## wpiea2023069-print-pdf - Section 6 concludes.

## Source details

**Canonical URL:** [wpiea2023069-print-pdf - Section 6 concludes.](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023069-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2023/english/wpiea2023069-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2023/english/wpiea2023069-print-pdf.pdf.json)

---

### Model: summary and timing
- Two periods T∈ {1, 2}. Households maximize utility over two periods and choose savings s and borrowing b for investment in production technology F(.). Credit markets are imperfect and there is no consumer credit (i.e. s≥0).
- Household types t∈ {g,b}: "good" households succeed with probability p_g and "bad" households with probability p_b, where p_b < p_g.
- Households endowed with wealth ω. Opening a bank account entails fixed cost C and gives access to a deposit account and CBDC wallet.
- Period 1 (T=1): consumption c_1 paid using CBDC with convenience value v if they have a bank account or cash. Save in deposit (deposit interest rate r_d and liquidity risk ℓ with r_d − ℓ > −d), CBDC wallet (remuneration r_c), or cash (cost of storage −d). Borrow at loan interest r_n for no CBDC usage or r_t by type t for CBDC users.
- Period 2 (T=2): household success with probability p_t; consume profits and remaining savings c_2; payments using CBDC again (convenience value v) if they have a bank account or cash.
- Commercial bank sets deposit rate r_d, learns household type t if household uses CBDC for consumption, and sets loan rates r_n (no CBDC), r_g (CBDC, type g), r_b (CBDC, type b). Bank constrained by reserve requirement m and free entry.
- Central bank decides whether to issue CBDC and sets CBDC remuneration r_c and reserve requirement m. Timing and sequencing illustrated in Figure 1 (in source).

### Households: baseline (no CBDC)
- Household problem: U^0 = max{U^n, U^b}.
- Utility from no bank account U^n: constrained by
  - c_1 = ω − s
  - c_s^2 = s(1 − d) + F(b) − r_n b
  - c_b^2 = s(1 − d)
- Utility from opening bank account U^b: constrained by
  - c_1 = ω − s − C
  - c_s^2 = s(1 + r_d − ℓ) + F(b) − r_n b
  - c_b^2 = s(1 + r_d − ℓ)
- Key result: richer households (higher ω) open bank accounts; existence of threshold ̄ω_0 such that households open a bank account i.f.f. ω ≥ ̄ω_0 (Proposition 2.1).
- Comparative statics: poorer households open bank account if r_d is high, d is high, ℓ is low, and C is low. Lower loan interest rates increase lending and household utility (Proposition A.2).

### Households: CBDC scenario (two-tier model assumed)
- Household problem: U^c = max{U^n, U^b,dd, U^b,cd, U^b,dc, U^b,cc}. Subscripts dd, cd, dc, cc denote saving/payment choices in deposit (d) or CBDC (c).
- When opening bank account and saving in deposits but using CBDC for payments (U^b,dd), convenience value v scales consumption: c_1(1 − v) = ω − s − C; analogous scaling in period 2 constraints.
- When saving in CBDC (U^b,cd) constraints replace r_d − ℓ with r_c.
- Cash can be used to avoid revealing type to bank (U^b,dc and U^b,cc).
- Bank disintermediation: conditional on bank account, households save in CBDC instead of deposits if r_c > r_d − ℓ (Proposition 2.2).
- Payment behavior by type:
  - g-type: always make payments in CBDC if v > 0 or r_g < r_n (Proposition 2.3).
  - b-type: there exists threshold ̄v such that they make payments in CBDC iff v ≥ ̄v (Proposition 2.4).
- CBDC incentivizes previously unbanked households to open bank accounts (access to v > 0) and gives g-types incentive to open accounts to build credit and obtain lower loan rate r_g < r_n.
- Wealth thresholds for account opening under CBDC:
  - Existence of thresholds ̄ω_{c,g} and ̄ω_{c,b}: g-type open iff ω ≥ ̄ω_{c,g}; b-type open iff ω ≥ ̄ω_{c,b} (Proposition 2.5).
  - ̄ω_{c,g} < ̄ω_0 if v > 0 or r_g < r_b.
  - ̄ω_{c,b} ≤ ̄ω_0.
  - ̄ω_{c,g} < ̄ω_{c,b} if r_g < r_b.
- If banks can identify account ownership and non-use of CBDC, concealment via cash is impossible and all account holders will use CBDC for payment (Proposition A.3); this may deter some b-types from opening accounts (Proposition 2.6).
- If b-types can hide type (prevent bank access to CBDC data), then b-types face no downside from opening accounts and all with accounts use CBDC for payment (Proposition A.4; Proposition 2.7).
- Poorer households open accounts under CBDC if v is high, loan interest for type-t is low, and r_c is high (Proposition A.5).

### Commercial bank: pricing, reserves, and market clearing
- Bank faces population {ω(i), ℓ(i), t(i)} and production technologies F_i(.); reserve requirement m: total lending ≤ m × total deposits.
- Total deposits D(r_d) = Σ_i d_i(r_d) where d_i(r_d) is optimal deposit savings.
- Free-entry implies zero profits; break-even loan rates:
  - r_n^* = r_d / (m E[p_t | no CBDC use])
  - r_g^* = r_d / (m p_g)
  - r_b^* = r_d / (m p_b)
- Deposits flowing to CBDC (disintermediation) forces higher deposit rates to attract deposits, reducing lending.
- Total lending L(r_d) = Σ_i b_i(r_d) and market clearing requires L(r_d) ≤ m D(r_d).
- Existence: there exist prices {r_d^*, r_n^*, r_g^*, r_b^*} such that households maximize utility, bank maximizes profits, and markets clear (Proposition 2.8).
- Wholesale funding extension: with wholesale funding W at rate r_w, market clearing becomes L(r_d) ≤ m D(r_d) + W.

### Central bank: CBDC issuance and design choices
- Central bank decides whether to issue CBDC; calibration assumes r_c = 0.
- Central bank sets reserve requirement m.
- Two-tier model discussed; alternative with non-bank PSPs distributing CBDC considered in Section 4.
- Policy choice: central bank may prohibit commercial banks from using CBDC payment data for credit-building for privacy/legal reasons; implications explored in Subsection 3.1.

### Optimal policy: CBDC effects on lending and welfare
- Equilibrium deposit interest under CBDC and baseline denoted r_d^{c*} and r_d^{0*}; loan rates pinned down by break-even expressions above.
- CBDC issuance always increases share of population with bank accounts (financial inclusion) (follows from Proposition 2.5).
- Change in lending ∆L between baseline and CBDC scenarios expressed as (Equation (37)) with three terms:
  - Inflows from previously unbanked who open accounts and deposit (first integral term).
  - Outflows from previously banked who shift savings to CBDC (second integral term).
  - Change in savings among previously banked (third integral term).
- Main drivers: (1) inflows from previously unbanked, (2) outflows from previously banked (disintermediation), (3) change in savings by previously banked. First two drivers typically larger.
- Conditions affecting ∆L:
  - Lower ℓ(i) reduces disintermediation and increases deposits available for lending.
  - Size of deposit inflows from previously unbanked depends on population with ω(i) ∈ [̄ω_{c,t}, ̄ω_0] and their wealth since s_i^c(r_d^{c*}) increases in ω(i).
  - CBDC value v increases number opening accounts (̄ω_{c,t} decreases in v).
  - Larger r_n − r_g (greater benefit for g-types from credit-building via CBDC) increases account openings and deposit inflows; r_n − r_g larger when p_g − p_b is larger.
- Welfare effects:
  - Increased lending and lower interest rates raise aggregate production and profits (Proposition A.2), increasing aggregate welfare.
  - CBDC can increase aggregate welfare even when ∆L < 0: there exist parameter sets {r_c, d, v, C}, population {ω(i), ℓ(i), t(i)}, and {F_i} such that ∆L < 0 and aggregate welfare increases with CBDC issuance (Proposition 3.1).
  - Three direct welfare channels from CBDC issuance:
    1. Convenience value v enters utility directly.
    2. CBDC as safe savings vehicle for households with high ℓ(i): they can earn r_c vs r_d − ℓ(i).
    3. CBDC reduces credit-risk information asymmetry: banks can price loans by observed type for CBDC users, offering r_g = r_d / (m p_g) < r_n = r_d / (m E[p_t]) for g-types and r_b = r_d / (m p_b) > r_n for b-types, improving social surplus by correcting under- and over-investment relative to pooled pricing.
- Privacy/legal design note: if central bank prohibits commercial bank use of CBDC payments data for credit-building, then r_n = r_g = r_b = r_d / (m E[p_t]) and there is no welfare gain from reduced credit-risk information asymmetry.

### 3.1 CBDC payments data use and credit building
- Disallowing CBDC payments data use for credit building:
  - Policy: central bank may prohibit commercial banks from using CBDC payments data for credit building.
  - Implication for lending: all households face pooled loan interest r_n^* = r_g^* = r_b^* = r_d m E[p_t]; fewer account openings; more negative lending impact relative to allowing data use.
  - Implication for welfare: lower lending and higher interest rates reduce aggregate production and profits; welfare losses arise from disappearance of gains from reducing credit-risk information asymmetry.
  - Quantified comparison: when commercial banks cannot use CBDC payments data, the welfare impact of CBDC issuance is 0.14%, which is 0.05 p.p. lower than the baseline welfare impact of 0.19%.
- Two-tier CBDC model with non-bank PSPs (no data sharing):
  - Non-bank PSPs distribute CBDC; cost to open a PSP CBDC wallet is C′ < C.
  - Households can open PSP CBDC wallets, save in CBDC, and use CBDC for payments without sharing data with banks.
  - Behavioral effects: more households obtain CBDC access; fewer open bank accounts; only g-types have incentive to open bank accounts to build credit; PSP wallets induce some previously banked households to switch.
  - Proposition 4.1: thresholds ̄ω_p1,t and ̄ω_p2,t exist such that type-t households open PSP wallet if ̄ω_p1,t < ω < ̄ω_p2,t, open bank accounts if ω ≥ ̄ω_p2,t, and do not open either if ω ≤ ̄ω_p1,t; ̄ω_p2,t ≥ ̄ω_c,t and ̄ω_p1,t ≤ ̄ω_c,t; ̄ω_p2,g < ̄ω_p2,b.
- Two-tier CBDC model with non-bank PSPs (with data sharing):
  - PSP-collected CBDC payments data can be shared with banks, allowing credit history building via PSP CBDC usage.
  - Modification: households can open PSP wallet and share CBDC data with banks to access interest r_t; only g-types will share their credit history (Proposition A.6).
  - Behavioral effects: households need not open bank accounts to build credit; fewer may own bank accounts; g-types have additional incentive to open PSP wallets to build shareable credit histories.
  - Proposition 4.2: thresholds ̄ω_s1,t and ̄ω_s2,t exist with ̄ω_s1,g < ̄ω_p1,g and ̄ω_s1,b = ̄ω_p1,b.

### Calibration and baseline results (developing country context)
- Model setup:
  - ω ~ Pareto with shape α and bounds L and H.
  - ℓ ~ exponential with rate λ (1/λ is mean).
  - Households type g with probability q, else b.
  - Production F_i = A_i b_i^φ with A_i = (ω − L) ε, ε uniform on [L′, H′].
  - Rich households with ω > Ω > ̄ω_0 already have credit history and can borrow at rate r_t.
  - CBDC remuneration r_c = 0.
- Baseline calibration targets:
  - Share of population with a bank account = 75%.
  - Equilibrium deposit interest rate = 3%.
- Baseline quantitative results (Table 1):
  - Change in Total Lending (%) 2.2%
  - Share of Population w/ Bank Account 92.4% (from 75% w/o CBDC)
  - Share of Population Saving in Deposits 79.7%
  - Share of Population Saving in a CBDC Wallet 12.8%
  - Share of Population Making Payments with CBDC 46.8%
  - Change in Production Profits (%) 4.5%
  - Change in Welfare (%) 0.19%
- Baseline interpretation:
  - Issuing CBDC increases total lending by 2.2%, driven by a 17 p.p. increase in bank account share (75% → 92%).
  - 14% of bank account holders (or 13% of overall population) choose to save in CBDC instead of deposits.
  - Share saving in deposits increases from 75% to 80% (a 5% increase) boosting lending.
  - 47% of the population choose to make payments in CBDC.
  - Aggregate profits from investing in household production technologies increase by 5%.
  - Total household welfare increases by 0.19%.

### Comparative statics
- Value of CBDC as means of payment (v):
  - Total lending impact increases in v with diminishing returns.
  - At v = 0, lending impact is negative because fewer previously unbanked households open bank accounts.
  - Population banked and saving in deposits increase in v.
  - Population using CBDC for payments increases in v.
  - Production profits and welfare increase in v.
  - At v = 0 the welfare impact is still positive due to credit-building and alternative savings benefits.
- Extent CBDC usage reduces credit-risk information asymmetry (p_g − p_b):
  - Parameterized p_g = p + e and p_b = p − e; vary e.
  - Total lending impact increases in e.
  - Shares with bank accounts and saving in deposits increase in e.
  - Share using CBDC for payments sharply drops after e = 0 as b-types avoid CBDC payments to hide type.
  - Production profits and welfare increase in e.
  - If commercial banks are not allowed to use CBDC payments data, all households face pooled loan rate r_n = r_d m E[p_t], equivalent to e = 0; welfare impact in this no-data-use scenario = 0.14% (5 p.p. lower than baseline 0.19%).
- Population features:
  - Liquidity risk ℓ: CBDC impact on total lending decreases with greater ℓ (more disintermediation).
  - Wealth distribution α (Pareto): increasing α implies a poorer population with larger unbanked share; as α increases more previously unbanked households open accounts in response to CBDC; CBDC impact on total lending increases in α. CBDC impact on lending is negative when α is small (wealthier countries with smaller unbanked populations more likely to see contraction in lending).

### Comparing CBDC designs (quantitative summary)
- Designs compared: (1) only commercial banks distribute CBDC; (2) commercial banks + non-bank PSPs (no data sharing); (3) commercial banks + non-bank PSPs (with data sharing).
- Key outcomes (Table 2):
  - Change in Total Lending (%) : 2.2% | -0.5% | -1.6%
  - % w/ Bank Account : 92.4% | 81.2% | 76.6%
  - % w/ Non-bank PSP Account : 0.0% | 17.8% | 22.4%
  - % Saving in a Bank Deposit Account : 79.7% | 74.4% | 73.6%
  - % Saving in a CBDC Wallet : 12.8% | 21.6% | 22.4%
  - % Making Payments with CBDC : 46.8% | 61.0% | 61.0%
  - Change in Production Profits (%) : 4.5% | 3.0% | 3.5%
  - Change in Welfare (%) : 0.19% | 0.20% | 0.21%
- Interpretation:
  - Allowing non-bank PSPs reduces bank account ownership and total lending (negative lending impacts: -0.5% without data sharing; -1.6% with data sharing).
  - Fewer households open bank accounts because they can access CBDC via PSP wallets.
  - Data sharing slightly boosts production profits (3.5% vs 3.0%) by reducing credit-risk asymmetry.
  - Two-tier models yield greater payments inclusion and CBDC savings access (61% payments usage; 22% saving in CBDC wallet under PSP distribution) and can slightly increase welfare (0.20% and 0.21%) relative to commercial-bank-only design (0.19%).
  - Trade-off: greater payments inclusion vs reduced bank account ownership and lending.

### Conclusion — policy-relevant takeaways
- CBDC issuance affects financial inclusion through two channels:
  1. Increasing bank deposits by incentivizing previously unbanked households to open bank accounts to access CBDC wallets.
  2. Allowing CBDC usage to build credit and reduce credit-risk information asymmetry.
- Conditions under which CBDC increases overall lending:
  - Bank deposit liquidity risk (disintermediation risk) is low.
  - Size and relative wealth of the previously unbanked population is large.
  - CBDC has value to households as a means of payment or for credit building.
- CBDC can be welfare-improving even when overall lending decreases because:
  - Value from using CBDC for payments.
  - CBDC as an alternative "safe" savings vehicle.
  - Surplus gains from reduced credit-risk information asymmetry.
- Two-tier models with non-bank PSP distribution:
  - Lead to fewer bank accounts and potentially lower lending.
  - If PSP-collected CBDC data is shareable, unbanked households can still build credit and access lower-rate loans.
  - Optimality depends on whether gains from broader access to CBDC outweigh lending contractions.

*Source: wpiea2023069-print-pdf - Section 6 concludes.*

### Section 6 concludes.

### wpiea2023069-print-pdf - Section 6 concludes.

### Model: summary and timing
- Two periods T∈ {1, 2}. Households maximize utility over two periods and choose savings s and borrowing b for investment in production technology F(.). Credit markets are imperfect and there is no consumer credit (i.e. s≥0).
- Households types t∈ {g,b}: "good" households succeed with probability p_g and "bad" households with probability p_b, where p_b < p_g.
- Households endowed with wealth ω. Opening a bank account entails fixed cost C and gives access to a deposit account and CBDC wallet.
- Period 1 (T=1): households pay for consumption c_1 using CBDC with convenience value v if they have a bank account or cash. Save in deposit (deposit interest rate r_d and liquidity risk ℓ with r_d − ℓ > −d), CBDC wallet (remuneration r_c), or cash (cost of storage −d). Borrow at loan interest r_n for no CBDC usage or r_t by type t for CBDC users.
- Period 2 (T=2): household success with probability p_t; consume profits and remaining savings c_2, payments using CBDC again (convenience value v) if they have a bank account or cash.
- Commercial bank sets deposit rate r_d, learns household type t if household uses CBDC for consumption, and sets loan rates r_n (no CBDC), r_g (CBDC, type g), r_b (CBDC, type b). Bank constrained by reserve requirement m and free entry.
- Central bank decides whether to issue CBDC and sets CBDC remuneration r_c and reserve requirement m. Figure 1 illustrates timing and sequencing.

### Households: baseline (no CBDC)
- Household problem: U^0 = max{U^n, U^b} (Equation (1)).
- Utility from no bank account U^n: maximize over s,b subject to
  - c_1 = ω − s (Equation (3))
  - c_s^2 = s(1 − d) + F(b) − r_n b (Equation (4))
  - c_b^2 = s(1 − d) (Equation (5))
- Utility from opening bank account U^b: maximize over s,b subject to
  - c_1 = ω − s − C (Equation (7))
  - c_s^2 = s(1 + r_d − ℓ) + F(b) − r_n b (Equation (8))
  - c_b^2 = s(1 + r_d − ℓ) (Equation (9))
- Key result: richer households (higher ω) open bank accounts, poorer households do not.
  - Proposition 2.1: Holding other parameters fixed, there exists threshold \bar{ω}_0 such that households open a bank account i.f.f. ω ≥ \bar{ω}_0.
- Comparative statics: poorer households open bank account if r_d is high, d is high (cost of storing cash), ℓ is low (liquidity risk), and C is low. Lower loan interest rates increase lending and household utility (Proposition A.2).

### Households: CBDC scenario (two-tier model assumed)
- Household problem: U^c = max{U^n, U^b,dd, U^b,cd, U^b,dc, U^b,cc} (Equation (10)).
- Notation: subscripts dd, cd, dc, cc correspond to choices of saving/payment in deposit (d) or CBDC (c) and payment method (d or c).
- When opening a bank account and saving in deposits but using CBDC for payments (U^b,dd), budget constraints incorporate convenience value v by scaling consumption (Equations (15)-(18)):
  - c_1(1 − v) = ω − s − C (Equation (16))
  - c_s^2(1 − v) = s(1 + r_d − ℓ) + F(b) − r_t b (Equation (17))
  - c_b^2(1 − v) = s(1 + r_d − ℓ) (Equation (18))
- When saving in CBDC (U^b,cd) constraints use r_c instead of r_d − ℓ (Equations (19)-(22)).
- Using cash to avoid revealing type to bank is possible (U^b,dc and U^b,cc; Equations (23)-(30)).
- Bank disintermediation: conditional on bank account, households save in CBDC instead of deposits if r_c > r_d − ℓ.
  - Proposition 2.2: U^b,dd < U^b,cd and U^b,dc < U^b,cc iff r_c > r_d − ℓ.
- Payment behavior by type:
  - Proposition 2.3: g-type households always make payments in CBDC if v > 0 or r_g < r_n (i.e. U^b,dd > U^b,dc and U^b,cd > U^b,cc).
  - Proposition 2.4: For b-type households there exists threshold \bar{v} such that they make payments in CBDC iff v ≥ \bar{v}.
- CBDC incentivizes previously unbanked households to open bank accounts (access to v > 0). Additional incentive for g-types to open accounts to build credit and receive lower loan rate r_g < r_n.
- Existence of wealth thresholds for account opening under CBDC:
  - Proposition 2.5: thresholds \bar{ω}_{c,g} and \bar{ω}_{c,b} exist such that g-type open iff ω ≥ \bar{ω}_{c,g} and b-type open iff ω ≥ \bar{ω}_{c,b}. Further:
    - \bar{ω}_{c,g} < \bar{ω}_0 if v > 0 or r_g < r_b.
    - \bar{ω}_{c,b} ≤ \bar{ω}_0.
    - \bar{ω}_{c,g} < \bar{ω}_{c,b} if r_g < r_b.
- If banks can identify account ownership and non-use of CBDC, concealment via cash is impossible; all households with accounts will use CBDC for payment (Proposition A.3). This may incentivize some b-types to avoid opening accounts.
  - Proposition 2.6: analogous thresholds exist; parts (2) and (3) indicate \bar{ω}_{c,b} can be < \bar{ω}_0 or > \bar{ω}_0 depending on parameters.
- If households can prevent bank access to CBDC data (b-types can hide type), then b-types face no downside from opening accounts; all with accounts will use CBDC for payment (Proposition A.4).
  - Proposition 2.7: thresholds exist and \bar{ω}_{c,t} < \bar{ω}_0 if v > 0; \bar{ω}_{c,g} < \bar{ω}_{c,b} if r_g < r_b.
- Poorer households open accounts under CBDC if v is high, loan interest for type-t is low, and r_c is high (Proposition A.5).

### Commercial bank: pricing, reserves, and market clearing
- Bank faces population of households i with draws {ω(i), ℓ(i), t(i)} and production technologies F_i(.); reserve requirement m: total lending ≤ m × total deposits.
- Total deposits D(r_d) = Σ_i d_i(r_d) (Equation (31)), where d_i(r_d) is optimal savings in deposits for household i.
- Free-entry (competition) implies zero profits. Expected profit rate on a loan is p_t r_t − r_d / m, yielding break-even loan rates:
  - r_n^* = r_d / (m E[p_t | no CBDC use]) (Equation (32))
  - r_g^* = r_d / (m p_g) (Equation (33))
  - r_b^* = r_d / (m p_b) (Equation (34))
- Deposits flowing to CBDC (disintermediation) forces higher deposit rates to attract deposits, reducing lending.
- Total lending L(r_d) = Σ_i b_i(r_d) (Equation (35)). Market clearing requires L(r_d) ≤ m D(r_d) (Equation (36)).
- Existence result:
  - Proposition 2.8: For any parameter set {r_c, d, v, C}, population {ω(i), ℓ(i), t(i)}, and {F_i}, there exist prices {r_d^*, r_n^*, r_g^*, r_b^*} such that households maximize utility, bank maximizes profits, and markets clear.
- Wholesale funding extension: expected profit rate and market clearing adjust when wholesale funding W at rate r_w is available; market clearing becomes L(r_d) ≤ m D(r_d) + W.

### Central bank: CBDC issuance and design choices
- Central bank decides whether to issue CBDC; calibration exercise assumes r_c = 0 (most countries not considering interest-bearing CBDC).
- Central bank sets reserve requirement m.
- CBDC "design": two-tier model discussed; alternative with non-bank PSPs distributing CBDC considered in Section 4.
- Policy choice: central bank may prohibit commercial banks from using CBDC payment data for credit-building for privacy/legal reasons; implications explored in Subsection 3.1.

### Optimal policy: CBDC effects on lending and welfare
- Define equilibrium deposit interest under CBDC and baseline as r_d^{c*} and r_d^{0*} respectively; these pin down loan rates via Equations (32)-(34).
- CBDC issuance always increases share of population with bank accounts (financial inclusion) (follows from Proposition 2.5).
- Change in lending ∆L between baseline and CBDC scenarios:
  - ∆L = m [ ∫_i s_i^c(r_d^{c*}) 1{r_d^{c*} − ℓ(i) ≥ r_c} 1{ω(i) ∈ [\bar{ω}_{c,t(i)}, \bar{ω}_0]} di  (New deposits from previously unbanked)
    − ∫_i s_i^0(r_d^{0*}) 1{r_d^{c*} − ℓ(i) < r_c} 1{ω(i) ≥ \bar{ω}_0} di  (Previously banked saving in CBDC: Disintermediation)
    + ∫_i (s_i^c(r_d^{c*}) − s_i^0(r_d^{0*})) 1{r_d^{c*} − ℓ(i) ≥ r_c} 1{ω(i) ≥ \bar{ω}_0} di  (Change in savings among previously banked) ] (Equation (37))
- Main drivers: (1) inflows of deposits from previously unbanked who open accounts, (2) outflows from previously banked who shift savings to CBDC, (3) change in savings by previously banked. First two drivers typically larger.
- Conditions affecting ∆L:
  - Lower ℓ(i) (liquidity risk) reduces disintermediation and increases deposits available for lending.
  - Size of deposit inflows from previously unbanked depends on (a) size of previously unbanked population ω(i) ∈ [\bar{ω}_{c,t}, \bar{ω}_0], and (b) wealth of those opening accounts since s_i^c(r_d^{c*}) increases in ω(i) (Lemma B.2).
  - CBDC value v increases number opening accounts ( \bar{ω}_{c,t} decreases in v; Proposition A.5), increasing deposit inflows.
  - If r_n − r_g increases (greater benefit for g-types from credit-building via CBDC), more households open accounts and deposit inflows rise; r_n − r_g larger when p_g − p_b is larger.
- Welfare effects:
  - Increased lending and lower interest rates raise aggregate production and profits (Proposition A.2), increasing aggregate welfare.
  - CBDC can increase aggregate welfare even when ∆L < 0:
    - Proposition 3.1: There exist parameter sets {r_c, d, v, C}, population {ω(i), ℓ(i), t(i)}, and {F_i} such that ∆L < 0 and aggregate welfare increases with CBDC issuance.
  - Three direct welfare channels from CBDC issuance:
    1. Convenience value v enters utility directly: households gain from using CBDC for payments.
    2. CBDC provides a safe savings vehicle for households with high liquidity risk aversion ℓ(i): they can earn r_c vs r_d − ℓ(i).
    3. CBDC reduces credit-risk information asymmetry: banks can price loans by observed type for CBDC users, offering r_g = r_d / (m p_g) < r_n = r_d / (m E[p_t]) for g-types and r_b = r_d / (m p_b) > r_n for b-types, improving social surplus by correcting under- and over-investment relative to pooled pricing.
  - Figure 4 (illustration): deadweight loss from pooled lending at r_n to b-types and g-types is reduced when banks learn types via CBDC payments data.
- Privacy/legal design note: if central bank prohibits commercial bank use of CBDC payments data for credit-building, then r_n = r_g = r_b = r_d / (m E[p_t]) and there is no welfare gain from reduced credit-risk information asymmetry.

*Italic: Source: wpiea2023069-print-pdf - Section 6 concludes.*

### 3.1  CBDC payments data use and credit building

### 3.1  CBDC payments data use and credit building

### Disallowing CBDC payments data use for credit building
- Policy: Central bank may prohibit commercial banks from using CBDC payments data for credit building due to privacy, political, or restrictive data-sharing legislation.
- Implication for lending:
  - All households face a pooled loan interest rate: r_n^* = r_g^* = r_b^* = r_d m E[p_t].
  - Disallowing data use reduces the incentive for households to open bank accounts to access CBDC for credit building, resulting in fewer account openings and smaller inflows of deposits for lending.
  - Result: a more negative lending impact compared to allowing CBDC data use.
- Implication for welfare:
  - Lower lending and higher interest rates reduce aggregate production and profits.
  - Welfare losses also arise from the disappearance of gains from reducing credit-risk information asymmetry: banks cannot offer lower loan interest rates or larger loans to "good" types, reducing aggregate household profits and aggregate welfare.
- Quantified comparison:
  - When commercial banks cannot use CBDC payments data, the welfare impact of CBDC issuance is 0.14%, which is 0.05 p.p. lower than the baseline welfare impact of 0.19%.

### Two-tier CBDC model with non-bank PSPs (no data sharing)
- Design: Non-bank payment service providers (PSPs) can distribute CBDC; cost to open a PSP CBDC wallet is C′ < C (cheaper than bank account).
- Households gain an additional choice to open a PSP CBDC wallet, save in CBDC, and use CBDC for payments without sharing data with banks.
- Behavioral effects:
  - More households obtain access to CBDC (payments and savings) because C′ < C.
  - Fewer households open bank accounts; only "good" (g-type) households have incentive to open bank accounts to build credit.
  - PSP CBDC wallet offers remuneration r_c at cheaper fixed cost C′, inducing some previously banked households to switch to PSP CBDC wallets.
- Proposition 4.1 (thresholds):
  - There exist thresholds ̄ω_p1,t and ̄ω_p2,t such that households of type t open a PSP CBDC wallet if ̄ω_p1,t < ω < ̄ω_p2,t, open bank accounts if ω ≥ ̄ω_p2,t, and do not open either if ω ≤ ̄ω_p1,t.
  - ̄ω_p2,t ≥ ̄ω_c,t for t ∈ {g,b} (fewer households will open bank accounts).
  - ̄ω_p1,t ≤ ̄ω_c,t for t ∈ {g,b} (more households have access to CBDC).
  - ̄ω_p2,t ≶ ̄ω_0 (households may open non-bank PSP CBDC wallets instead).
  - ̄ω_p2,g < ̄ω_p2,b (only g-type households have incentive to open bank accounts to build credit).

### Two-tier CBDC model with non-bank PSPs (with data sharing)
- Design: PSP-collected CBDC payments data can be shared with commercial banks (e.g., open banking), allowing credit history building via PSP CBDC usage.
- Household problem modification:
  - Households can open a PSP wallet and share CBDC data with banks to access interest r_t.
  - Only "good" (g-type) households will share their credit history (Proposition A.6).
- Behavioral effects:
  - Households need not open bank accounts to build credit or access CBDC wallets.
  - Fewer households may own bank accounts compared to the baseline without CBDC; PSP CBDC wallets are a cheaper alternative.
  - "Good" types have an additional incentive to open PSP CBDC wallets to build shareable credit histories.
- Proposition 4.2 (thresholds):
  - There exist thresholds ̄ω_s1,t and ̄ω_s2,t such that households of type t open a PSP CBDC wallet if ̄ω_s1,t ≤ ω < ̄ω_s2,t and open bank accounts if ω ≥ ̄ω_p2,t.
  - ̄ω_s1,g < ̄ω_p1,g (additional incentive for g-types to open PSP CBDC wallet).
  - ̄ω_s1,b = ̄ω_p1,b (no additional incentive for b-types).

### Calibration and baseline results (developing country context)
- Model setup:
  - ω follows a Pareto distribution with shape parameter α and minimal and maximal values L and H.
  - ℓ ~ exponential with rate λ (1/λ is mean).
  - Households are type g with probability q, b otherwise.
  - Production F_i = A_i b_i^φ with productivity correlated with wealth: A_i = (ω − L) ε, ε uniform on [L′, H′].
  - Rich households with ω > Ω > ̄ω_0 already have a credit history and can borrow at rate r_t.
  - CBDC remuneration r_c = 0.
- Baseline calibration targets:
  - Share of population with a bank account = 75% (World Bank Global Findex Database 2021).
  - Equilibrium deposit interest rate = 3%.
- Baseline quantitative results (Table 1):
  - Change in Total Lending (%) 2.2%
  - Share of Population w/ Bank Account 92.4% (from 75% w/o CBDC)
  - Share of Population Saving in Deposits 79.7%
  - Share of Population Saving in a CBDC Wallet 12.8%
  - Share of Population Making Payments with CBDC 46.8%
  - Change in Production Profits (%) 4.5%
  - Change in Welfare (%) 0.19%
- Baseline interpretation:
  - Issuing CBDC increases total lending by 2.2%, driven by a 17 p.p. increase in bank account share (75% → 92%).
  - 14% of bank account holders (or 13% of overall population) choose to save in CBDC instead of deposits.
  - Share saving in deposits increases from 75% to 80% (a 5% increase) boosting lending.
  - 47% of the population choose to make payments in CBDC.
  - Aggregate profits from investing in household production technologies increase by 5%.
  - Total household welfare increases by 0.19%.

### Comparative statics
- Value of CBDC as a means of payment (v):
  - Total lending impact increases in v with diminishing returns.
  - At v = 0, lending impact is negative (CBDC issuance causes contraction in lending) because fewer previously unbanked households open bank accounts.
  - Population banked and saving in deposits increase in v.
  - Population using CBDC for payments increases in v.
  - Production profits and welfare increase in v.
  - At v = 0 the welfare impact is still positive due to credit-building and alternative savings benefits.
- Extent CBDC usage reduces credit-risk information asymmetry (p_g − p_b):
  - Parameterized p_g = p + e and p_b = p − e; vary e.
  - Total lending impact increases in e.
  - Shares of population with bank accounts and saving in deposits increase in e.
  - Share using CBDC for payments sharply drops after e = 0 because bad types avoid CBDC payments to hide type.
  - Production profits and welfare increase in e.
  - If commercial banks are not allowed to use CBDC payments data, all households face pooled loan rate r_n = r_d m E[p_t], equivalent to e = 0.
  - Welfare impact in this no-data-use scenario = 0.14% (5 p.p. lower than baseline 0.19%).
- Population features:
  - Liquidity risk ℓ:
    - Impact of CBDC on total lending decreases with greater ℓ (more bank disintermediation; more saving in CBDC).
  - Wealth distribution α (Pareto parameter):
    - Increasing α implies a poorer population with a larger unbanked population in baseline without CBDC.
    - As α increases, more poor and previously unbanked households open bank accounts in response to CBDC; CBDC impact on total lending increases in α.
    - CBDC impact on lending is negative when α is small; wealthier countries with smaller unbanked populations are more likely to experience contraction in lending from CBDC issuance.

### Comparing CBDC designs (Table 2 summary)
- Three designs compared:
  1. Only commercial banks distribute CBDC.
  2. Commercial banks + non-bank PSPs distribute CBDC (no data sharing).
  3. Commercial banks + non-bank PSPs distribute CBDC (with data sharing).
- Key quantitative comparisons (Table 2):
  - Change in Total Lending (%) : 2.2% | -0.5% | -1.6%
  - % w/ Bank Account : 92.4% | 81.2% | 76.6%
  - % w/ Non-bank PSP Account : 0.0% | 17.8% | 22.4%
  - % Saving in a Bank Deposit Account : 79.7% | 74.4% | 73.6%
  - % Saving in a CBDC Wallet : 12.8% | 21.6% | 22.4%
  - % Making Payments with CBDC : 46.8% | 61.0% | 61.0%
  - Change in Production Profits (%) : 4.5% | 3.0% | 3.5%
  - Change in Welfare (%) : 0.19% | 0.20% | 0.21%
- Interpretation:
  - Allowing non-bank PSPs to distribute CBDC reduces bank account ownership and total lending (negative lending impacts: -0.5% without data sharing; -1.6% with data sharing).
  - Fewer households open bank accounts because they can access CBDC via PSP wallets.
  - Data sharing slightly boosts production profits (3.5% vs 3.0%) by reducing credit-risk asymmetry.
  - Two-tier models yield greater payments inclusion and CBDC savings access (61% payments usage; 22% saving in CBDC wallet under PSP distribution) and can slightly increase welfare (0.20% and 0.21%) relative to commercial-bank-only design (0.19%).
  - Trade-off: greater payments inclusion vs reduced bank account ownership and lending.

### Conclusion (policy-relevant takeaways)
- CBDC issuance affects financial inclusion through two channels:
  1. Increasing bank deposits by incentivizing previously unbanked households to open bank accounts to access CBDC wallets.
  2. Allowing CBDC usage to build credit and reduce credit-risk information asymmetry.
- Conditions under which CBDC increases overall lending:
  - Bank deposit liquidity risk (disintermediation risk) is low.
  - Size and relative wealth of the previously unbanked population is large.
  - CBDC has value to households as a means of payment or for credit building.
- CBDC can be welfare-improving even when overall lending decreases because:
  - Value from using CBDC for payments.
  - CBDC as an alternative "safe" savings vehicle.
  - Surplus gains from reduced credit-risk information asymmetry.
- Two-tier models with non-bank PSP distribution:
  - Lead to fewer bank accounts and potentially lower lending.
  - If PSP-collected CBDC data is shareable, unbanked households can still build credit and access lower-rate loans.
  - Optimality depends on whether the gains from broader access to CBDC outweigh lending contractions.

*Source: wpiea2023069-print-pdf - 3.1  CBDC payments data use and credit building.*

### References

### wpiea2023069-print-pdf - References

### References (selected citations as listed in source)
- Adrian, T., Grinberg, F., Mancini-Griffoli, T., Townsend, R., & Zhang, N. (2022). The rise of digital money: A strategic plan to continue delivering on the imf’s mandate.IMF Working Paper 22/217.
- Adrian, T., & Mancini-Griffoli, T. (2019). The rise of digital money.Annual Review of Financial Economics,13.
- Agur, I., Ari, A., & Dell’Ariccia, G. (2022). Designing central bank digital currencies.Journal of Monetary Economics,125, 62–79.
- Andolfatto, D. (2021). Assessing the impact of central bank digital currency on private banks.The Economic Journal,131. https://doi.org/10.1093/ej/ueaa073
- Auer, R., & Böhme, R. (2020). The technology of retail central bank digital currency.
- Auer, R., Cornelli, G., & Frost, J. (2020). Covid-19, cash, and the future of payments.BIS Bulletin,25.
- Bank of Denmark. (2017). Central bank digital currency in denmark?Report, Danmarks Nationalbank.
- Bank of England. (2021). New forms of digital money.Bank of England Discussion Paper.
- Bank of Israel. (2018). Report of the team to examine the issue of central bank digital currencies.Bank of Israel Report.
- Brunnermeier, M., & Payne, J. (2022). Platforms, tokens, and interoperability.
- Carapella, F., & Flemming, J. (2020). Central bank digital currency: A literature review. FEDS Notes 2020-11-09. Washington: Board of Governors of the Federal Reserve System.
- Chang, H., Gornicka, L., Grinberg, F., Miccoli, M., & Tan, B. (2023). Cbdc and banking disintermediation in a portfolio choice model.Forthcoming IMF Working Paper.
- Chiu, J., Davoodalhosseini, M., Jiang, J., & Zhu, Y. (2022). Bank market power and central bank digital currency: Theory and quantitative assessment.Journal of Political Economy. https://doi.org/10.1086/722517
- Demirguc-Kunt, A., Klapper, L., & Singer, D. (2017). Financial inclusion and inclusive growth: A review of recent empirical evidence. https : / / doi . org / 10 . 1596 / 1813 -9450-8040
- Demirgüç-Kunt, A., Klapper, L., Singer, D., & Ansar, S. (2022). The global findex database 2021: Financial inclusion, digital payments, and resilience in the age of covid-19. Washington, DC: World Bank.
- European Central Bank. (2020). Report on a digital euro.European Central Bank.
- Garratt, R., Yu, J., & Zhu, H. (2022). How central bank digital currency design choices impact monetary policy pass-through and market composition.SSRN Electronic Journal. https://doi.org/10.2139/ssrn.4004341
- IMF. (2021). The rise of digital money: A strategic plan to continue delivering on the imf’s mandate.IMF Policy Paper.
- Infante, S., Kim, K., Orlik, A., Silva, A., & Tetlow, R. (2022). The macroeconomic implications of cbdc: A review of the literature.Federal Reserve Board, Finance and Economics Discussion Series.
- Keister, T., & Sanches, D. (2022). Should central banks issue digital currency?The Review of Economic Studies. https://doi.org/10.1093/restud/rdac017
- Kosse, A., & Mattei, I. (2022). Gaining momentum–results of the 2021 bis survey on central bank digital currencies.BIS Papers No.125.
- Mancini-Griffoli, T., Peria, M. S. M., Agur, I., Ari, A., Kiff, J., Popescu, A., & Rochon, C. (2018). Casting light on central bank digital currency.IMF Staff Discussion Notes,18(08).
- Maniff, J. L. (2020).Inclusion by design: Crafting a central bank digital currency to reach all americans.
- Mester, L., Nakamura, L., & Renault, M. (2007). Transactions accounts and loan monitoring.The Review of Financial Studies 20 (3), 529–556.
- Murakami, D., Shchapov, I., & Viswanath-Natraj, G. (2022). Cbdcs, financial inclusion, and optimal monetary policy.
- Norden, L., & Weber, M. (2010). Credit line usage, checking account activity, and default risk of bank borrowers.The Review of Financial Studies 23 (10), 3665–9.
- Norges Bank. (2019). Central bank digital currencies.Norges Bank.
- Piazzesi, M., & Schneider, M. (2020). Credit lines, bank deposits or cbdc? competition efficiency in modern payment systems.Working Paper.
- Puri, M., Rocholl, J., & Steffen, S. (2017). What do a million observations have to say about loan defaults? opening the black box of relationships.Journal of Financial Intermediation 31, 1–15.
- Riksbank. (2018). The riksbank’s e-krona project.Sveriges Riksbank.
- Soderberg, G., Bechara, M., Bossu, W., Che, N. X., Kiff, J., Lukonga, I., Griffoli, T. M., Sun, T., & Yoshinaga, A. (2022). Behind the scenes of central bank digital currency: Emerging trends, insights, and policy lessons.FinTech Notes,2022(004).
- Wang, X., & Hu, X. (2022). Financial development, non-bank e-money, and central bank digital currency.manuscript.
- Whited, T., Wu, Y., & Xiao, K. (2022). Central bank digital currency and banks.

### Appendix — Propositions (statements preserved)
- A.1 Baseline Scenario – No CBDC
  - Proposition A.1.  ̄ω0 is increasing in C andℓ, and decreasing in rd and d.
  - Proposition A.2. Household borrowing which maximizes utility b∗ and household profits F(b∗)− b∗ rn are decreasing in rn.
- A.2 CBDC Scenario - without non-bank PSPs
  - Proposition A.3. If banks are able to identify that a household owns a bank account and chooses not to use CBDC when making a loan, all households always make payments in CBDC, i.e. Ub,dd ≥ Ub,dc and Ub,cd ≥ Ub,cc.
  - Proposition A.4. If "bad" b-types can always hide their type from the bank when getting a loan (choose not to allow the bank to use their CBDC data), all households always make payments in CBDC, i.e. Ub,dd ≥ Ub,dc and Ub,cd ≥ Ub,cc.
  - Proposition A.5. ̄ωc,t is increasing in rt, C andℓ, and decreasing in v, rc, rd and d.
- A.3 CBDC Scenario - with non-bank PSPs
  - Proposition A.6. (1) Up,s > Up for g-type households if rn > rg. (2) Up,s = Up for b-type households.

### Appendix — Key Lemmas and Proof outcomes (high-level)
- Lemma B.1. Existence and uniqueness of solutions maximizing Un and Ub for ω>C; respective solutions s_n(ω) and s_b(ω) strictly increasing in ω. Condition ω>C required for well-defined utility maximization for those with a bank account.
- Lemma B.2. s_ci(r) is increasing in ω(i) for CBDC savers; FOC with respect to savings for CBDC savers preserves monotonicity in ω.
- Proposition proofs establish comparative statics:
  - ̄ω0 increases with C andℓ, decreases with rd and d (Proposition A.1 and proof B.2).
  - Optimal borrowing b∗ decreases in rn; F(b∗)−b∗ rn decreases in rn (Proposition A.2 and proof B.3).
  - Comparative results on choices between CBDC and deposits depend on interest rate orderings (rc vs rd−ℓ) and on parameters v and π(r) = max_b F(b) − br (Propositions 2.2–2.6 with proofs in B.4–B.9).
  - Existence of equilibrium rates {r∗d, r∗n, r∗g, r∗b} where deposit supply D(rd) and lending demand L(rd) clear; L(rd) decreasing in loan interest rates, D(rd) increasing in rd (Proposition 2.8, proof B.13).
  - Welfare impact: with u(x)=log(x), aggregate household utility under CBDC increases in v and baseline utility is constant in v; adjusting v can raise aggregate welfare in equilibria with ∆L<0 (Proposition 3.1, proof B.15).
  - Threshold comparisons when non-bank PSPs and PSP distribution exist:
    - Thresholds ̄ωp1,t, ̄ωp2,t relative to ̄ωc,t and ̄ω0 characterized; results depend on v, C′ vs C, and rate equalities (Proposition 4.1, proof B.16).
    - Incentives differ by household types g and b for opening PSP CBDC wallets; Up,s > Up for g-types when rn > rg, Up,s = Up for b-types (Proposition A.6, proof B.17).
- Proof techniques: first-order conditions (FOCs), monotonicity, continuity arguments, construction of threshold cutoffs ̄ω and comparative statics in parameters C, ℓ, rd, d, rc, rt, v, and returns π(r).

*Italic: Source — wpiea2023069-print-pdf - References (PDF chapter/section).*

### 4.2 impliesU

### 4.2 impliesU

### Key analytical result
- For a household of type g, the PSP CBDC wallet utility when data sharing is allowed is:
  - U_p,s = max{U_n, U_p, U_p,s, U_b,dd, U_b,cd, U_b,dc, U_b,cc}
- The g-type household with ω̄_p1,g always opens a PSP CBDC wallet when data sharing is allowed.
  - Therefore, ω̄_s1,g < ω̄_p1,g.
- Equation (2) follows from identical utility maximization problems.

### Calibration parameters (Table A1: Baseline Parameters)
- p_g 0.95
- p_b 0.55
- C 0.2
- r_c 0
- β 0.9
- d 0.03
- v 0.003
- m 1
- φ 0.5
- C′ 0.175
- α 3
- L 10
- H 50
- λ 50
- q 0.5
- L′ 0.17
- H′ 0.34
- r_w 0.02
- δ 0.01
- Ω 1
- 40
- W 1
- 10000

### Notes on calibration
- This table presents the baseline parameters for the calibration exercise.
- For robustness. Similar results for all Ω∈[40,H] and W∈[0, 10000].

*Central Bank Digital Currency and Financial Inclusion Working Paper No. WP/2023/069*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023069-print-pdf.pdf_