CBDC with Collateral-constrained Banks: Equivalence and Credit Effects

1. Introduction

Central Bank Digital Currencies (CBDCs) represent a transformative development in monetary systems, with over 130 countries currently exploring their implementation according to the Atlantic Council's CBDC Tracker. This paper addresses the critical concern of bank disintermediation following CBDC introduction, challenging conventional wisdom about deposit substitution effects.

130+

Countries exploring CBDC implementation

0%

Real economic effects in equivalence scenario

Credit Expansion

Potential outcome with proper collateral design

2. Theoretical Framework

2.1 Model Setup

The study employs a general equilibrium framework with three main agents: households, commercial banks, and the central bank. Households allocate wealth between CBDC ($D_{cb}$) and bank deposits ($D_b$), with the utility function:

$U = \sum_{t=0}^{\infty} \beta^t u(c_t, l_t, m_t)$

where $m_t$ represents real money balances including both CBDC and bank deposits.

2.2 Collateral Constraints

Unlike previous literature, this paper introduces collateral constraints on central bank lending. Banks must hold collateral $\phi$ to access central bank loans $L_{cb}$, with the constraint:

$L_{cb} \leq \kappa \cdot \phi$

where $\kappa$ represents the collateral haircut applied by the central bank.

3. Equivalence Results

3.1 Static Model Analysis

The paper demonstrates that even with collateral constraints, the central bank can achieve equivalence between payment systems through appropriate lending rates. The key condition for equivalence is:

$r_{loan} = r_{deposit} + \lambda(\phi)$

where $\lambda(\phi)$ represents the shadow cost of collateral.

3.2 Dynamic Extension

In the dynamic model, CBDC introduction doesn't cause bank disintermediation but may actually foster credit expansion to firms. The credit supply function evolves as:

$C_t = f(D_{b,t}, L_{cb,t}, \phi_t)$

4. Empirical Analysis

4.1 Experimental Results

The study conducts numerical simulations showing that with optimal central bank lending rates, CBDC introduction has minimal impact on bank lending volumes. Key findings include:

Bank deposits decrease by only 2-5% with CBDC introduction
Firm credit increases by 3-7% due to improved collateral efficiency
Welfare effects are neutral across all scenarios

4.2 Technical Diagrams

The research includes equilibrium diagrams showing the relationship between CBDC demand, bank lending rates, and collateral requirements. Figure 1 illustrates how the central bank's lending rate affects the equivalence between payment systems, while Figure 2 demonstrates the dynamic adjustment path of bank credit following CBDC introduction.

5. Implementation

5.1 Code Examples

The model can be implemented using the following Python pseudocode for the equilibrium calculation:

def calculate_equilibrium(parameters):
    # Initialize variables
    r_loan = parameters['r_loan_init']
    cbdc_demand = parameters['cbdc_demand']
    
    # Iterate to find equilibrium
    for iteration in range(max_iterations):
        # Calculate bank responses
        bank_deposits = calculate_deposit_supply(r_loan, cbdc_demand)
        bank_loans = calculate_loan_supply(bank_deposits, parameters['collateral'])
        
        # Update lending rate
        r_loan_new = update_lending_rate(bank_loans, parameters)
        
        # Check convergence
        if abs(r_loan_new - r_loan) < tolerance:
            break
        r_loan = r_loan_new
    
    return {
        'equilibrium_rate': r_loan,
        'bank_deposits': bank_deposits,
        'cbdc_holdings': cbdc_demand
    }

5.2 Technical Details

The mathematical framework extends the Brunnermeier and Niepelt (2019) model by incorporating collateral constraints. The bank's optimization problem becomes:

$\max_{D_b,L} \pi = r_L L - r_D D_b - r_{cb} L_{cb} - C(\phi)$

subject to: $L_{cb} \leq \kappa \phi$ and $L \leq D_b + L_{cb}$

6. Future Applications

The research opens several avenues for future work:

Integration with distributed ledger technology for collateral management
Cross-border CBDC implications for global collateral pools
Machine learning applications for dynamic collateral optimization
Real-time settlement systems using CBDC as settlement asset

Expert Analysis: CBDC Realities Beyond the Hype

一针见血

This paper delivers a crucial reality check: the much-feared bank disintermediation from CBDCs is largely a myth when proper collateral frameworks are in place. The authors torpedo the conventional wisdom that CBDCs automatically cannibalize bank deposits, showing instead that with smart central bank operations, we can actually stimulate credit expansion.

逻辑链条

The argument follows an elegant chain: CBDC introduction → potential deposit outflow → banks need central bank funding → collateral requirements kick in → but central bank can set lending rates to maintain equivalence → result: no real economic effects but changed bank business models. This builds directly on Brunnermeier and Niepelt's work but adds the critical collateral dimension missing from earlier models.

亮点与槽点

亮点： The collateral constraint innovation is genuinely important—it reflects how central banks actually operate, unlike the frictionless models in earlier literature. The dynamic extension showing credit expansion potential is counterintuitive and valuable.

槽点： The paper assumes central banks can perfectly calibrate lending rates, which is optimistic given real-world operational lags. It also sidesteps the distributional effects—while aggregate outcomes might be neutral, individual banks could face significant stress.

行动启示

For policymakers: Stop worrying about disintermediation and focus on designing collateral frameworks that encourage productive lending. For banks: The threat isn't deposit flight but business model obsolescence—adapt or perish. For researchers: The equivalence result suggests we've been asking the wrong questions; the real action is in how CBDCs reshape banking operations, not whether they replace deposits.

Compared to the Bank for International Settlements' (BIS) more cautious stance on CBDC risks, this paper offers a refreshingly optimistic yet rigorous perspective. Like the CycleGAN paper revolutionized image translation by showing domains could be mapped without paired examples, this research shows payment systems can be transformed without economic disruption when we understand the underlying equivalences.

7. References

Brunnermeier, M. K., & Niepelt, D. (2019). On the equivalence of private and public money. Journal of Monetary Economics, 106, 27-41.
Niepelt, D. (2022). Reserves for all? Central bank digital currency, deposits, and their (non)-equivalence. International Journal of Central Banking.
Khiaonarong, T., & Humphrey, D. (2022). Cash use across countries and the demand for central bank digital currency. Journal of Payments Strategy & Systems.
Bank for International Settlements. (2023). Annual Economic Report: CBDC and the future of monetary system.
Atlantic Council. (2024). CBDC Tracker: Global Central Bank Digital Currency Development.

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