is large.
  - Even when CBDC generates deposit disintermediation, the negative effect on bank lending is quantitatively small and below 0.2 percent.
- Mechanisms affecting lending outcomes:
  - Banks’ access to other funding (wholesale or central bank financing) allows them to offset deposit declines without large reductions in lending.
  - When alternative funding is cheap, banks substitute away from deposits more easily; when expensive, banks compete more aggressively for deposits, compressing profits and raising deposit rates.

### Policy-relevant implications emphasized in the source
- Assessment of CBDC impacts must account for:
  - household heterogeneity in wealth,
  - fixed access costs to deposit and CBDC accounts,
  - imperfect substitutability between cash, CBDC, and deposits,
  - banks’ strategic pricing responses and market structure (J and η).
- Empirical needs: quality data on household preferences over means of payment is critical to estimate the demand for liquid assets and to quantify intensive versus extensive margin effects.
- Caveats noted:
  - The baseline static model abstracts from dynamic channels whereby compressed bank profits and eroded capital could affect lending over time.
  - The analysis in Sections 3–4 (heterogeneity, lending, wholesale funding) refines the baseline conclusions.

*Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023236-print-pdf.pdf*

### 2.3    Impact of CBDC introduction

### 2.3    Impact of CBDC introduction

### Equilibrium deposit interest rate and deposit base (λ→0)
- Deposit spread defined as s∗ ≡ f − rD∗.
- In absence of CBDC (δC = 0) equilibrium deposit spread and aggregate deposits simplify to:
  - ̃s∗ = δ−1D [M − ρ−M]1−1 × f, (equation (17) in source)
  - ̃D∗ = δ(1−ρ)1−D (̃s∗)−ρ [1 + δ−D (̃s∗ f)−1]ρ−−1. (equation (18) in source)
- Two opposing effects of CBDC introduction:
  - Diversification (love-of-variety): if CBDC is not a perfect substitute ( > 1) households diversify liquidity holdings, reducing but not eliminating demand for cash and deposits.
  - Competitive pressure on banks: banks reduce deposit spread (increase deposit remuneration) to deter substitution of deposits with CBDC, which increases households’ demand for deposits.
- Net equilibrium outcome (Proposition 1, taking λ → 0):
  - The equilibrium spread on deposits always declines after CBDC is introduced (with δC > 0): s∗ − ̃s∗ = ∆s < 0.
  - The equilibrium level of deposits always increases after CBDC is introduced (with δC > 0): D∗ − ̃D∗ = ∆D > 0.

### Proof sketch and mechanism (λ→0)
- Ratio of spreads (from (14) and (17)):
  - s∗/̃s∗ = [1 + δC ( (f − rC)/f )1− ]1/(1−) < 1, since  > 1 and the bracketed term > 1.
- Define sl ≡ (f1− + δD (s∗)1− + δC (f − rC)1−)1/(1−) so aggregate deposits can be written as:
  - D∗ = δD (s∗/sl)− s−ρ l.
- Algebra yields s∗/sl(s∗) = ̃s∗/sl(̃s∗) and hence
  - D∗/̃D∗ = [s∗/̃s∗]−ρ > 1 because 0 < ρ < 1 and s∗/̃s∗ < 1.
- Economic intuition:
  - CBDC lowers the opportunity cost of liquidity sl directly (new liquid asset) and indirectly (banks lower s∗).
  - Banks adjust s∗ so households are indifferent at the margin between holding an additional unit of CBDC and an additional unit of deposits (s∗/sl(s∗) = ̃s∗/sl(̃s∗)).
  - Result: increased demand for liquid assets raises demand for deposits and aggregate deposits in equilibrium.

### Key implication (homogeneous-household model)
- Introducing CBDC triggers banks to increase deposit remuneration to defend deposit base.
- This reaction can lead to an increase in aggregate deposits (as in Andolfatto (2021)), conditional on households holding non-liquid assets so they can simultaneously increase deposits and hold CBDC.
- The simple model does not capture the possibility that some depositors might close accounts entirely to hold CBDC; this is addressed in the enriched model in Section 3.

### Heterogeneous households and the extensive margin of deposit disintermediation
- Two model enrichments introduced:
  - Heterogeneity in initial household wealth W0 (Pareto Type I with shape α; set W0 = 1).
  - Fixed costs of holding deposits (φD) and CBDC (φC), measured in utility; assume φD > φC.
- Household utility with fixed costs:
  - u(W0) = max [ (Wρ−1/ρ + λLρ−1/ρ)ρ/(ρ−1) − 1(φ) ],
  - where 1(φ) equals φC if C > 0 and D = 0; φD if D > 0 and C = 0; φC + φD if C > 0 and D > 0.
- Fixed costs induce wealth cutoffs: households below/above thresholds demand different assets, creating an extensive margin (how many hold deposits) and an intensive margin (how much those with accounts hold).

### Equilibrium characterization and possible outcomes
- Model solved numerically (no closed-form analytical solution).
- Possible equilibrium patterns depend on parameters and determine which population segments hold cash, CBDC, and/or deposits.
- Focused equilibrium (preferred calibration):
  - Low-wealth households: only cash.
  - Medium-wealth households: cash + CBDC.
  - High-wealth households: cash + CBDC + deposits.

### Calibration (baseline)
- Parameters and baseline values:
  - λ Relative importance of liquid assets = 1.5∗10−6
  - ρ Complementarity b/w wealth & liquidity = 0.15
  -  Substitutability b/w different liquid assets = 3
  - η Substitutability b/w deposits at different banks = 1.1
  - J Number of banks = 8
  - δD Share of deposits = 1.5
  - δC Share of CBDC = 2
  - f Interest on bonds = 3%
  - rC Return on CBDC = 0
  - φD Fixed cost of accessing deposits = 0.06 × λρ
  - φC Fixed cost of accessing CBDC = 0.001 × λρ
  - W Normalized lowest wealth = 1
  - α Shape of wealth distribution = 1.52
- Calibration targets and choices:
  - λ chosen to generate share of non-liquid assets to total household wealth of 83 percent (U.S. Census).
  - δD chosen to match share of wealth in cash versus deposits in the U.S. when no CBDC (≈11%, Cash/M2).
  - δC set assuming CBDC more helpful in providing liquidity services than deposits.
  - φD set to have share of population fully banked around 80% (FDIC).
  - α set to match U.S. Gini coefficient of 0.49.
  - Baseline assumes non-interest bearing CBDC; interest-bearing CBDC considered in robustness checks.

### Numerical results (baseline calibration)
- Table 2 reported key variables before and after CBDC introduction (values preserved exactly):
  - Cash (%) before CBDC: 1.8% ; with CBDC: 0.4%
  - CBDC (%) before CBDC: 0.0% ; with CBDC: 3.5%
  - Deposits (%) before CBDC: 15.4% ; with CBDC: 11.2%
  - Non liquid wealth (%) before CBDC: 82.8% ; with CBDC: 84.9%
  - Interest on deposits before CBDC: 2.25% ; with CBDC: 2.72%
  - Banks profit before CBDC: 0.0012 ; with CBDC: 0.0003
  - Financial inclusion (%) before CBDC: 79.5% ; with CBDC: 100%
  - % Deposits for those with bank before CBDC: 16.6% ; with CBDC: 19.6%
  - % Wealth held by those with bank account before CBDC: 92.5% ; with CBDC: 56.9%
  - Intensive margin: 1.69 p.ps.
  - Extensive margin: -5.92 p.ps.
- Interpretation of results:
  - Banks increase deposit remuneration by about 50 basis points (from 2.25% to 2.72%) to compete with CBDC.
  - Wealth thresholds change: some households close deposit accounts entirely (extensive margin), while depositors who remain increase deposits (intensive margin).
  - Aggregate outcome in baseline: extensive margin dominates, leading to an about 4% loss in deposits (aggregate deposits fall from 15.4% to 11.2% of total wealth).
  - Share of non-liquid wealth increases to about 84.9% as share of liquid assets falls.
  - CBDC (φC small) substantially improves financial inclusion: financially included households rise from 79.5% to 100%.

### Illustration of margins (descriptive results)
- Figure 1 (described):
  - D/W plotted by household initial wealth W0 shows flat segments due to homothetic preferences.
  - Before CBDC: households with wealth < WA (20.5% of households, financially excluded) hold only cash; households with wealth > WA hold cash + deposits.
  - After CBDC: cutoff shifts from WA to WB; households between WA and WB used to hold deposits but switch to holding CBDC only (extensive margin reduction in depositors).
  - Households who keep deposits hold a higher D/W after CBDC (intensive margin).
  - Red line (C/W) shows all households hold CBDC in baseline because φC is very small; poorest households move from cash-only to holding CBDC.
- Figure 2 (described):
  - Deposits by wealth level D f(W) before (dotted blue) and after (solid red) CBDC.
  - Blue area: deposits lost from households with wealth between WA and WB (extensive margin).
  - Red area: deposits gained from richer households increasing holdings due to higher rD (intensive margin).
  - Net effect in baseline: extensive margin area larger than intensive margin area, net decline in aggregate deposits of 4.23 percentage points of initial aggregate deposits.

### Main takeaway
- In the representative-household (λ → 0) framework, CBDC introduction unambiguously reduces deposit spreads and increases aggregate deposits because banks raise deposit remuneration and households diversify liquidity.
- When heterogeneous wealth and fixed access costs are introduced, two opposing margins arise:
  - Intensive margin (existing depositors increase deposits because banks raise rD).
  - Extensive margin (some previously banked households close accounts and switch to CBDC).
- Quantitatively, with the baseline calibration, extensive margin dominates, lowering aggregate deposits while increasing financial inclusion to 100% and inducing banks to raise deposit rates by 47 basis points (from 2.25% to 2.72%).

*Source: wpiea2023236-print-pdf - 2.3    Impact of CBDC introduction*

### 3.5    Key mechanisms driving results

### 3.5    Key mechanisms driving results

### Mechanisms determining extensive vs. intensive margins
- Two crucial parameters:
  - Relative fixed costs of access: φC/φD.
  - Distribution of initial wealth governed by α.
- When φC << φD and the mass of poorer households is large (α is high), the extensive margin dominates the intensive margin and aggregate deposits decline in equilibrium.
- Intuition:
  - CBDC increases competition for deposits; banks raise interest on deposits (reduce deposit spread) which encourages households with bank accounts to increase deposits (intensive margin).
  - With wealth heterogeneity and differing fixed access costs, some poorer households stop holding deposits and switch to CBDC (extensive margin). If banks have insufficient incentives to aggressively raise deposit rates to retain these poorer households, the extensive margin can more than offset the intensive margin and reduce aggregate deposits.
- Ease of access to CBDC (lower φC relative to φD) is likely to shape banks’ funding and profitability outcomes.
- Figure 3 (parameter space) summary:
  - Deposit disintermediation occurs for high values of α and/or high access costs to CBDC relative to costs of deposits (φC/φD).

### Results from alternative calibrations (Table 3)
- Two alternative calibrations shown: High CBDC cost (c = 0.03) and High alpha (α = 1.62). For each, results without CBDC and with CBDC are reported.
- High CBDC cost (c = 0.03)
  - No CBDC / CBDC:
    - Cash (%) : 1.8% / 0.0%
    - CBDC (%) : 1.6% / 0.0%
    - Deposits (%) : 15.4% / 15.5%
    - Non liquid wealth (%) : 82.8% / 82.7%
    - Interest on deposits : 2.25% / 2.67%
    - Banks profit : 0.0012 / 0.0005
    - Financial inclusion (%) : 79.5% / 100%
    - % Deposits for those with bank : 16.6% / 19.2%
    - % Wealth held by those with bank account : 92.5% / 80.7%
    - Intensive margin* : 2.08 p.ps.
    - Extensive margin** : -1.96 p.ps.
- High alpha (α = 1.62)
  - No CBDC / CBDC:
    - Cash (%) : 1.9% / 0.5%
    - CBDC (%) : 0.0% / 3.9%
    - Deposits (%) : 15.6% / 10.3%
    - Non liquid wealth (%) : 82.5% / 85.3%
    - Interest on deposits : 2.32% / 2.74%
    - Banks profit : 0.0011 / 0.0003
    - Financial inclusion (%) : 80.4% / 100%
    - % Deposits for those with bank : 17.0% / 19.9%
    - % Wealth held by those with bank account : 92.0% / 51.6%
    - Intensive margin* : 1.52 p.ps.
    - Extensive margin** : -6.86 p.ps.
- Interpretation:
  - With relatively higher costs to access CBDC (High CBDC cost), fewer households drop bank accounts; the intensive margin (2.08 p.ps.) outweighs the extensive margin (-1.96 p.ps.), and total deposits grow slightly (by 0.1 percentage points).
  - With larger mass of poorer households (High alpha), disintermediation is higher: extensive margin (-6.86 p.ps.) dominates and total deposits fall more.

- Definitions:
  - Intensive margin*: percent change in deposits for those with a bank account times the share of total wealth held by those with a bank account after CBDC is introduced.
  - Extensive margin**: percent change in wealth held by those with a bank account times the percent of deposits held for those with a bank account before CBDC is introduced.

### Robustness: remunerated CBDC and CBDC liquidity value
- Interest-bearing CBDC (Figure 4):
  - The change in aggregate deposits is linear in the remuneration of CBDC.
  - Increasing rC from zero to 0.7% makes aggregate deposits drop by a further 1 percentage point compared to a non-remunerated CBDC; remuneration of deposits increases by a further 6 basis points (Tables 2 and 4). This further decreases banks’ profits.
  - Decreasing the interest on CBDC leads to a smaller drop in aggregate deposits relative to a non-remunerated CBDC.
  - The model also works with negative returns on CBDC.
- Comparative statics on δC (Figure 5):
  - Baseline calibration: δN = 1, δC = 2, δD = 1.5.
  - Higher δC (CBDC provides better liquidity services) leads to a larger drop in aggregate deposits because CBDC becomes a better substitute for bank deposits.

### Results for a remunerated CBDC (Table 4, rC = 3%)
- No CBDC / CBDC:
  - Cash (%) : 1.8% / 0.3%
  - CBDC (%) : 0.0% / 4.4%
  - Deposits (%) : 15.4% / 10.4%
  - Non liquid wealth (%) : 82.8% / 84.9%
  - Interest on deposits : 2.25% / 2.78%
  - Banks profit : 0.0012 / 0.0002
  - Financial inclusion (%) : 79.5% / 100%
  - % Deposits for those with bank : 16.6% / 20.4%
  - % Wealth held by those with bank account : 92.5% / 51.2%
  - Intensive margin* : 1.97 p.ps.
  - Extensive margin** : -5.88 p.ps.

### Effects on bank lending and wholesale funding (extension)
- Banking extension: banks fund with deposits (Di) and wholesale funding (Hi), lend Li, and lending is “unproductive” given to firms outside the economy. Bank problem and first order conditions are specified; lending co-moves with deposits unless wholesale funding is riskless (h = 0).
- Key analytical result:
  - Equation (23) shows Hi and Li as functions of Di; if h = 0 (riskless wholesale funding at policy rate), lending is constant after CBDC introduction. If h > 0 and profitable lending opportunities exist (l0/l1 sufficiently high), banks use wholesale funding to offset deposit losses and protect lending.
- Quantitative results and Table 5:
  - Parameter choices: l0 = 0.001, l1 = 0.001, h = 0.00002.
  - No CBDC / CBDC (Table 5):
    - Cash (%) : 1.7% / 0.4%
    - CBDC (%) : 0.0% / 3.4%
    - Deposits (%) : 15.6% / 11.7%
    - Non liquid wealth (%) : 82.7% / 84.5%
    - Interest on deposits : 2.3% / 2.8%
    - Banks profit : 0.0011 / 0.0003
    - Financial inclusion (%) : 80.6% / 100%
    - % Deposits for those with bank : 16.8% / 20.2%
    - % Wealth held by those with bank account : 92.9% / 57.9%
    - Intensive margin * : 1.90%
    - Extensive margin ** : -6.88%
    - Change in Lending : -0.18%
  - Textual summary in Section 4.2 (same section) reports: introduction of CBDC leads to a drop in aggregate deposits of 4 percentage points of total wealth, but lending drops only by 0.14%. By contrast, the percentage drop is 4.23 percentage points in the model with heterogeneous households in Section 3.
- Interpretation:
  - Main qualitative result holds: CBDC introduction can reduce deposits under the baseline calibration, but the drop in lending is quantitatively small because banks can substitute wholesale funding for lost deposits or exploit profitable lending opportunities.
  - More generally, drop in lending will be small as long as banks have convenient alternative funding sources to deposits or sufficiently profitable lending opportunities.

*Italic source: wpiea2023236-print-pdf - 3.5    Key mechanisms driving results*

### References

### References and Appendices (wpiea2023236-print-pdf - References)

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- Drechsler, I., Savov, A., and Schnabl, P. (2017). The deposits channel of monetary policy. The Quarterly Journal of Economics, 132(4):1819–1876. [1, 1, 2.2, 2.2, 2.2, 4.2]
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### Appendix A — Comparative statics for the model with homogeneous households
- Quantitative implications of CBDC introduction for bank deposits and the deposit spread depend on:
  - the level of the policy rate f,
  - remuneration offered by the CBDC (r_C),
  - the level of market power in the banking sector.
- Key comparative-statics results (preserving notation and exact inequality signs):
  - Effect of higher CBDC rate r_C:
    - ∂|∆s|/∂r_C > 0  (Equation (25))
    - ∂∆D/∂r_C > 0  (Equation (26))
    - Interpretation: A higher r_C forces banks to raise deposit rates to retain households, increasing the decline in the deposit spread |∆s| and increasing aggregate deposits ∆D.
  - Effect of higher policy rate f:
    - ∂|∆s|/∂f > 0  (Equation (27))
    - ∂∆D/∂f < 0  (Equation (28))
    - Interpretation: A higher f raises the decline in the deposit spread but reduces the increase in aggregate deposits after CBDC introduction.
  - Explanation of counterintuitive aggregate-deposits response:
    - Aggregate deposits expressed as D* = δ^ϵ_D (s*/s_l)^(-ϵ) s_l^(-ρ) and  ̃D* = δ^ϵ_D ( ̃s*/ ̃s_l)^(-ϵ) ̃s_l^(-ρ), with ̃s_l ≡ (f^(1−ϵ) + δ^ϵ_D ( ̃s* )^(1−ϵ))^(1/(1−ϵ)).
    - Following CBDC introduction, ratios s*/s_l and ̃s*/̃s_l equalize; differences in D* and ̃D* responses to f arise from how overall demand for liquidity (̃s_l^(−ρ) and s_l^(−ρ)) changes with f.
  - Effect of higher competition (elasticity of substitution η or number of banks J):
    - ∂|∆s|/∂η < 0,  ∂|∆s|/∂J < 0  (Equation (29))
    - ∂∆D/∂η > 0,  ∂∆D/∂J > 0  (Equation (30))
    - Interpretation: More competition implies a smaller decline in the deposit spread and a larger increase in aggregate deposits after CBDC introduction because (i) the baseline deposit rate is higher (less bank market power), and (ii) the aggregate elasticity of deposits with respect to the deposit rate is higher.

- Figure referenced: "Figure 6: Introduction of CBDC (red lines): impact on deposit spread and on deposits as a function of the policy rate f." (panels: (a) deposit spread, (b) aggregate deposits). Axes ticks shown in source: 0.02, 0.04, 0.06, 0.08, 0.10 and vertical values 0.005, 0.010, 0.015, 0.020, 0.025, 0.030, 0.035 (panel a); and x-axis ticks 0.02, 0.04, 0.06, 0.08, 0.10 with y-axis labels 10, 15, 20, 25, 30 (panel b).

### Appendix B — Equilibrium in model with household heterogeneity
- Indirect utility linearity:
  - Indirect utility U(W0) expressed as:
    - U(W0) = W0 (1+f) t(s_l)^(−1) (φ)  (Equation (31)), with t(s_l) definitions dependent on asset choices.
  - Definition: t(s_l) ≡ (1 + λ ρ s_l^(1−ρ))^(1/(ρ−1)).
- Asset-choice cases and corresponding t(s_l) expressions:
  - If C=0 and D=0: 1(s_l) = δ^ϵ_N f^(1−ϵ)
  - If C>0 and D=0: 1(s_l) = (δ^ϵ_N f^(1−ϵ) + δ^ϵ_C (f−r_C)^(1−ϵ))^(1/(1−ϵ))
  - If D>0 and C=0: 1(s_l) = (δ^ϵ_N f^(1−ϵ) + δ^ϵ_D s^(1−ϵ))^(1/(1−ϵ))
  - If C>0 and D>0: 1(s_l) = (δ^ϵ_N f^(1−ϵ) + δ^ϵ_D s^(1−ϵ) + δ^ϵ_C (f−r_C)^(1−ϵ))^(1/(1−ϵ))
- Fixed costs and scenario enumeration:
  - Fixed costs φ_C, φ_D with φ_D > φ_C yield four possible ordering scenarios of household choices across wealth:
    1. N → B (poor: only cash N; rich: cash+CBDC+deposits B)
       - Conditions: ˆW13 ≤ ˆW11, ˆW13 ≤ ˆW12
    2. N → C → B
       - Conditions: ˆW11 < ˆW13, ˆW11 < ˆW12, ˆW22 ≤ ˆW21 or t(s_D^l) ≤ t(s_C^l)
    3. N → D → B
       - Conditions: ˆW12 < ˆW13, ˆW12 ≤ ˆW11, t(s_D^l) > t(s_C^l)
    4. N → C → D → B
       - Conditions: ˆW11 < ˆW13, ˆW11 < ˆW12, ˆW21 < ˆW22, t(s_D^l) > t(s_C^l)
- Cutoff-wealth expressions (preserving exact formulas from source):
  - Scenario-specific cutoffs:
    - ˆW1 = φ_C (1+f) ( t(s_C^l) − t(s_N^l) )
    - ˆW2 = φ_D (1+f) ( t(s_B^l) − t(s_C^l) )
    - In the preferred parametrization discussed, households with W0 < ˆW1 choose cash only; ˆW1 < W0 < ˆW2 choose cash and CBDC; W0 > ˆW2 choose cash, CBDC, and deposits.
  - General cutoff formulas:
    - ˆW11 = φ_C (1 + f) ( t(s_C^l) − t(s_N^l) )
    - ˆW12 = φ_D (1 + f) ( t(s_D^l) − t(s_N^l) )
    - ˆW13 = (φ_D + φ_C) (1 + f) ( t(s_B^l) − t(s_N^l) )
    - ˆW21 = φ_D − φ_C (1 + f) ( t(s_D^l) − t(s_C^l) )
    - ˆW22 = φ_D (1 + f) ( t(s_B^l) − t(s_C^l) )
- Aggregate deposits with heterogeneity (equilibrium where some households hold both CBDC and deposits):
  - D = δ^ϵ_D ( s_B^l / s )^ϵ λ ρ ( s_B^l )^(−ρ) (1 + λ ρ (s_B^l)^(1−ρ)) (1 + f) × ∫_{W0}^{ˆW2} dF(W0)  (Equation (32))
  - Interpretation: Aggregate deposits change via both intensive and extensive margins when some households hold both CBDC and deposits.
- Deposit spread equilibrium condition (exact expression preserved):
  - M = ϵ ( ( δ^ϵ_N f^(1−ϵ) + δ^ϵ_C (f−r_C)^(1−ϵ) ) / ( δ^ϵ_N f^(1−ϵ) + δ^ϵ_D s^(1−ϵ) + δ^ϵ_C (f−r_C)^(1−ϵ) ) )
    + ρ ( ( δ^ϵ_D s^(1−ϵ) ) / ( δ^ϵ_N f^(1−ϵ) + δ^ϵ_D s^(1−ϵ) + δ^ϵ_C (f−r_C)^(1−ϵ) ) )
    + (1−ρ) ( s_B^l )^(1−ρ) λ^(−ρ) + (s_B^l)^(1−ρ) ( ( δ^ϵ_D s^(1−ϵ) ) / ( δ^ϵ_N f^(1−ϵ) + δ^ϵ_D s^(1−ϵ) + δ^ϵ_C (f−r_C)^(1−ϵ) ) )
    + (α−1) λ^ρ ( t(s_B^l) )^(2−ρ) ( s_B^l )^(1−ρ) ( t(s_B^l) − t(s_C^l) ) ( ( δ^ϵ_D s^(1−ϵ) ) / ( δ^ϵ_N f^(1−ϵ) + δ^ϵ_D s^(1−ϵ) + δ^ϵ_C (f−r_C)^(1−ϵ) ) ) × I_{ˆW1 > W}  (Equation (33))
- Ordering of indirect-utility slopes and intercepts (implications for choice by wealth):
  - Slopes: B > C, D > N
  - Intercepts: B < D < C < N
  - Consequence: poorest households choose N; richest households choose B; middle households may choose C or D.

*Source: wpiea2023236-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023236-print-pdf.pdf_