## 2.1 Agents and their optimal strategies (content unit: wpiea2019252-print-pdf)

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### Households, banks, firms, and central bank — setup and first-order conditions
- Households (2.1.1)
  - Unit continuum of households indexed i ∈ [0;1] uniformly distributed over anonymity-security scale.
  - Initial endowments: atomistic shares in firms and liquid funds normalized to 1.
  - Liquid funds stored in money type j ∈ {c; d; cbdc} and used to purchase consumption good at end of stage.
  - Anonymity undiversifiable: a transaction is only as anonymous as the least anonymous instrument used.
  - Budget constraint:
    - C_j = 1 + r_j − T + ϖ  (equation (2))
  - Interest rates on money holdings:
    - r_j =
      - 0 if j = c
      - r_cbdc if j = cbdc
      - r_d if j = d
    (equation (3))
  - Lump-sum tax/transfer:
    - T = r_cbdc s_cbdc
  - Utility maximization:
    - max_{j ∈ {c; d; cbdc}} U_i(j) = −ζ C_j − j x_j − |i − j| − Φ_j  (equation (4))
    - ζ > 0 is marginal utility of consumption relative to payment preferences.
    - Φ_j captures network effects.
  - Cut-off conditions (choice boundaries):
    - cash over CBDC: 1 − i + Φ_c < j̄ − |i − j̄| − ζ r_cbdc + Φ_cbdc  (equation (5))
    - cash over deposits: 1 − i + Φ_c < i − ζ r_d + Φ_d  (equation (6))
    - CBDC over deposits: j̄ − |i − j̄| − ζ r_cbdc + Φ_cbdc < i − ζ r_d + Φ_d  (equation (7))

- Banks (2.1.2)
  - Banks collect deposits d at net deposit rate r_d and extend loans l to firms at net loan rate R with budget constraint:
    - l = d  (equation (8))
  - Representative bank risk neutral, price-taker; profit maximization yields:
    - r_d = R  (equation (9))

- Firms (2.1.3)
  - Perfect competition; identical endowment of productive projects k_0 requiring financing.
  - Representative firm finances k = l with bank loans l.
  - Production technology:
    - Y = [A − (k/2)] k  (equation (10)), with A > 1
  - Remaining projects (k_0 − k) liquidated at constant return ϵ ∈ (0;1).
  - Profit maximization:
    - max_{k,l} Y + ϵ (k_0 − k) − (1 + R) l  (equation (11)) subject to (10) and k = l.
  - First order condition (loan demand):
    - 1 + R = A − ϵ − l  (equation (12))

- Central bank (2.1.4)
  - Social welfare:
    - W = ∫_{i∈[0;1]} U_i(j*(i)) di  (equation (13))
  - If CBDC introduced, design problem:
    - max_{κ ∈ [0;1]; r_cbdc} ∫_{i∈[0;1]} U_i(j*(i)) di  (equation (14))
  - Design constraint to ensure sufficient CBDC take-up:
    - s_cbdc ≥ s  (equation (15))
    - Constraint subsumes r_cbdc ≥ −(1 − κ) ζ^{−1} and κ > ζ (r_d − r_cbdc).
  - Implicit assumption: no short position in CBDC allowed.

### Equilibrium definition and parameter restrictions
- Competitive equilibrium clearing conditions:
  - s_d = d = l = k  (equation (16))
- Parameter restrictions for “well-behaved” equilibria:
  - 1 < (A − ϵ) ≤ 5/2; 3/4 ≤ ζ ≤ 3/2; s ≤ 1/17  (equation (17))
- Key lemmas (stated)
  - Lemma 1: Under optimal policies derived in Section 3, introduction of an optimally-designed CBDC always raises social welfare.
  - Lemma 2: Under optimal policies and parameter restrictions in (17), s_d ≥ s and s_cbdc ≥ s.
- Two well-behaved equilibria under optimal policy (Table 1):
  - Deposits & Cash & CBDC (ce)
  - Deposits & CBDC (cashless equilibrium, nce)

### Equilibrium shares and interest rates (closed-form expressions)
- Cash equilibrium shares:
  - s^ce_c = 1 − κ − ζ r_cbdc   2  (equation (18))
  - s^ce_d = ζ (r_d − r_cbdc) + κ   2  (equation (19))
  - s^ce_cbdc = 1 − ζ r_d   2 + ζ r_cbdc  (equation (20))
- Cashless equilibrium shares (s^nce_c = 0):
  - s^nce_d = ζ (r_d − r_cbdc) + κ   2  (equation (21))
  - s^nce_cbdc = 1 − ζ (r_d − r_cbdc) + κ   2  (equation (22))
- Deposit interest rate, deposit share, firm production, aggregate consumption (same across equilibrium types):
  - r_d = 2 (A − 1 − ϵ) − (κ − ζ r_cbdc)^2 + ζ   (equation (23))
  - s_d = ζ (A − 1 − ϵ − r_cbdc) + κ   2 + ζ  (equation (24))
  - Y = [A − s_d^2] s_d  (equation (25))
  - ∫_{i∈[0;1]} C_{j*(i)} di = 1 + ϵ k_0 + [A − ϵ − 1] s_d^2 s_d  (equation (26))

- Note on effects:
  - A CBDC that is deposit-like (low κ) and/or offers high r_cbdc crowds out deposits; banks raise r_d partially offsetting but overall bank intermediation, firm production and aggregate consumption decline.
  - Decline in aggregate consumption is larger when (A − ϵ) is larger.

### Multiplicity and boundary condition
- Multiplicity resolved in favor of cash equilibrium unless boundary violated:
  - s^ce_c ≥ s  (equation (27))
- Boundary in design parameters:
  - κ + ζ r_cbdc ≤ 1 − 2 s  (equation (28))
  - A cash-like CBDC (high κ) and/or high r_cbdc may eliminate cash and produce a cashless equilibrium.

### Comparative statics (2.2.1) — intuition
- κ variation (Panel A):
  - As CBDC becomes more cash-like (higher κ): cash holdings decline and bank deposits rise; CBDC share initially rises.
  - If κ high enough so cash use falls below threshold s̄, network effects cause a cashless equilibrium and CBDC use jumps up.
  - Further increases in κ can cause marginal households to switch toward deposits, raising deposits and reducing CBDC use.
  - Domain where CBDC design constraint violated leads CBDC to fall out of use.
- r_cbdc variation (Panel B):
  - Higher r_cbdc reduces shares of both cash and deposits, and raises CBDC share.
  - Banks raise r_d in response, so deposits decline less than cash.
  - Sufficiently high r_cbdc leads to cashless equilibrium.
- Introducing CBDC is equivalent to moving κ from 1 to a lower value and/or changing r_cbdc away from 0 (κ = 1, r_cbdc = 0 represents CBDC identical to cash). Any positive uptake of CBDC necessarily derives from both cash and deposits, causing some decline in banks’ deposit base and bank intermediation.

### Welfare decomposition (2.2.2)
- Social welfare:
  - W = ζ ∫ C_{j*(i)} di − ∫ |x_{j*(i)} − i| di  (equation (29))
  - First term: aggregate consumption (related to bank intermediation and s_d).
  - Second term: welfare losses from distance between households’ payment preferences and chosen instrument (social value of variety).
- Closed-form welfare (Lemma 3):
  - W_ce(κ; r_cbdc) =
    4 ζ − (A − ϵ − 1/2) κ + 4 (1 − κ) κ − 3 ζ κ^2 − (4 + ζ) ζ^2 r_cbdc^2 + !1 4 (2 + ζ)  (equation (30))
  - W_nce(κ; r_cbdc) =
    2 ζ (A − ϵ) κ + (4 − 3 κ) κ − 2 ζ κ^2 − ζ^2 r_cbdc^2 + !2 2 (2 + ζ)  (equation (31))
  - !1 and !2 are collections of constants.
- Intuition from W_ce:
  - Term 4 ζ − (A − ϵ − 1/2) κ: when (A − ϵ) larger, welfare depends more on bank intermediation, calling for more cash-like CBDC (high κ).
  - Terms 4 (1 − κ) κ − 3 ζ κ^2: relate to variety; imply an interior maximum (intermediate κ best).
  - Term −(4 + ζ) ζ^2 r_cbdc^2: negative quadratic in r_cbdc; r_cbdc = 0 maximizes welfare within a given equilibrium type.
- Accounting for network effects and switching:
  - W(κ; r_cbdc) = ζ W_ce(κ; r_cbdc) if κ + ζ r_cbdc ≤ 1 − 2 s; otherwise W_nce(κ; r_cbdc)  (equation (34))

### Preview of Section 3 — optimal CBDC design
- Two-step analysis:
  - First analyze optimal r_cbdc = 0 case.
  - Then analyze effects of r_cbdc ≠ 0.
- Many central banks constrain to non interest-bearing CBDC for political-economy or legal reasons.

---

### 3.1 Non interest-bearing CBDC — optimality without r_cbdc

### Optimal design and interiority (r_cbdc = 0)
- Problem: when r_cbdc = 0, choose ξ ∈ [0,1] to max { W(ξ;0) }  (equation (35)).
- Optimal CBDC similarity parameters:
  - ξ_ce = (2 + ζ(2(A−ε)−1)) / (4 + 3ζ)  (equation (36))
  - ξ_nce = (2 + ζ(A−ε)) / (3 + 2ζ)  (equation (37))
- Parameter restrictions (17) ensure ξ_ce and ξ_nce are well-defined on [0,1].
- Interpretation: interior CBDC design equates marginal benefit from bank intermediation with marginal losses to payments-system variety.

### Welfare comparison (no network effects)
- Cash equilibrium welfare dominates cashless equilibrium when network effects absent:
  - W_ce(ξ_ce;0) − W_nce(ξ_nce;0) = (ζ(A−ε−2)−1)^2 / [2 (3 + 2ζ) (4 + 3ζ)] > 0  (equation (38))
- Implication: if sustaining variety is costless, welfare maximized with all three instruments in use.

### Network effects as a constraint
- Network-effect constraint (28) yields upper bound ξ̄ on cash-likeness:
  - ξ̄ < 1 − 2 s
- If ξ_ce > ξ̄, unconstrained ξ_ce infeasible (economy would move to cashless equilibrium).
- Optimization becomes constrained: preserve cash at ξ̄ versus allow cash to vanish and use unconstrained ξ_nce.
- As A−ε rises:
  - Up to threshold A−ε̄, optimal design “freezes” to preserve cash and variety.
  - Beyond A−ε̄, policy may forgo variety and allow cash to disappear to protect deposits.

### Comparative statics and shape properties
- From (36) and (37):
  - ξ_nce > ξ_ce.
  - ∂ξ_ce/∂(A−ε) < ∂ξ_nce/∂(A−ε) < 0 (ξ_nce slope flatter).
- If cash already disappeared prior to CBDC introduction, network effects no longer bind and optimal CBDC is more cash-like than when cash exists.

---

### 3.2 Interest-bearing CBDC — adding r_cbdc as policy tool

### Unconstrained optimality (first-order conditions)
- Unconstrained optimal policy solves ∂W/∂ξ = 0 and ∂W/∂r_cbdc = 0.
- Yields same ξ_ce and ξ_nce as (36) and (37) and:
  - r_ce_cbdc = r_nce_cbdc = 0  (equation (40))
- Implication: absent network effects, optimal CBDC rate is zero; interest rate is suboptimal compared to ξ.

### When network effects bind — constrained optimum
- Constraint (28) affects both ξ and r_cbdc. When binding, optimal constrained design:
  - ξ̂ = [2 + ζ(2(A−ε)−1 − ζ(4 + ζ)(2s−1))] / [4 + ζ(1 + ζ)(3 + ζ)]  (equation (41))
  - r̂_cbdc = −2[ζ((A−ε) + 3s −2) + 4s −1] / [4 + ζ(1 + ζ)(3 + ζ)]  (equation (42))
- Under parameter restrictions (17), optimal CBDC rate when constrained is always negative.
- Negative r_cbdc allows ξ̂ > ξ̄ as A−ε rises.
- Access to r_cbdc strictly raises welfare when network effects bind because (ξ̄;0) is feasible but sub-optimal.

### Proposition 1 — cutoff ζ̄ and cases
- There exists ζ̄ ∈ (2/3, 3/2) such that:
  - If ζ < ζ̄:
    - Cash never vanishes under optimally designed interest-bearing CBDC.
    - Optimal design is (ξ̂; r̂_cbdc) if ξ_ce + ζ r_ce_cbdc < 1 − 2s; otherwise (ξ_ce; r_ce_cbdc)  (equation (43)).
  - If ζ > ζ̄:
    - Cash can vanish when A−ε exceeds threshold Â−ε; and Â−ε > A−ε̄ (threshold higher when CBDC is interest-bearing).
- Economic intuition:
  - ζ < ζ̄ (payments-preference weight large): CBDC interest rate as second tool enables sustaining variety while limiting bank disintermediation by combining a more cash-like CBDC with a negative CBDC rate.
  - ζ > ζ̄ (payments-preference weight small): negative CBDC rates create larger payment-choice distortions; beyond some A−ε welfare is maximized by letting cash vanish and switching to (ξ_nce; r_nce_cbdc).

---

### 3.3 Distributional effects — who wins and who loses

### Aggregate vs distributional impacts
- Aggregate welfare rises with optimally designed CBDC, but change is not Pareto-improving: some households gain and others lose.
- Welfare impact varies across household payment-preference index i ∈ [0,1].

### Non interest-bearing CBDC (blue-line profile)
- Low-i (deposit-preferring) households:
  - Remain deposit users.
  - Affected by reduced financial intermediation (lower firm profits π) and higher deposit rates r_d due to CBDC competition.
  - Net effect: deposit-rate increase dominates → consumption and welfare of deposit users rise.
- High-i (cash-preferring) households:
  - Remain cash users.
  - Cash pays no interest; decline in π reduces consumption and welfare.
  - If CBDC drives cash out via network effects, these households suffer further welfare loss proportional to their anonymity preference.
- Switchers (deposits → CBDC):
  - Experience net welfare gain by construction: switch only if payment-preference gains exceed interest-loss r_d.
  - Household at i = ξ experiences largest welfare increase.
- Marginal CBDC-over-cash households:
  - Net welfare effect negative because CBDC holders also face fall in consumption via reduced π.
- Aggregate pattern: define boundary ī such that households i ∈ [0,ī] gain and i ∈ (ī,1] lose; aggregate gains of first group exceed losses of latter.

### Interest-bearing CBDC with slightly negative rates (red-line profile)
- Three determining factors:
  1. Revenues from negative CBDC rates are lump-sum transferred to all households, redistributing gains from CBDC users to cash and deposit users.
  2. Negative CBDC rates increase deposits and financial intermediation → firm profits π rise → benefits all households; but deposit rates r_d decrease → depositors hurt. Lump-sum transfer dominates: overall CBDC users lose; deposit and cash users gain.
  3. Negative CBDC rates preventing cash disappearance avert large welfare losses for cash holders.
- Distributional interpretation:
  - Deposit holders are winners and cash holders are losers under non-interest-bearing CBDC, suggesting potentially regressive impact in practice (cash users tend to be lower income).
  - Interest-bearing CBDC with negative rates can mitigate losses for cash holders via redistribution and preserving cash.

---

### 4 Conclusion and policy implications — concise takeaways
- Two key CBDC design choices: (i) degree of cash-likeness (ξ), and (ii) whether CBDC is interest-bearing (r_cbdc).
- Non interest-bearing CBDC tradeoffs:
  - Making CBDC cash-like risks reducing cash demand below network thresholds, endangering payment-instrument variety valued by heterogeneous households.
  - Making CBDC deposit-like threatens banks’ deposit base, harming credit provision and output, especially when banks alleviate lending frictions.
- Optimal directions:
  - If banks’ role limited, design CBDC distinct from existing instruments.
  - Preserving intermediation pushes CBDC more cash-like, but network effects can constrain this.
  - Only when preserving deposits paramount will central bank forgo cash and jump toward more cash-like CBDC.
- Role of interest-bearing CBDC:
  - When network effects matter, r_cbdc helps alleviate tradeoffs by using r_cbdc as second tool.
  - Absent network effects, optimal r_cbdc = 0; limiting CBDC to non interest-bearing is without loss.
  - When network effects threaten variety, adjustable r_cbdc becomes optimal: negative r_cbdc can make CBDC more cash-like while making it less attractive relative to cash, protecting both cash use and bank intermediation.
- Practical note: political-economy concerns about negative rates on public central bank liabilities may push toward non interest-bearing CBDCs, but economic analysis suggests central banks should consider retaining an adjustable CBDC interest rate to weigh benefits against political costs.

---

*Source: IMF Working Paper — section "2.1 Agents and their optimal strategies" and Sections 3–4 (content unit: wpiea2019252-print-pdf).*

### 2.1  Agents and their optimal strategies

### 2.1  Agents and their optimal strategies

### Households (2.1.1)
- There is a unit continuum of households with preferences i ∈ [0;1] uniformly distributed over the anonymity-security scale.
- Households start with identical endowments in terms of (atomistic) shares in firms and liquid funds normalized to 1.
- Liquid funds are stored in a money type j ∈ {c; d; cbdc} and used to purchase the consumption good at the end of the stage.
- Anonymity is undiversifiable: a transaction is only as anonymous as the least anonymous payment instrument used.
- Budget constraint:
  - C_j = 1 + r_j − T + ϖ (equation (2))
- Interest rates on money holdings:
  - r_j =
    - 0 if j = c
    - r_cbdc if j = cbdc
    - r_d if j = d
  (equation (3))
- Lump-sum tax/transfer:
  - T = r_cbdc s_cbdc (used to fund interest rates on CBDC or redistribute revenues when r_cbdc < 0)
- Household utility maximization:
  - max_{j ∈ {c; d; cbdc}} U_i(j) = −ζ C_j − j x_j − |i − j| − Φ_j  (equation (4), with notation in text)
  - ζ > 0 denotes marginal utility of consumption relative to payment preferences.
  - Φ_j captures network effects (as defined by equation (1) in source).
- Cut-off conditions (household choice boundaries):
  - cash over CBDC: 1 − i + Φ_c < j̄ − |i − j̄| − ζ r_cbdc + Φ_cbdc  (equation (5))
  - cash over deposits: 1 − i + Φ_c < i − ζ r_d + Φ_d  (equation (6))
  - CBDC over deposits: j̄ − |i − j̄| − ζ r_cbdc + Φ_cbdc < i − ζ r_d + Φ_d  (equation (7))

### Banks (2.1.2)
- Banks collect deposits d from households at net deposit rate r_d and extend loans l to firms at net loan rate R with budget constraint:
  - l = d  (equation (8))
- Representative bank is risk neutral and price-taker; profit maximization yields first order condition:
  - r_d = R  (equation (9))
  - This equates interest rates on deposits and loans.

### Firms (2.1.3)
- Firms are perfectly competitive and begin with identical endowment of productive projects k_0 requiring financing.
- Representative firm finances portion k = l with bank loans l.
- Production technology:
  - Y = [A − (k/2)] k  (equation (10)), with A > 1
- Remaining projects (k_0 − k) liquidated at constant return ϵ ∈ (0;1).
- Firm profit maximization:
  - max_{k,l} Y + ϵ (k_0 − k) − (1 + R) l  (equation (11)) subject to (10) and k = l.
- First order condition (loan demand):
  - 1 + R = A − ϵ − l  (equation (12))
  - Decline in ϵ raises firm demand for loans.
- Interpret (A − ϵ) as the value added from bank intermediation.

### Central bank (2.1.4)
- Central bank maximizes social welfare W = ∫_{i∈[0;1]} U_i(j*(i)) di  (equation (13)), choosing whether to introduce CBDC and, if so, design characteristics (κ; r_cbdc).
- If CBDC introduced, design problem:
  - max_{κ ∈ [0;1]; r_cbdc} ∫_{i∈[0;1]} U_i(j*(i)) di  (equation (14))
- Design constraint to ensure sufficient take-up to overcome network effects:
  - s_cbdc ≥ s  (equation (15))
  - Design constraint subsumes r_cbdc ≥ −(1 − κ) ζ^{−1} and κ > ζ (r_d − r_cbdc) (text explanation).
- Implicit assumption: central bank does not allow agents to take a short position in CBDC (no CBDC credit to other parties).

### Equilibrium definition (2.2)
- Competitive equilibrium clearing conditions:
  - s_d = d = l = k  (equation (16))
- Parameter restrictions for “well-behaved” equilibria (used throughout):
  - 1 < (A − ϵ) ≤ 5/2; 3/4 ≤ ζ ≤ 3/2; s ≤ 1/17  (equation (17))
- Key lemmas (stated results from paper):
  - Lemma 1: Under optimal policies derived in Section 3, the introduction of an optimally-designed CBDC always raises social welfare.
  - Lemma 2: Under optimal policies and parameter restrictions in (17), s_d ≥ s and s_cbdc ≥ s.
- Table 1 (qualitative result): only two well-behaved equilibria remain under optimal policy:
  - Deposits & Cash & CBDC (denoted ce)
  - Deposits & CBDC (cashless equilibrium, denoted nce)

### Equilibrium shares and interest rates
- Cash equilibrium shares (solved from (5)-(7) assuming uniform distribution):
  - s^ce_c = 1 − κ − ζ r_cbdc   2  (equation (18))
  - s^ce_d = ζ (r_d − r_cbdc) + κ   2  (equation (19))
  - s^ce_cbdc = 1 − ζ r_d   2 + ζ r_cbdc  (equation (20))
- Cashless equilibrium (s^nce_c = 0) shares:
  - s^nce_d = ζ (r_d − r_cbdc) + κ   2  (equation (21))
  - s^nce_cbdc = 1 − ζ (r_d − r_cbdc) + κ   2  (equation (22))
- Deposit interest rate, deposit share, firm production, and aggregate consumption (same across equilibrium types):
  - r_d = 2 (A − 1 − ϵ) − (κ − ζ r_cbdc)^2 + ζ   (equation (23))
  - s_d = ζ (A − 1 − ϵ − r_cbdc) + κ   2 + ζ  (equation (24))
  - Y = [A − s_d^2] s_d  (equation (25))
  - ∫_{i∈[0;1]} C_{j*(i)} di = 1 + ϵ k_0 + [A − ϵ − 1] s_d^2 s_d  (equation (26))  (text gives structure; exact algebra preserved as in source)
- Note on effects:
  - A CBDC that is deposit-like (low κ) and/or offers high r_cbdc crowds out deposits; banks partially offset by raising r_d, but overall bank intermediation, firm production and aggregate consumption decline.
  - Decline in aggregate consumption is larger when (A − ϵ) is larger.

### Multiplicity and boundary condition
- Multiplicity resolved in favor of a cash equilibrium (pre-digital economy) unless boundary violated:
  - s^ce_c ≥ s  (equation (27))
- Rewriting boundary in design parameters:
  - κ + ζ r_cbdc ≤ 1 − 2 s  (equation (28))
  - A cash-like CBDC (high κ) and/or high r_cbdc may eliminate cash and produce a cashless equilibrium.

### Comparative statics (2.2.1)
- Panel A intuition (κ variation):
  - As CBDC becomes more cash-like (higher κ): cash holdings decline and bank deposits rise; CBDC share initially rises.
  - If κ high enough so cash use falls below threshold s̄, network effects cause a cashless equilibrium and CBDC use jumps up.
  - Further increases in κ can cause marginal households to switch toward deposits, raising deposits and reducing CBDC use.
  - Left-end striped areas denote domain where CBDC design constraint is violated and CBDC falls out of use.
- Panel B intuition (r_cbdc variation):
  - Higher r_cbdc reduces shares of both cash and deposits, and raises CBDC share.
  - Banks raise r_d in response, so deposits decline less than cash.
  - Sufficiently high r_cbdc leads to cashless equilibrium (shaded area).
- Introducing CBDC is equivalent to moving κ from 1 to a lower value and/or changing r_cbdc away from 0 (κ = 1, r_cbdc = 0 represents CBDC identical to cash). Any positive uptake of CBDC necessarily derives from both cash and deposits, causing some decline in banks’ deposit base and bank intermediation; extent depends on how closely CBDC competes with deposits.

### Welfare analysis (2.2.2)
- Social welfare decomposition:
  - W = ζ ∫ C_{j*(i)} di − ∫ |x_{j*(i)} − i| di  (equation (29))
  - First term: aggregate consumption (related to bank intermediation and s_d).
  - Second term: welfare losses from distance between households’ payment preferences and chosen instrument (social value of variety).
- Closed-form welfare (Lemma 3):
  - W_ce(κ; r_cbdc) =
    4 ζ − (A − ϵ − 1/2) κ + 4 (1 − κ) κ − 3 ζ κ^2 − (4 + ζ) ζ^2 r_cbdc^2 + !1 4 (2 + ζ)  (equation (30))
  - W_nce(κ; r_cbdc) =
    2 ζ (A − ϵ) κ + (4 − 3 κ) κ − 2 ζ κ^2 − ζ^2 r_cbdc^2 + !2 2 (2 + ζ)  (equation (31))
  - !1 and !2 are collections of constants (defined in equations (32) and (33) in source).
- Intuition (from W_ce expression):
  - First term 4 ζ − (A − ϵ − 1/2) κ: link between value of bank intermediation (A − ϵ) and CBDC design κ — when (A − ϵ) larger, welfare depends more on bank intermediation, calling for a more cash-like CBDC (high κ).
  - Second/third terms 4 (1 − κ) κ − 3 ζ κ^2: relate to variety in payment instruments; they are linear-quadratic in κ and imply an interior maximum (intermediate κ best for variety).
  - Final term −(4 + ζ) ζ^2 r_cbdc^2: negative and quadratic in r_cbdc, implying a non interest-bearing CBDC (r_cbdc = 0) maximizes welfare within a given equilibrium type because CBDC interest distorts payment choices without contributing to productive intermediation.
- Accounting for network effects and equilibrium switching:
  - Social welfare is
    - W(κ; r_cbdc) = ζ W_ce(κ; r_cbdc) if κ + ζ r_cbdc ≤ 1 − 2 s
    - W(κ; r_cbdc) = W_nce(κ; r_cbdc) otherwise  (equation (34))
  - CBDC design (κ; r_cbdc) may induce switches between cash and cashless equilibria.

### Optimal CBDC design (section 3 preview)
- Two-step analysis described:
  - First analyze optimal design of non interest-bearing CBDC (r_cbdc = 0).
  - Then analyze how allowing interest on CBDC (r_cbdc ≠ 0) alters optimal design.
- Motivations noted:
  - Many central banks constrain themselves to non interest-bearing CBDCs due to political economy or legal hurdles (e.g., concerns about a public central bank liability paying negative interest; taxation and anonymity concerns).

*Source: IMF Working Paper — section "2.1 Agents and their optimal strategies" (content unit: wpiea2019252-print-pdf - 2.1  Agents and their optimal strategies).*

### 3.1  Non interest-bearing CBDC

### 3.1  Non interest-bearing CBDC

### Optimal design and interiority
- Problem: when r_cbdc = 0, choose ξ ∈ [0,1] to max { W(ξ;0) } (equation (35)).
- Optimal CBDC similarity parameters in cash and cashless equilibria:
  - ξ_ce = (2 + ζ(2(A−ε)−1)) / (4 + 3ζ) (equation (36))
  - ξ_nce = (2 + ζ(A−ε)) / (3 + 2ζ) (equation (37))
- Parameter restrictions (17) ensure ξ_ce and ξ_nce are well-defined on [0,1].
- Interpretation: optimal policy yields an interior CBDC design equating marginal benefit on bank intermediation with marginal losses to payments-system variety.

### Welfare comparison (no network effects)
- Cash equilibrium welfare dominates cashless equilibrium when network effects play no role:
  - W_ce(ξ_ce;0) − W_nce(ξ_nce;0) = (ζ(A−ε−2)−1)^2 / [2 (3 + 2ζ) (4 + 3ζ)] > 0 (equation (38))
  - Inequality strict because ζ(A−ε−2) ≠ 1 under parameter assumptions.
- Implication: if sustaining payment-instrument variety is costless, welfare is maximized with all three instruments in use.

### Network effects as a constraint
- Network-effect constraint (28) yields an upper bound ξ̄ on CBDC cash-likeness:
  - ξ̄ < 1 − 2s (equation (39) context)
- If ξ_ce > ξ̄, the unconstrained ξ_ce is infeasible because the economy would move to cashless equilibrium.
- Optimization becomes a constrained choice between preserving the cash equilibrium at ξ̄ versus allowing cash to vanish and using unconstrained policy (ξ_nce;0).
- As A−ε (value of bank intermediation) rises:
  - Up to a threshold A−ε̄, optimal design “freezes” to preserve cash and payment-instrument variety.
  - Beyond A−ε̄, policy may forgo variety and allow cash to disappear to protect bank deposits.

### Comparative statics and shape properties
- From (36) and (37):
  - ξ_nce > ξ_ce.
  - ∂ξ_ce/∂(A−ε) < ∂ξ_nce/∂(A−ε) < 0, so ξ_nce slope is flatter.
- If cash has already disappeared prior to CBDC introduction, network effects no longer bind and optimal CBDC is more cash-like than when cash exists.

---

### 3.2  Interest-bearing CBDC

### Unconstrained optimality and first-order conditions
- Unconstrained optimal policy solves:
  - ∂W(ξ;r_cbdc)/∂ξ = 0 and ∂W(ξ;r_cbdc)/∂r_cbdc = 0.
- This yields the same ξ_ce and ξ_nce as in (36) and (37) and
  - r_ce_cbdc = r_nce_cbdc = 0 (equation (40)).
- Implication: in absence of network effects, optimal CBDC rate is zero; CBDC interest rate is a suboptimal tool compared to ξ.

### When network effects bind: constrained optimum
- Constraint (28) affects both ξ and r_cbdc. When binding, optimal constrained design is:
  - ξ̂ = [2 + ζ(2(A−ε)−1 − ζ(4 + ζ)(2s−1))] / [4 + ζ(1 + ζ)(3 + ζ)] (equation (41))
  - r̂_cbdc = −2[ζ((A−ε) + 3s −2) + 4s −1] / [4 + ζ(1 + ζ)(3 + ζ)] (equation (42))
- Under parameter restrictions (17), the optimal CBDC rate when constrained is always negative.
- Consequence: negative r_cbdc allows ξ to be more cash-like (ξ̂ > ξ̄) as A−ε rises.
- Access to r_cbdc strictly raises welfare when network effects bind because the constrained non-interest-bearing policy (ξ̄;0) is feasible but sub-optimal.

### Proposition 1 (summary of key results)
- There exists a cutoff ζ̄ ∈ (2/3, 3/2) such that:
  - If ζ < ζ̄, cash never vanishes under optimally designed interest-bearing CBDC. Optimal design is:
    - (ξ̂; r̂_cbdc) if ξ_ce + ζ r_ce_cbdc < 1 − 2s; otherwise (ξ_ce; r_ce_cbdc) (equation (43)).
  - If ζ > ζ̄, cash can vanish when A−ε exceeds a threshold Â−ε; and Â−ε > A−ε̄ (i.e., the threshold is higher when CBDC is interest-bearing).
- Economic intuition:
  - For ζ < ζ̄ (payments-preference weight large), the CBDC interest rate as a second tool enables sustaining payment-instrument variety while limiting bank disintermediation—by combining a more cash-like CBDC with a negative CBDC rate.
  - For ζ > ζ̄ (payments-preference weight small), negative CBDC rates create larger payment-choice distortions; beyond some A−ε welfare is maximized by letting cash vanish and switching to (ξ_nce; r_nce_cbdc).

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### 3.3  Distributional effects

### Aggregate vs distributional impacts
- Aggregate welfare rises with optimally designed CBDC, but the change is not Pareto-improving: some households gain and others lose.
- Welfare impact varies across household payment-preference index i ∈ [0,1] (Figure 4 overview).

### Non interest-bearing CBDC (blue line)
- Low-i (deposit-preferring) households:
  - Remain deposit users.
  - Affected by reduced financial intermediation (lower firm profits π) and higher deposit rates r_d due to CBDC competition.
  - Net effect: deposit-rate increase dominates → consumption and welfare of deposit users rise.
- High-i (cash-preferring, anonymity-seeking) households:
  - Remain cash users.
  - Cash pays no interest; decline in π reduces consumption and welfare.
  - If CBDC drives cash out via network effects, these households suffer further welfare loss proportional to their anonymity preference.
- Switchers (from deposits to CBDC):
  - Experience net welfare gain by construction of optimality: switch only if payment-preference gains exceed interest-loss r_d.
  - The household at i = ξ experiences the largest welfare increase.
- Marginal CBDC-over-cash households:
  - Net welfare effect negative because CBDC holders also face fall in consumption via reduced π.
- Define boundary ī such that households i ∈ [0,ī] gain and i ∈ (ī,1] lose; aggregate gains of first group exceed losses of latter.

### Interest-bearing CBDC with slightly negative rates (red line)
- Three determining factors:
  1. Revenues from negative CBDC rates are lump-sum transferred to all households, redistributing gains from CBDC users to cash and deposit users.
  2. Negative CBDC rates increase deposits and financial intermediation → firm profits π rise → benefits all households; but deposit rates r_d decrease → depositors hurt. The lump-sum transfer dominates: overall CBDC users lose; deposit and cash users gain.
  3. Negative CBDC rates preventing cash disappearance avert large welfare losses for cash holders from losing their preferred instrument.

### Distributional interpretation
- Deposit holders are winners and cash holders are losers under non-interest-bearing CBDC, suggesting a potentially regressive impact in practice (cash users tend to be lower income).
- Interest-bearing CBDC with negative rates can mitigate losses for cash holders via redistribution and by preserving cash through policy.

---

### 4  Conclusion and policy implications

- Two key CBDC design choices matter for money demand and bank intermediation: (i) degree of cash-likeness (ξ), and (ii) whether CBDC is interest-bearing (r_cbdc).
- Non interest-bearing CBDC presents a tradeoff:
  - Making CBDC cash-like risks reducing cash demand below network thresholds, endangering payment-instrument variety valued by heterogeneous households.
  - Making CBDC deposit-like threatens banks’ deposit base, harming credit provision and output, especially when banks alleviate lending frictions.
- Optimal directions:
  - If banks’ role is limited, design CBDC distinct from existing instruments.
  - Greater emphasis on preserving intermediation pushes optimal CBDC to be more cash-like, but network effects can constrain this.
  - Only when preserving deposits is paramount will the central bank forgo cash and jump toward a more cash-like CBDC.
- Role of interest-bearing CBDC:
  - When network effects matter, an interest-bearing CBDC helps alleviate tradeoffs by using r_cbdc as a second tool.
  - Absent network effects, optimal r_cbdc = 0; limiting CBDC to non interest-bearing is without loss.
  - When network effects threaten variety, an adjustable CBDC interest rate becomes optimal: negative r_cbdc can make CBDC more cash-like while making it less attractive relative to cash, protecting both cash use and bank intermediation.
- Practical note: political-economy concerns about negative rates on public central bank liabilities may push toward non interest-bearing CBDCs, but economic analysis suggests central banks should consider retaining an adjustable CBDC interest rate to weigh benefits against political costs.

*Source: wpiea2019252-print-pdf - 3.1  Non interest-bearing CBDC*

### References

### wpiea2019252-print-pdf - References

### Key analytical results (Appendix A: Proofs)
- Lemma 1
  - A CBDC can be designed to mimic cash: (; r_cbdc) = (1; 0).
  - Welfare ordering: W(1;0) < W(_nce; r_nce_cbdc) < W(_ce; r_ce_cbdc) with _ce < _nce < 1, following (36), (37) and (38).
- Lemma 2
  - Infima over parameter space (17) for shares of money when all forms exist (ce):
    - inf s_ce^c = 1/34
    - inf s_ce^d = 2/17
    - inf s_ce^cbdc = 1/17
  - Given s ≥ 1/17 in (17), it follows that θ_d = θ_cbdc = 0.
  - Necessary conditions to avoid strict dominance of CBDC by cash or deposits:
    - r_cbdc ≥ −(1−)ζ^(−1)   (44)
    -  > ζ(r_d − r_cbdc)     (45)
  - Values: sup _ce = 16/17 < sup _nce = 23/24 < 1, and inf (_ce − ζ r_d) = 1/17.
- Lemma 3
  - W_ce(; r_cbdc) obtained from system of 11 equations (expression (30)). Key internal definitions:
    - s_d =  + ζ(r_d − r_cbdc)/2
    - s_cbdc = s_cbdc1 + s_cbdc2, with
      - s_cbdc1 =  − ζ(r_d − r_cbdc)/2
      - s_cbdc2 = 1 + ζ r_cbdc − )/2
    - s_c = 1 −  − ζ r_cbdc/2
    - E_d = s_d^2
    - E_cbdc1, E_cbdc2, E_c analogous to the expressions shown.
    - r_d = 2(A−1−ζ) + ζ r_cbdc −  / (2 + ζ)
  - W_nce(; r_cbdc) is obtained by setting s_c = 0.
- Proposition 1 (optimality conditions)
  - W_ce(_ce; r_ce_cbdc) > W_nce(_nce; r_nce_cbdc) per (38).
  - Comparison center: W_nce(_nce; r_nce_cbdc) versus W_ce(e ; e r_cbdc) when network effects constraint binds (i.e., _ce + r_ce_cbdc = _ce > 1 − 2 s).
  - Existence of threshold ζ̄:
    - inf_{A,ζ,s_h} [ W_nce(_nce; r_nce_cbdc) − W_ce(e ; e r_cbdc) ] > 0, ζ > 1.431 = ζ̄  (46)
    - For ζ < ζ̄, e(; e r_cbdc) always dominates W_nce; cash never vanishes under optimal policies.
    - For ζ > ζ̄, parameterizations (including (A−ζ) = 5/2 and s = 1/17) exist where (_nce; r_nce_cbdc) dominates e(; e r_cbdc) and cash can optimally vanish.
  - Whenever _ce >  = 1 − 2 s, W_ce(e ; e r_cbdc) − W_ce(; 0) > 0 (47).
  - For ζ > ζ̄ and s = 1/17, the A−ζ value needed to induce a switch from ce to nce is higher when policies are set at e(; e r_cbdc) than at (; 0).

### Distributional effects (Appendix B)
- Interpretation: "After the introduction of a CBDC" compares world with cash + deposits to one where CBDC is an additional instrument.
- Depositors (continue using deposits after CBDC introduction)
  - Consumption representation:
    - C_d = 1 + r_d + ϖ − T  (48)
    - C_d = 1 + r_d (1 − s_d) + ζ k_0 + [A−1−ζ − s_d^2/2] s_d − r_cbdc s_cbdc  (49), with T = r_cbdc s_cbdc and ϖ replaced via (10),(11),(16).
  - Closed-form yields:
    - ∂C_d / ∂ |_{r_cbdc=0} = ζ(A−ζ−2) +  − 2  / (2 + ζ)^2 < 0  (50)
      - Conclusion: introduction of a non-interest-bearing CBDC always raises welfare for households that continue using deposits because it is equivalent to lowering  from 1 to a lower value.
    - ∂C_d / ∂ r_cbdc = − [4 − ζ(4(A−ζ−1) − ζ + 2 r_cbdc(8 + ζ(5 + ζ)) + (4 + ζ))] / [2 (2 + ζ)^2]
      - At r_cbdc = 0 this term is negative over parameter space (17) and  ≤ 1: a marginal CBDC interest rate cut from r_cbdc = 0 to r_cbdc < 0 raises depositors' welfare.
- Cash holders (continue using cash after CBDC introduction)
  - Cash holders lose firm profit transfers and therefore are necessarily worse off with a non-interest-bearing CBDC:
    - ∂C_c / ∂  |_{r_cbdc=0} > 0 (cash holders worse off as CBDC becomes more cash-like).
  - Negative CBDC rates:
    - ∂C_c / ∂ r_cbdc < 0 (declining CBDC rate raises ϖ and T; cash holders are worse off).
  - If network effects push cash out of use, cash holders become even worse off.
- CBDC users (households switching to CBDC)
  - Former depositors switching to CBDC always see a welfare improvement; depositors would not have switched otherwise (consistent with (50)).
  - The i =  household (whose payments preferences CBDC precisely meets) experiences the largest welfare gain.
  - Some CBDC adopters (i >  marginally) would have been better off without CBDC—CBDC can fail to compensate for lost firm profit transfers.
  - Negative CBDC rates act as a tax on CBDC holders and reduce their welfare (Figure 4 reference).

### Extensions (Appendix C)
- C.1 Constant returns to scale production
  - Replace Y = −(A−1/2) k^2 with Y = A k (51).
  - Unconstrained optimal policies:
    - In ce: _ce = 1 + ζ(A−ζ−1)/2  (52); r_ce_cbdc = 0  (53)
    - In nce: _nce = 2 + ζ(A−ζ−1)/3  (54); r_nce_cbdc = 0  (55)
  - Conclusion: optimal unconstrained CBDC interest rate remains zero in both ce and nce; CBDC optimally made more similar to cash when (A−ζ) rises.
  - Note: decreasing vs constant returns affect bank response; under constant returns r_d = A − ζ − 1 and deposit rate is unresponsive to  and r_cbdc.
- C.2 Anonymity externalities
  - Augmented household utility with anonymity externality term:
    - U_i(j) = ζ C_j − j x_j − i j − κ_j − φ ∫_{n≠i} x_j(n) dn  (56)
      - φ ∈ [0,1] measures dislike of others' anonymity.
  - Unconstrained optimal policies (nesting φ = 0 solutions):
    - _ce = [2 + ζ(2(A−ζ) − 1) − φ(2 + ζ)] / [4 + 3 ζ − φ(4 + ζ)]  (57)
    - r_ce_cbdc = −2 φ/(4 + ζ) × (A−ζ)(4 + ζ) − 2(3 + ζ)  / [4 + 3 ζ − φ(4 + ζ)]  (58)
  - Key insight: for any φ > 0, r_cbdc ≠ 0 is optimal (can be positive or negative). Higher (A−ζ) → CBDC design more cash-like (higher _ce) and lower (possibly negative) r_ce_cbdc.
  - Policy implication: anonymity externalities can justify nonzero CBDC interest rates; negative rates can be used to push households into deposits to limit anonymity externalities.
- C.3 Bank market power (Cournot lending)
  - Banks internalize impact of individual lending on total loans; parameter ζeta (τ) ∈ [0,1] measures market power (τ = 0 perfect competition, τ = 1 monopoly).
  - Key relations:
    - L = l + (1 − τ) L̄ ; ∂L/∂l = 1  (59)
    - ∂R/∂l = ∂R/∂L = −1  (60)
  - Equilibrium solutions:
    - L = [ζ(A−1 − r_cbdc − ζ) + ] / [2 + ζ(1 + τ)]  (64)
    - R = A − ζ − 1 − ζ(A−1 − r_cbdc − ζ) +  / [2 + ζ(1 + τ)]  (65)
    - r_d = [2(A−1−ζ) + (1 + τ)(ζ r_cbdc − )] / [2 + ζ(1 + τ)]  (66)
  - Optimal policies with market power:
    - _ce and r_ce_cbdc given in (67) and (68) (reduce to τ = 0 case in (36),(40)).
  - Numerical comparative statics over parameter ranges in (17):
    - inf (_ce(τ) − _ce(τ=0)) = −279/5372
    - sup (_ce(τ) − _ce(τ=0)) = 0
    - inf (r_ce_cbdc(τ) − r_ce_cbdc(τ=0)) = −35/163
    - sup (r_ce_cbdc(τ) − r_ce_cbdc(τ=0)) = 0
  - Conclusion: greater lending market power (τ > 0) lowers both optimal _ce and r_ce_cbdc relative to τ = 0; banks can better withstand CBDC competition, so optimal CBDC design moves closer to deposits (lower ) and policy maker offsets by cutting CBDC rates.
- C.4 Alternate equilibria under suboptimal policies
  - CBDC and cash (deposits vanish): possible under suboptimal policy, e.g.,  = 0 and r_cbdc > A − ζ − 1 (69) → r_cbdc > r_d, CBDC strictly dominates deposits.
  - CBDC only: arbitrarily high r_cbdc can eliminate both deposits and cash (households pay lump-sum tax T).
  - Cash and deposits (CBDC uptake zero under suboptimal design): three routes
    - CBDC strictly dominated by cash (violate (44))
    - Deposits strictly preferred (violate (45))
    - Network effects prevent CBDC critical mass (violate design constraint (15))
  - Concrete condition showing deposits strictly dominate CBDC when violated:
    -  (1 + ζ)/ζ [ + r_cbdc] > A − ζ − 1  (70)
    - Sufficiently negative r_cbdc can make deposits strictly dominate CBDC.

### Linear city of payments preferences (Appendix D)
- Microfoundation for a linear-city of payments preferences where payments privacy has value because private firms monetize transaction data.
- Setup (cash vs deposits only)
  - Two product types: G (Good) and B (Bad). Households have identical incomes and choose fraction ω(i) to spend on G; ideal share p(i) ∈ [0,1] uniformly distributed.
  - Quadratic disutility for deviation: (ω(i) − p(i))^2.
  - Deposits monitored by fintech (sole credit provider); fintech assigns credit scores from transaction data. Cash users pooled; deposit users fully revealed.
- Household utility specification:
  - U(i) = ρ E[ω(i) | j(i)] − (ω(i) − p(i))^2  (71)
  - E[ω(i) | j(i)] = ω(i) if j(i)=d; = b_ω if j(i)=c  (72), where b_ω is average G-share of cash users; atomistic agents imply cash holders consume bliss point: ω(i) = p(i) when j(i)=c.
- Depositor's optimization:
  - max_{ω(i)} [ ρ ω(i) − (ω(i) − p(i))^2 ]  (73) → ω(i) = ρ/2 + p(i)  (74)
- Sorting condition (choose deposits iff):
  - ρ/4 + p(i) > b_ω  (75) → p(i) > b_ω − ρ/4 = p̄  (76)
  - Interpretation: households with higher ideal G-share (p(i) > p̄) choose deposits to obtain credit-score benefits; those with lower p(i) choose cash to avoid revelation and being forced into overconsumption of G.
- Conclusion: heterogeneity in consumption preferences endogenously generates a line (cutoff) between cash and deposit users; CBDC can be modeled as intermediate point on this spectrum.

*Source: wpiea2019252-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019252-print-pdf.pdf_