## 1. The adoption of CBDC by households and firms

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### Model setup and context
- Population: overlapping generation households consuming a continuum of goods; firms heterogeneous across good value.
- Household incomes: wages, remittances, government payments.
- Payment/account options:
  - Bank account at a fixed cost to access card payments.
  - CBDC wallet at a lower fixed cost.
  - Otherwise financially excluded.
- Firms choose whether to accept CBDC and/or card payments at fixed costs, or remain informal and evade taxes.
- Two-sided market feedback: household adoption increases merchant acceptance and vice versa.
- Empirical snapshots:
  - 17 countries in pilot phase; 11 countries rolled out (Atlantic Council tracker).
  - eNaira adoption stagnated close to 0.5%.
  - Sand Dollar comprises 0.1% of total currency in circulation in the Bahamas.
  - JAM-DEX uptake low; large merchants yet to come on board (Bank of Jamaica).
  - Example policies: Morocco 2020 five-year tax exemption for mobile payments by small merchants; Egypt law requiring public-body salaries/fees be paid electronically.

### Household adoption — margins, thresholds, and comparative statics
- Two adoption margins:
  - Financial inclusion: unbanked open CBDC wallets when many firms accept CBDC; CBDC costs low; CBDC remuneration high; CBDC reduces remittance costs; CBDC facilitates government payments.
  - Disintermediation: banked households switch from bank accounts to CBDC when many firms accept only CBDC; CBDC remuneration high relative to bank deposit rates; bank account costs high relative to CBDC wallets; CBDC payment value high relative to card payments.
- Income sorting (baseline, government payments and remittances constant):
  - Lowest income: cash only.
  - Middle income: CBDC wallets.
  - Highest income: bank accounts.
- Household budget constraint components and key parameters preserved in the source: e_0, κ_0, d, e_b, κ_b, r_b, v_b, F_b, e_c, κ_c, r_c, v_c, F_c.
- Key comparative statics (preserve numeric inequality structure from source):
  - i_{b,0}^w decreases when j_{b,0} decreases, r_b is high, v_b is high, and F_b is low.
  - i_c^w decreases (more CBDC adoption) when r_c is high, e_c is low, v_c is high, κ_c is low, F_c is low, and j_c and j_{c,b} are low.
  - i_b^w decreases when r_b is high, v_b is high, F_b is low, and j_b and j_{c,b} are low.
- Factors increasing financial inclusion:
  - e_c is small; κ_c is small; r_c is large; v_c is large; F_c is small; j_c and j_{c,b} small and j_b high.
- Factors increasing disintermediation:
  - r_c − r_b is large; F_b − F_c is large; v_c − v_b is large; j_b − j_c is large.

### Firm adoption — acceptance thresholds, revenues, and margins
- Household payment groups: g0 (cash only), gc (pay in CBDC if accepted), gb (pay in card if accepted, else CBDC if accepted, else cash).
- Firm revenue expressions (payment-mode specific) preserve parameters: f0, fb, fc, τb, τc, Kb, Kc, Kc,b.
- Assumptions:
  - τc ≤ τb (possible CBDC tax exemptions).
  - Kc < Kb (policy objective of low-cost CBDC acceptance).
  - Kc,b > Kb and Kc,b ≤ Kb + Kc.
- Firm acceptance segmentation:
  - Firms j ∈ [0, jc]: only cash.
  - Firms j ∈ [jc, jb]: cash and CBDC.
  - Firms j ∈ [jb, jc,b]: cash and card.
  - Firms j ∈ [jc,b, 1]: cash, CBDC and card.
- Comparative statics on jc, jb, jc,b (preserved inequality directions from source):
  - jc decreases (more firms accept CBDC) if many households prefer CBDC, these households are high income, Kc and Kc,b decrease, fc + τc decreases.
  - jb and jc,b decrease (more firms accept cards) if many households prefer cards, those households are high income, Kb and Kc,b decrease, fb + τb decreases.
- Two firm-level impacts of CBDC:
  - Formalization: informal (cash-only) firms adopt CBDC/card and enter formal sector when household demand for non-cash payments is high, Kc low, CBDC fees low, and tax/subsidy incentives exist.
  - Displacement: firms stop accepting cards and accept CBDC instead; small or zero when households preferring CBDC over cards are small or when card costs relative to CBDC remain small.

### Aggregate outcomes, coordination failures, and trade-offs
- Aggregate effects:
  - Greater CBDC adoption increases financial inclusion and formalization.
  - CBDC can cause disintermediation of banks and displacement of card payments.
  - Trade-off: design for greater adoption versus limiting disintermediation.
  - Distributional: gains from inclusion and formalization more likely to outweigh disintermediation risks in lower income economies with larger unbanked populations and informal sectors.
- Path dependence and coordination failure:
  - Economy may remain in low-CBDC adoption steady state despite CBDC being welfare-improving due to merchant–household coordination externality.
  - Non-merchant-dependent CBDC features (remuneration r_c, lower remittance costs e_c, efficient government payments κ_c) are crucial to avoid low-adoption traps.

### Policy levers and calibration / simulation findings
- Policy levers highlighted:
  - Temporary subsidies or tax exemptions to spur initial adoption.
  - Use CBDC for government payments, or make CBDC the cost-effective way to receive government payments.
  - Targeted tax exemptions for CBDC transactions for small firms.
- Calibration environment:
  - Representative developing economy with large informal sector and unbanked population.
  - Overlapping generations: k = 2; utility u(x) = log(x).
  - Banked population share = 57%.
  - 80% of lowest-value goods selling firms informal (jb,0 = 0.2), implying informal sector ≈ 30% of total output.
  - Baseline CBDC: improves cross-border and government payments efficiency, more valuable than cash but less than cards, non-remunerated, no tax exemptions, lower fees than cards, lower fixed costs than bank account ownership.

### Simulation results — baseline steady state (Pre-CBDC → Post-CBDC)
- Households (%):
  - Cash only: 43% → 8%
  - CBDC wallet: 0% → 39%
  - Bank account: 57% → 53%
  - Financial Inclusion (CBDC or Bank): 57% → 92%
  - % Change in welfare: −0.39%
- Firms (%):
  - Cash only (Informal): 80% → 26%
  - Accept CBDC: 0% → 55%
  - Accept Card: 20% → 19%
  - Formal (CBDC and/or Card): 20% → 74%
- Savings (%):
  - Cash: 13% → <1%
  - CBDC wallet: 0% → 14%
  - Bank deposits: 87% → 86%
- Interpretation:
  - Transition to steady state with 39% household and 55% firm CBDC adoption.
  - Financial inclusion increases by 35 percentage points (57% → 92%).
  - Informal firms decline by 54 percentage points (80% → 26%).
  - Disintermediation low: 1 p.p. decrease in bank deposits share; 4 p.p. decrease in bank account ownership.
  - Card payment displacement small: 1 p.p. decrease in firms accepting cards.
  - Reported household welfare change recorded as −0.39% (measurement note in source: gains exclude potential tax revenue gains from formalization).

### Low adoption steady state and role of temporary incentives
- Low-adoption scenario (no improvements to cross-border/government payments: e_c = e0 and κ_c = κ0), no incentives:
  - Households (%):
    - Cash only: 43% → 42%
    - CBDC wallet: 0% → 1%
    - Bank account: 57% → 57%
    - Financial Inclusion: 57% → 58%
    - % Change in welfare: −0.01%
  - Firms (%):
    - Cash only (Informal): 80% → 80%
    - Accept CBDC: 0% → <1%
    - Accept Card: 20% → 20%
    - Formal: 20% → 20%
  - Savings (%):
    - Cash: 13% → 13%
    - CBDC wallet: 0% → 0%
    - Bank deposits: 87% → 87%
  - Interpretation: CBDC adoption stalls at ≈1% households; coordination failure prevents meaningful uptake.
- Temporary incentive package (applied first period) — policy components and outcomes:
  - Policy measures in first period:
    - 20% decrease in F_c for below median income households.
    - τc = 0 (CBDC transactions tax-exempt).
    - Increase κ0 by 1 p.p. and decrease κc by 0.5 p.p. (government payments administered through CBDC / more cost effective).
  - Outcomes (Pre-CBDC → Post-CBDC w/ incentives):
    - Households (%):
      - Cash only: 43% → 21%
      - CBDC wallet: 0% → 23%
      - Bank account: 57% → 56%
      - Financial Inclusion: 57% → 79%
      - % Change in welfare: −0.01% → 0.25%
    - Firms (%):
      - Cash only (Informal): 80% → 31%
      - Accept CBDC: 0% → 49%
      - Accept Card: 20% → 20%
      - Formal (CBDC and/or Card): 20% → 69%
    - Savings (%):
      - Cash: 13% → 4%
      - CBDC wallet: 0% → 9%
      - Bank deposits: 87% → 87%
  - Interpretation:
    - Temporary incentives trigger a transition out of the low-adoption steady state.
    - Steady state with 23% households and 49% firms adopting CBDC.
    - Household welfare increases by 0.25%.
    - Financial inclusion increases by 22 p.p. (57% → 79%).
    - Informal firms drop by 49 p.p. (80% → 31%).
    - Disintermediation small: <1 p.p. decrease in bank deposit share; 1 p.p. decrease in bank account ownership.
    - No displacement of card payments observed in this scenario.

### Additional simulation findings (summary)
- Higher income countries: smaller financial inclusion and formalization gains; greater disintermediation and displacement risks.
- Interest-bearing CBDC: can increase adoption with slightly greater inclusion and formality gains, but disintermediation and displacement impacts are large.
- Two-tier intermediation: if CBDC wallet fixed cost F_c is high relative to bank account F_b, previously unbanked may open bank accounts to access CBDC.
- If CBDC does not reduce firms' fixed cost barriers to accepting card payments, overall adoption remains low even if some households open CBDC wallets.

*Source: wpiea2023127-print-pdf - 1. The adoption of CBDC by households and firms; 2.1 Households; 4. Cash, CBDC, and Card payments (Formal); 4.3 Country income level.*

### 1.  The adoption of CBDC by households and firms

### 1.  The adoption of CBDC by households and firms

### Model setup
- Population: overlapping generation households consuming a continuum of goods (low to high value) sold by heterogeneous firms.
- Household incomes: wages, remittances, and government payments.
- Payment/account options:
  - Bank account at a fixed cost to access card payments.
  - CBDC wallet at a lower fixed cost.
  - Otherwise financially excluded.
- Firms choose whether to accept CBDC and/or card payments at fixed costs to avoid variable cash-handling costs, or remain informal and evade taxes.
- Feedback loop: household adoption increases merchant acceptance and vice versa (two-sided market dynamics).

### Context and empirical observations (from source)
- According to the Atlantic Council’s tracker, 17 countries currently have their CBDC in the pilot phase and 11 countries have rolled it out.
- eNaira’s adoption has stagnated close to 0.5%.
- Only 0.1% of total currency in circulation in the Bahamas comprises Sand Dollar.
- The Bank of Jamaica reports low uptake of JAM-DEX; large merchants yet to come on board.
- Examples of policy measures:
  - Morocco introduced in 2020 for 5 years a total exemption from taxes for transactions made by mobile payment with small size merchants.
  - Egypt implemented a law requiring that salaries and fees out paid by public bodies be made electronically.

### Household adoption: two margins
- Financial inclusion margin:
  - Unbanked households open CBDC wallets.
  - This margin is large when:
    - many firms/merchants accept CBDC;
    - CBDC is valuable as a means of payment;
    - CBDC costs are low;
    - CBDC remuneration is high;
    - CBDC reduces cost of remittances (cross-border CBDC);
    - CBDC facilitates government payments.
- Disintermediation margin:
  - Banked households switch from bank accounts to CBDC wallets.
  - This margin is large when:
    - many firms accept only CBDC and not card payments;
    - CBDC remuneration is high relative to bank deposit rates;
    - costs of bank accounts are high relative to CBDC wallets;
    - CBDC’s payment value is high relative to card payments.
- Income sorting in baseline (holding government payments and remittances constant):
  - Lowest income households use cash.
  - Middle income households own CBDC wallets.
  - Highest income households own bank accounts.
- If lower income households receive more government payments and remittances, designing CBDC to reduce cross-border costs and facilitate government payments is particularly effective in boosting CBDC adoption among the unbanked and reducing disintermediation among the banked.

### Firm adoption: two margins
- Formalization margin:
  - Informal firms that only accepted cash choose to accept CBDC (or card) and enter the formal sector.
  - Formalization is greater when:
    - many households have bank accounts and CBDC wallets preferring CBDC over cash;
    - these households have high income (large revenue share);
    - CBDC payment costs are low;
    - tax implications of adopting CBDC and formalization are not too large.
- Displacement margin:
  - Firms that accepted cards stop accepting cards and accept CBDC instead.
  - Displacement is small (or zero) when:
    - the size and income of households preferring CBDC over cards is small;
    - costs and taxes of card payments relative to CBDC payments are small.
- Goods-value sorting:
  - Firms selling the lowest value goods remain informal and use cash.
  - Firms selling middle value goods accept CBDC.
  - Firms selling high value goods accept card payments.

### Aggregate outcomes and trade-offs
- Greater CBDC adoption tends to increase financial inclusion and formalization, but can also cause disintermediation of banks and displacement of card payments.
- Trade-off: designing for greater adoption versus limiting disintermediation.
- Distributional note: gains from financial inclusion and formalization are more likely to outweigh disintermediation risks in lower income economies with larger unbanked populations and informal sectors.
- Path dependence and coordination failure:
  - An economy may not transition to a high-CBDC adoption steady state even if CBDC is welfare-improving because of the merchant–household coordination externality.
  - If CBDC features that do not rely on merchant acceptance (remuneration, lower remittance costs, efficient government payments) are not sufficiently attractive, the economy can be stuck in a low-adoption equilibrium.

### Policy levers and numerical/calibration findings
- Temporary subsidies or tax exemptions can spur initial adoption and transition the economy to a welfare-improving steady state with high CBDC usage.
- Using CBDC for government payments, or making it more cost-effective (or mandatory) to receive government payments through CBDC, will spur adoption.
- Calibration to a representative developing economy with a large informal sector and unbanked population yields:
  - Large gains in financial inclusion and formalization.
  - Low levels of disintermediation and displacement in the baseline.
- Scenarios where economies get stuck at low adoption:
  - No efficiency gains in cross-border and government payments (i.e., CBDC features not attractive without merchant acceptance).
  - Introducing temporary measures (subsidies, tax exemptions, government payments in CBDC) can enable transition to high adoption with significant welfare, inclusion, and formalization gains.
- Higher income countries: smaller financial inclusion and formalization gains, and greater disintermediation and displacement risks.
- Interest-bearing CBDC:
  - Can increase adoption with slightly greater financial inclusion and formality gains.
  - But the disintermediation and displacement impact is large.
- Two-tier intermediation possibility:
  - If CBDC wallet costs are high relative to bank accounts, previously unbanked may open bank accounts to access CBDC, generating bank intermediation.
- If CBDC does not lower fixed cost barriers firms face in accepting card payments, overall adoption remains low even if some households are willing to open CBDC wallets.

### Novel contributions to literature
- First two-sided payment model endogenizing CBDC utility as a payments instrument by linking acceptance by merchants to CBDC’s usefulness for households.
- Joint analysis of households and firms to study:
  - CBDC adoption dynamics.
  - Financial inclusion.
  - Formalization of the informal sector.
  - Disintermediation of banks and card payments.
- Incorporates understudied CBDC use cases: cross-border payments and government payments, and novel government policies to encourage CBDC take-up.
- Builds on two-sided market theory and heterogeneous agents literature.

*Source: wpiea2023127-print-pdf - 1.  The adoption of CBDC by households and firms.*

### 2.1    Households

### 2.1    Households

### Model setup and household preferences
- Model populated by k-period-lived overlapping generations. Time is denoted by t = 1,2,...,T.
- Households value consumption of a continuum of goods indexed by j ∈ [0,1].
- Utility over goods: U_t = u( min_j c_t(j)^{(1−w)j − w} ), w ∈ R− (equation (1) in source).
- Leontief ordering implies c_t(j)^{(1−w)j − w} = c_t (equation (2) in source). Integrating j ∈ [0,1] gives total consumption c_t.
- Households born in period T maximize Σ_{t=T}^{T+k} β^{t−T} u(c_t) subject to budget constraints and lifecycle income.
- Households receive wage income i_w, remittance income i_r, and government payments i_g each period. In last period (t = T + k): i_w = 0. Initial period (t = T): s_t = 0.

### No CBDC scenario — household choices and budget constraints
- Household payment options: (1) cash only, (2) bank account with card payments.
- Merchant acceptance: firms j ≥ j_{b,0} accept card payments; firms j < j_{b,0} accept only cash.
- Cash-only household budget constraint:
  - i_w + i_r (1 − e_0) + i_g (1 − κ_0) + s_{t−1} (1 − d) = c_t + s_t  (equation (4))
  - Parameters: e_0 = fees to receive remittance in cash; κ_0 = costs for government payments in cash; d = cost of storing cash.
- Bank-account household budget constraint pre-CBDC issuance:
  - i_w + i_r (1 − e_b) + i_g (1 − κ_b) + s_{t−1} (1 + r_b) = ∫_0^{j_{b,0}} c_t(j) dj + (1 − v_b) ∫_{j_{b,0}}^1 c_t(j) dj + F_b + s_t  (equation (5))
  - Parameters: e_b = remittance fees with bank account; κ_b = government payment costs with bank account; r_b = interest on bank deposits; v_b = (convenience) value of card payments relative to cash; F_b = fixed cost to own a bank account.
  - Assumptions and comparative statics:
    - {e_b, κ_b} are likely similar or slightly less than {e_0, κ_0}.
    - v_b > 0; r_b ≥ 0.
    - Households open bank accounts if benefits (r_b > −d and v_b > 0) exceed fixed cost F_b.
    - Holding all else fixed, higher wage income i_w increases likelihood of opening a bank account; there exists income threshold i_{b,0}^w: use cash if i_w < i_{b,0}^w, open bank account if i_w ≥ i_{b,0}^w.
    - i_{b,0}^w decreases (more households open bank accounts) when j_{b,0} decreases (more firms accept cards), r_b is high, v_b is high (if j_{b,0} < 1), and F_b is low.

### CBDC issuance scenario — household choices and budget constraints
- Household payment options post-CBDC: (1) cash only, (2) CBDC wallet, (3) bank account with card payments and access to CBDC.
- Merchant acceptance by firm groups:
  - firms j ∈ [0, j_c] accept cash only;
  - firms j ∈ [j_c, j_{b}] accept cash and CBDC;
  - firms j ∈ [j_{b}, j_{c,b}] accept cash and card;
  - firms j ∈ [j_{c,b}, 1] accept cash, CBDC and card.
- Cash-only household budget constraint remains:
  - i_w + i_r (1 − e_0) + i_g (1 − κ_0) + s_{t−1} (1 − d) = c_t + s_t  (equation (6))
- CBDC-wallet household budget constraint:
  - i_w + i_r (1 − e_c) + i_g (1 − κ_c) + s_{t−1} (1 + r_c) =
    ∫_0^{j_c} c_t(j) dj + ∫_{j_c}^{j_{c,b}} c_t(j) dj  [cash payments]
    + (1 − v_c) [ ∫_{j_b}^{j_c} c_t(j) dj + ∫_{1}^{j_{c,b}} c_t(j) dj ] + F_c + s_t  (equation (7))
  - Parameters: e_c = remittance fees with CBDC; κ_c = government payment costs with CBDC; r_c = CBDC remuneration; v_c = value of CBDC versus cash; F_c = fixed cost to own CBDC wallet.
  - Assumptions and comparative statics:
    - e_c ≤ e_0; κ_c ≤ κ_0; v_c > 0; r_c ≥ 0 (r_c = 0 if non-interest bearing); r_c ≥ 0 > −d.
    - Households open CBDC wallets if benefits (r_c > −d, v_c > 0, lower e_c, lower κ_c) exceed fixed cost F_c.
    - Higher i_w increases likelihood of CBDC wallet ownership; those with higher remittance i_r or government payments i_g are also likelier to adopt.
    - CBDC wallet adoption increases as more firms accept CBDC (j_c and j_{c,b} decrease).
- Bank-account household budget constraint post-CBDC:
  - i_w + i_r (1 − {e_b, e_c}) + i_g (1 − {κ_b, κ_c}) + s_{t−1} (1 + max{r_b, r_c}) =
    ∫_0^{j_c} c_t(j) dj [cash]
    + (1 − v_c) ∫_{j_b}^{j_c} c_t(j) dj [CBDC]
    + (1 − v_b) ∫_{j_c,b}^{j_b} c_t(j) dj [card]
    + (1 − max{v_b, v_c}) ∫_{1}^{j_{c,b}} c_t(j) dj [CBDC or card]
    + F_b + s_t  (equation (8))
  - Assumptions and comparative statics:
    - e_c is likely less than e_b; κ_c may be less than κ_b.
    - v_b > 0; v_c may be > or < v_b.
    - r_b ≥ r_c and r_b ≥ 0; F_b > F_c (objective of CBDC is accessibility and low cost).
    - Households open bank accounts if higher fixed cost F_b is outweighed by higher return (r_b ≥ r_c) and potentially higher payments value (v_b > v_c).
    - Two income thresholds arise: i_c^w and i_b^w. Poorest use cash (i_w < i_c^w), middle-income own CBDC wallets (i_c^w ≤ i_w < i_b^w), richest own bank accounts (i_w ≥ i_b^w).
    - i_c^w decreases (more CBDC adoption) when r_c is high, e_c is low, v_c is high (if j_c < 1), κ_c is low, F_c is low, and many firms accept CBDC (j_c and j_{c,b} low).
    - i_b^w decreases (more bank-account adoption) when r_b is high, v_b is high, F_b is low, and many firms accept card payments (j_b and j_{c,b} low).

### Impact of CBDC on households: financial inclusion and disintermediation
- Two margins of impact illustrated in Figure 3:
  1. Financial Inclusion: households switch from cash to CBDC wallets (or bank accounts).
  2. Disintermediation: banked households open CBDC wallets instead of relying on bank deposits.
- Factors increasing financial inclusion (more households switch to CBDC):
  - e_c is small (CBDC reduces remittance costs), especially benefiting high remittance income households.
  - κ_c is small (cheaper/more efficient government payments).
  - r_c is large (CBDC remuneration high).
  - v_c is large (CBDC valuable as a means of payment).
  - F_c is small (low fixed cost to own CBDC wallet).
  - j_c and j_{c,b} are small and j_b is high (many firms accept CBDC).
- Factors increasing disintermediation (banked households switch to CBDC wallets):
  - r_c − r_b is large (CBDC remuneration high relative to bank deposit rate).
  - F_b − F_c is large (bank accounts cost much more than CBDC wallets).
  - v_c − v_b is large (CBDC payments more valuable than card payments).
  - j_b − j_c is large (many firms accept only CBDC).
- Aggregate margins depend on income distribution:
  - Size of financial inclusion margin depends on density of lower-income previously unbanked households adopting CBDC.
  - Size of disintermediation margin depends on density of higher-income banked households switching to CBDC.
  - In lower-income economies: larger financial inclusion margin and smaller disintermediation margin; reverse in higher-income economies.
- Policy design note:
  - If low-income unbanked households receive more government payments and remittances, designing CBDC to reduce remittance costs and facilitate efficient government payments is more effective at boosting financial inclusion with less disintermediation.

*Source: Excerpt from "2.1 Households" and related subsections in the provided chapter.*

### 4.  Cash, CBDC, and Card payments (Formal)

### 4.  Cash, CBDC, and Card payments (Formal)

### Household payment groups and firm revenue functions
- Three household groups from optimization:
  - g0: households that only pay in cash.
  - gc: households that pay in CBDC if accepted and pay in cash otherwise.
  - gb: households that pay in card payments if accepted, if not pay in CBDC if accepted, and pay in cash otherwise.
- Group membership conditions expressed in terms of vc and vb (preserved in text).
- Firm revenue by accepted payment modes (period t):
  - Cash only firms:
    - Revenue = (1−f0) Σi c_{t,i}(j)
    - f0 is the cost of transacting in and handling cash. “Only Cash” firms are informal and evade taxes.
  - Cash and card:
    - Revenue = (1−fb−τb) Σ_{i∈gb} c_{t,i}(j) + (1−f0) Σ_{i∈g0,gc} c_{t,i}(j) − Kb
    - fb: fees for card transactions; τb: taxes on card transactions; Kb: fixed cost to accept cards.
  - Cash and CBDC:
    - Revenue = (1−fc−τc) Σ_{i∈gc} c_{t,i}(j) + (1−f0) Σ_{i∈g0,gb} c_{t,i}(j) − Kc
    - fc: fees for CBDC (incurred by firm and not passed on to households); τc: taxes on CBDC transactions; Kc: fixed cost to accept CBDC.
    - Note: government can exempt (targeted/small) firms from taxes for CBDC transactions, so τc ≤ τb.
    - Assumption: Kc < Kb (low cost is a policy objective of CBDC).
  - Cash, CBDC and card:
    - Revenue = (1−fb−τb) Σ_{i∈gb} c_{t,i}(j) + (1−fc−τc) Σ_{i∈gc} c_{t,i}(j) + (1−f0) Σ_{i∈g0} c_{t,i}(j) − Kc,b
    - Kc,b is fixed cost to accept both CBDC and card payments. Assumed Kc,b > Kb and Kc,b ≤ Kb + Kc.
- Firms choose acceptance by trading off revenue from groups gx ∈ {g0,gc,gb} (Σ_{i∈gx} c_{t,i}(j)) against fixed adoption costs. Benefits are zero if no households prefer that mode.
- Σ_{i∈gx} c_{t,i}(j) is large when:
  - |gx| is large (many households prefer mode x).
  - Households using mode x are high income (higher c_{t,i}(j)).
  - j is large (firm sells higher value good).

### Firm acceptance thresholds and comparative statics
- Firms indexed j segmented by acceptance:
  - Firms j ∈ [0, jc]: only accept cash.
  - Firms j ∈ [jc, jb]: accept cash and CBDC.
  - Firms j ∈ [jb, jc,b]: accept cash and card.
  - Firms j ∈ [jc,b, 1]: accept cash, CBDC, and card.
- Boundary cases:
  - If f0 < fb + τb and f0 < fc + τc → all firms only accept cash, jc, jb, jc,b = 1.
  - If f0 > fb + τb > fc + τc → no benefit to cards over CBDC, 0 < jc ≤ 1 = jb, jc,b. High-value goods firms accept CBDC; lower-value firms accept only cash.
    - jc decreases (more firms accept CBDC) if: many households have CBDC wallets or bank accounts preferring CBDC; these households are high income; Kc and Kc,b decrease; fc + τc decreases.
  - If f0 > fc + τc > fb + τb → card payments preferred over CBDC. Two cases:
    - Case 1: (1−fb−τb) Σ_{i∈gb} c_{t,i}(j) + (1−f0) Σ_{i∈gc} c_{t,i}(j) < (1−fc−τc) [Σ_{i∈gb} c_{t,i}(j) + Σ_{i∈gc} c_{t,i}(j)]. Firms prefer CBDC over cards overall (since Kb > Kc). Then 0 < jc < jb = jc,b < 1.
    - Case 2: above inequality reversed or equal. Firms may prefer cards over CBDC. Then 0 < jc ≤ jb ≤ jc,b < 1; CBDC may not be adopted by firms if jc = jb.
    - jb and jc,b decrease (more firms accept cards) if: many households have bank accounts and prefer cards; those households are high income; Kb and Kc,b decrease; fb + τb decreases.

### Impact of CBDC on firms: formalization and displacement
- Two potential margins of impact illustrated:
  1. Formalization: informal firms (cash-only) choose to accept CBDC (or card) after CBDC issuance.
  2. Displacement: firms that accepted card payments stop accepting cards and accept CBDC instead.
- Formalization increases when:
  - Many households have bank accounts and CBDC wallets and prefer CBDC over cash.
  - These households are high income (bring in revenue).
  - Fixed cost to accept CBDC (Kc) is low.
  - Fees associated with CBDC are low relative to cash handling.
  - Tax exemptions or subsidies for CBDC transactions exist.
- Displacement of card payments by CBDC is small or zero if:
  - Size and income of households preferring CBDC over cards is small (vc > vb holds for small group).
  - Fixed and variable costs of cards relative to CBDC for firms are small.
  - Tax exemptions or subsidies for CBDC payments are small.
- CBDC issuance could increase card acceptance if Kc,b − Kc is small, making accepting both modes worthwhile.

### Dynamics, timing, and coordination problems
- Transition dynamics:
  - Start at steady state “no CBDC” at t = 0; introduce CBDC from t = 1 onwards.
  - In each period t: firms choose acceptance based on household choices in t−1; households then choose bank/CBDC wallet ownership after observing firm decisions; economy converges to new steady state.
- Coordination failure / low adoption steady state:
  - Households cannot benefit from CBDC’s payments value (vc) if merchants do not accept it; merchants have no incentive to pay fixed cost Kc if households do not adopt.
  - If non-merchant-dependent CBDC benefits are weak (rc low/zero; costs for remittances and government payments ec, κc high and close to e0, κ0) or beneficiary household populations are too low income, positive feedback between firms and households may never start → economy stuck at low adoption steady state.
- Policy tools to spur adoption:
  - Temporary subsidies to spur initial adoption and move economy to welfare-improving steady state with high CBDC usage.
  - Temporary tax exemptions for transactions in CBDC for targeted firms.
  - Using CBDC for government payments or making receiving government payments via CBDC more cost effective (increase κ0 and decrease κc).
- Model remarks:
  - Baseline assumes rb fixed. Appendix allows banks market power raising interest rates in response to deposit outflows, reducing switching to CBDC (less disintermediation).
  - Results robust to introducing switching frictions/costs δh for households and δf for firms.

### Calibration and simulation setup
- Calibrated to a representative developing country with large informal sector and unbanked population.
- Households receive wage income iw, remittances ir, government payments ig using administrative household survey data (Household Income Sources Survey (2019)).
- Firm distribution w calibrated using administrative firm survey data (Office of Industrial and Commercial Property Firms Survey).
- Overlapping generations: k = 2 (two-period-lived), constant population. Utility u(x) = log(x).
- Assumptions:
  - Banked population share = 57% (World Bank and central bank data).
  - 80% of lowest-value goods selling firms are informal (jb,0 = 0.2), corresponding to informal sector representing 30% of total output (Quarterly Informal Economy Survey).
  - Back out Fb (fixed cost of owning bank account) and Kb (fixed cost of accepting cards) in “no CBDC” equilibrium.
  - Baseline CBDC parameters (Table A2): CBDC improves efficiency of cross-border and government payments, more valuable than cash but less than cards, non-remunerated, no tax exemptions, lower fees than cards, lower fixed costs than owning bank account.

### Simulation results — overview and baseline
- Baseline steady state results (Table 1; Pre-CBDC → Post-CBDC):
  - Households (%):
    - Cash only: 43% → 8%
    - CBDC wallet: 0% → 39%
    - Bank account: 57% → 53%
    - Financial Inclusion (CBDC or Bank): 57% → 92%
    - % Change in welfare: −0.39%
  - Firms (%):
    - Cash only (Informal): 80% → 26%
    - Accept CBDC: 0% → 55%
    - Accept Card: 20% → 19%
    - Formal (CBDC and/or Card): 20% → 74%
  - Savings (%):
    - Cash: 13% → <1%
    - CBDC wallet: 0% → 14%
    - Bank deposits: 87% → 86%
- Interpretation:
  - Economy transitions from no-CBDC to steady state with 39% of households and 55% of firms adopting CBDC.
  - Household welfare increases by 0.39% (note measured gains exclude potential tax revenue gains from formalization).
  - Financial inclusion increases by 35 percentage points (57% → 92%).
  - Informal firms drop by 54 percentage points (80% → 26%).
  - Disintermediation is relatively low: 1 p.p. decrease in total savings share in bank deposits; 4 p.p. decrease in bank account ownership.
  - Displacement of card payments is small: 1 p.p. decrease in firms accepting cards.

### Low adoption steady state and role of temporary incentives
- Scenario: CBDC with no improvements to cross-border and government payments (ec = e0 and κc = κ0). CBDC offers non-remunerated savings vehicle and payment means only.
- Without incentives (Pre-CBDC → Post-CBDC w/o incentives):
  - Households (%):
    - Cash only: 43% → 42%
    - CBDC wallet: 0% → 1%
    - Bank account: 57% → 57%
    - Financial Inclusion: 57% → 58%
    - % Change in welfare: −0.01%
  - Firms (%):
    - Cash only (Informal): 80% → 80%
    - Accept CBDC: 0% → <1%
    - Accept Card: 20% → 20%
    - Formal (CBDC and/or Card): 20% → 20%
  - Savings (%):
    - Cash: 13% → 13%
    - CBDC wallet: 0% → 0%
    - Bank deposits: 87% → 87%
- Interpretation:
  - CBDC wallet adoption remains persistently low at 1% → near zero firm take-up and near zero impact on welfare, inclusion, and formalization.
  - Coordination failure: benefits not sufficient to trigger firm-household adoption feedback loop.
- Temporary incentive package (applied in first period) and effects (Post-CBDC w/ incentives):
  - Policy measures in first period:
    - 20% decrease in Fc for below median income households.
    - τc = 0 (CBDC transactions exempt from taxes).
    - Increase κ0 by 1 p.p. and decrease κc by 0.5 p.p. (government payments administered through CBDC / more cost effective).
  - Outcomes (Pre-CBDC → Post-CBDC w/ incentives):
    - Households (%):
      - Cash only: 43% → 21%
      - CBDC wallet: 0% → 23%
      - Bank account: 57% → 56%
      - Financial Inclusion: 57% → 79%
      - % Change in welfare: −0.01% → 0.25%
    - Firms (%):
      - Cash only (Informal): 80% → 31%
      - Accept CBDC: 0% → 49%
      - Accept Card: 20% → 20%
      - Formal (CBDC and/or Card): 20% → 69%
    - Savings (%):
      - Cash: 13% → 4%
      - CBDC wallet: 0% → 9%
      - Bank deposits: 87% → 87%
  - Interpretation:
    - Temporary incentives spur high initial take-up, which declines slightly after measures expire but successfully pushes economy out of low adoption steady state.
    - Steady state with 23% households and 49% firms adopting CBDC.
    - Household welfare increases by 0.25%.
    - Financial inclusion increases by 22 p.p. (57% → 79%).
    - Informal firms drop by 49 p.p. (80% → 31%).
    - Disintermediation small: <1 p.p. decrease in savings held in banks and 1 p.p. decrease in bank account ownership.
    - No displacement of card payments observed in this scenario.

### Additional simulation findings (summary points in text)
- Higher income countries: smaller financial inclusion and formalization gains; greater disintermediation and displacement risks.
- Interest-bearing CBDC: can increase adoption with slightly greater inclusion and formality gains but large disintermediation and displacement impacts.
- If fixed cost to own a CBDC wallet is high relative to bank account, previously unbanked may open bank accounts to gain CBDC access in a two-tier system (bank intermediation).
- If CBDC does not reduce fixed cost barriers firms face in accepting card payments, overall adoption is low despite some household willingness to open CBDC wallets.

*Source: wpiea2023127-print-pdf - 4.  Cash, CBDC, and Card payments (Formal)*

### 4.3    Country income level

### 4.3    Country income level

### Summary
- The analysis increases household incomes by 10%, producing a higher income population with a smaller unbanked population (18%) and a smaller informal sector (j_b,0 = 0.3) in the “no CBDC” equilibrium.
- Results for the higher income country scenario are reported in Table 3 and Figure 10.
- Main qualitative finding: financial inclusion and formalization gains from CBDC are smaller in higher income economies, while disintermediation and displacement risks are greater.

### Quantitative impacts — Households (Table 3)
- Pre-CBDC / Post-CBDC:
  - Cash only: 18% → 5%
  - CBDC wallet: 0% → 25%
  - Bank account: 82% → 60%
  - Financial Inclusion (CBDC or Bank): 82% → 95%
- % Change in welfare: -0.25%
- Reported comparative margins versus lower-income baseline:
  - Financial inclusion gain: 13 p.p. (vs 35 p.p. in lower-income baseline)
  - Formality gain: 21 p.p. (vs 54 p.p. in lower-income baseline)
- Disintermediation (household-level):
  - Decrease in share of households with a bank account who switch to owning a CBDC wallet: 22 p.p. (compared to 4 p.p. in baseline)

### Quantitative impacts — Firms and Payments (Table 3)
- Pre-CBDC / Post-CBDC:
  - Cash only (Informal): 30% → 9%
  - Accept CBDC: 0% → 91%
  - Accept Card: 70% → 69%
  - Formal (CBDC and/or Card): 70% → 91%
- Displacement of card payments:
  - Decrease in share of firms accepting card payments: 1 p.p.

### Quantitative impacts — Savings (Table 3)
- Pre-CBDC / Post-CBDC:
  - Cash: 3% → <1%
  - CBDC wallet: 0% → 8%
  - Bank deposits: 97% → 91%
- Disintermediation (savings-level):
  - Decrease in share of savings in bank deposit accounts: 6 p.p. (compared to 1 p.p. in baseline)

### Comparison with baseline (lower-income calibration)
- Financial inclusion increase is smaller: 13 p.p. (higher income) vs 35 p.p. (lower income).
- Formalization increase is smaller: 21 p.p. (higher income) vs 54 p.p. (lower income).
- Disintermediation larger in higher income scenario:
  - Bank account ownership decrease: 22 p.p. switching to CBDC wallets (higher income) vs 4 p.p. (baseline).
  - Share of savings in bank deposits falls by 6 p.p. (higher income) vs 1 p.p. (baseline).
- Card payment displacement modest: 1 p.p. in higher income case.

### Implications and interpretation
- In higher income economies with smaller unbanked populations and informal sectors, the margin for gains in financial inclusion and formalization from CBDC issuance is limited.
- The risks of bank disintermediation and shifts in savings away from bank deposits can be substantially larger in higher income settings.
- Trade-off: designing CBDC to promote greater adoption may amplify disintermediation and payment-system displacement in higher income economies, while the benefits from inclusion and formalization are comparatively smaller.

*Source: 4.3 Country income level — Central Bank Digital Currency Adoption: A Two-Sided Model, Working Paper No. WP/2023/127*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023127-print-pdf.pdf_