## _wp1330

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---

### I. Introduction — purpose and contribution
- Purpose: enrich debate on Basel capital requirements by adding bank funding mode as a dimension.
- Observations motivating analysis:
  - Pre-crisis: unsecured wholesale bank finance boomed; insured retail deposits share declined.
  - Post-2007: share of collateralized funding in wholesale bank funding increased considerably.
- Methodological contribution:
  - Extends Stiglitz and Weiss (1981) (SW) credit-rationing model to allow for bank default and differentiation between debt and equity financing.
  - Capital requirements modeled as fixing a minimum Q/X (equity share), implying a maximum debt level D for given equity Q.
- Key mechanism (informal):
  - Higher capital requirements → smaller balance sheets + greater internalization of default risk by banks → banks lower loan rates to improve borrower pool quality → credit demand rises while supply falls → credit rationing increases.

### II. Model — structure and definitions
- Agents and objects:
  - N projects (indexed i = 1..N), returns R_i with identical means but differing volatilities (sorted by risk).
  - Borrower needs amount B to start project; loan rate br_b; borrower collateral C.
  - Bank assets X = Q + D (equation (7)); funding cost (1 + br_d)D; equity fixed at Q̄.
- Borrower net return and default condition:
  - Default if C + R_i < (1 + br_b) B (equation (3)).
  - Borrower net return: max{R_i − (1 + br_b)B; −C} (equation (4)).
- Bank revenues and profits:
  - Per-loan bank receipt: min{R_i + C; (1 + br_b)B} (equation (5)).
  - Total revenues sum over lent borrowers (equation (6)).
  - Bank profit: Π = max( Σ min{R_i + C; (1 + br_b)B} − (1 + br_d)D , 0 ) (equation (8)).
  - Bank chooses br_b to maximize E[Π] (equation (9)).
- Credit rationing measure:
  - Ψ = B [N − b̄_i] − M X (equation (10)), where b̄_i is cutoff index: firms with i ≥ b̄_i apply for loans.
- Important equilibrium facts from SW:
  - d b̄_i / d br_b > 0: higher loan rates attract a riskier pool (adverse selection).

### III. Comparative statics: capital requirements and credit rationing
- Lemma 1: At a given bank funding rate (br_d), higher capital requirements imply lower loan rates (br_b) and more credit rationing:
  - ∂Ψ / ∂(Q/X) > 0.
- Intuition:
  - Equity holders are payoff residual claimants (option-like). Higher leverage increases value of volatility to shareholders; forcing higher Q/X reduces appetite for volatility → banks lower br_b → demand rises while supply falls → Ψ increases.

### IV. Retail funding (deposit-insured)
- Setup:
  - Depositors fully insured (100% coverage), risk-free rate set to zero so equilibrium funding rate br_d^M = 0 (equation (11)).
- Bank reaction:
  - Lemma 2: bank optimal loan rate increases in its funding rate:
    - ∂(br_b)^* / ∂ br_d > 0.
- Graphical intuition:
  - Bank reaction function upward sloping; deposit-insurer reaction function vertical at br_d = 0.
  - Higher capital requirements shift bank reaction function downward (lower br_b for any br_d).

### V. Unsecured wholesale funding
- Setup:
  - Uninsured wholesale financiers care about bank risk; market funding rate solves 0 = E[min( Σ min{R_i + C, (1 + br_b)B} − D ; br_d D )] (equation (13)).
- Endogenous feedback:
  - Lemma 3: with unsecured wholesale funding, ∂ br_d^M / ∂ br_b > 0.
- Main comparative result:
  - Proposition 1: The impact of capital requirements on credit rationing is larger with unsecured wholesale funding than with retail funding:
    - [∂Ψ / ∂(Q/X)]^UF > [∂Ψ / ∂(Q/X)]^DI.
- Intuition:
  - Feedback loop: lower br_b → lower bank risk → lower br_d → higher charter value → less risk-taking → further lower br_b → amplifies Ψ increase relative to retail-funded case.

### VI. Secured (collateralized) wholesale funding
- Setup:
  - Bank must post collateral amount λ ≤ D in high-quality, risk-free assets; invest fraction λ/X of balance sheet in risk-free asset; remaining X − λ goes to loans.
  - Funding market condition: 0 = E[min( Σ min{R_i + C, (1 + br_b)B} − D + λ ; br_d D )] (equation (14)).
- Comparative statics:
  - Lemma 4: ∂ br_d / ∂ λ < 0 (more collateral → lower funding costs).
  - Lemma 5: ∂ br_b / ∂ λ < 0 (more collateral → bank internalizes more risk → lowers loan rates).
  - Proposition 2: Increasing collateralization of wholesale funding expands credit rationing:
    - ∂Ψ / ∂ λ > 0.
- Channels through which collateralization raises credit rationing:
  - Direct supply reduction: higher collateral requirement uses balance sheet for risk-free assets → fewer loanable funds.
  - Bank risk-taking channel: loss of collateral on default increases bank exposure → lower br_b to mitigate adverse selection.
  - Financier channel: collateral reduces financier risk → lower br_d → higher charter value → further reduces bank risk-taking and br_b.
- Historical interpretation:
  - Pre-crisis: securitized loans functioned as collateral without reducing loan supply (λ ≈ 0).
  - Post-crisis: collateral backed by public debt increased λ and thus predicted increase in credit rationing.

### VII. Mixed funding (retail + unsecured wholesale)
- Setup:
  - Representative mixed financier holds fraction φ ∈ (0,1) in insured deposits and 1−φ in unsecured wholesale funding.
  - Market funding condition expressed in equation (16).
- Comparative statics:
  - Proposition 3: The larger the share of insured deposits, the greater the extent of credit rationing:
    - ∂Ψ / ∂ φ > 0.
  - Proposition 4: The larger the share of insured deposits, the smaller the impact of capital requirements on the degree of credit rationing:
    - ∂^2 Ψ / ∂(Q/X) ∂ φ < 0.
- Interpretation:
  - More insured deposits → cheaper funding → higher charter value → bank becomes more risk-averse → lower br_b → more rationing.
  - But larger φ weakens the feedback between br_b and br_d, so capital requirement changes have a dampened effect on Ψ when φ is higher.

### VIII. Conclusions — summary of substantive findings and policy relevance
- Core findings:
  - Higher capital requirements increase credit rationing (Lemma 1).
  - Unsecured wholesale funding amplifies the impact of capital requirements on credit rationing relative to retail (Proposition 1).
  - Collateralization of wholesale funding expands credit rationing via supply, bank-incentive, and financier channels (Proposition 2).
  - Greater shares of insured retail deposits increase baseline credit rationing but reduce the sensitivity of rationing to capital requirements (Propositions 3 and 4).
- Scope and limits:
  - Analysis is positive (descriptive) — does not assess welfare implications.
  - Model abstracts from monitoring/moral hazard decisions by depositors; results on deposit insurance are model-specific.
- Policy relevance:
  - Bank funding modes materially affect how capital requirements translate into credit supply outcomes; policymakers should account for funding composition and collateral practices when assessing capital regulation effects.

*Source: _wp1330 - References . . . . . . . . . . . . . . . . . . . . . . . . .   20*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   20

### _wp1330 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   20

### I. Introduction — purpose and contribution
- Purpose: enrich debate on Basel capital requirements by adding bank funding mode as a dimension.
- Observations motivating analysis:
  - Pre-crisis: unsecured wholesale bank finance boomed; insured retail deposits share declined.
  - Post-2007: share of collateralized funding in wholesale bank funding increased considerably.
- Methodological contribution:
  - Extends Stiglitz and Weiss (1981) (SW) credit-rationing model to allow for bank default and differentiation between debt and equity financing.
  - Capital requirements modeled as fixing a minimum Q/X (equity share), implying a maximum debt level D for given equity Q.
- Key mechanism (informal):
  - Higher capital requirements → smaller balance sheets + greater internalization of default risk by banks → banks lower loan rates to improve borrower pool quality → credit demand rises while supply falls → credit rationing increases.

### II. Model — structure and definitions
- Agents and objects:
  - N projects (indexed i = 1..N), returns R_i with identical means but differing volatilities (sorted by risk).
  - Borrower needs amount B to start project; loan rate br_b; borrower collateral C.
  - Bank assets X = Q + D (equation (7)); funding cost (1 + br_d)D; equity fixed at Q̄.
- Borrower net return and default condition:
  - Default if C + R_i < (1 + br_b) B (equation (3)).
  - Borrower net return: max{R_i − (1 + br_b)B; −C} (equation (4)).
- Bank revenues and profits:
  - Per-loan bank receipt: min{R_i + C; (1 + br_b)B} (equation (5)).
  - Total revenues sum over lent borrowers (equation (6)).
  - Bank profit: Π = max( Σ min{R_i + C; (1 + br_b)B} − (1 + br_d)D , 0 ) (equation (8)).
  - Bank chooses br_b to maximize E[Π] (equation (9)).
- Credit rationing measure:
  - Ψ = B [N − b̄_i] − M X (equation (10)), where b̄_i is cutoff index: firms with i ≥ b̄_i apply for loans.
- Important equilibrium facts from SW:
  - d b̄_i / d br_b > 0: higher loan rates attract a riskier pool (adverse selection).

### III. Comparative statics: capital requirements and credit rationing
- Lemma 1: At a given bank funding rate (br_d), higher capital requirements imply lower loan rates (br_b) and more credit rationing:
  - ∂Ψ / ∂(Q/X) > 0.
- Intuition summarized:
  - Equity holders are payoff residual claimants (option-like). Higher leverage increases value of volatility to shareholders; forcing higher Q/X reduces appetite for volatility → banks lower br_b → demand rises while supply falls → Ψ increases.

### IV. Retail funding (deposit-insured)
- Setup:
  - Depositors fully insured (100% coverage), risk-free rate set to zero so equilibrium funding rate br_d^M = 0 (equation (11)).
- Bank reaction:
  - Lemma 2: bank optimal loan rate increases in its funding rate:
    - ∂(br_b)^* / ∂ br_d > 0.
- Graphical intuition (Figure 1):
  - Bank reaction function upward sloping; deposit-insurer reaction function vertical at br_d = 0.
  - Higher capital requirements shift bank reaction function downward (lower br_b for any br_d).

### V. Unsecured wholesale funding
- Setup:
  - Uninsured wholesale financiers care about bank risk; market funding rate solves 0 = E[min( Σ min{R_i + C, (1 + br_b)B} − D ; br_d D )] (equation (13)).
- Endogenous feedback:
  - Lemma 3: with unsecured wholesale funding, ∂ br_d^M / ∂ br_b > 0.
- Main comparative result:
  - Proposition 1: The impact of capital requirements on credit rationing is larger with unsecured wholesale funding than with retail funding:
    - [∂Ψ / ∂(Q/X)]^UF > [∂Ψ / ∂(Q/X)]^DI.
- Intuition:
  - Feedback loop: lower br_b → lower bank risk → lower br_d → higher charter value → less risk-taking → further lower br_b → amplifies Ψ increase relative to retail-funded case.

### VI. Secured (collateralized) wholesale funding
- Setup:
  - Bank must post collateral amount λ ≤ D in high-quality, risk-free assets; invest fraction λ/X of balance sheet in risk-free asset; remaining X − λ goes to loans.
  - Funding market condition: 0 = E[min( Σ min{R_i + C, (1 + br_b)B} − D + λ ; br_d D )] (equation (14)).
- Comparative statics:
  - Lemma 4: ∂ br_d / ∂ λ < 0 (more collateral → lower funding costs).
  - Lemma 5: ∂ br_b / ∂ λ < 0 (more collateral → bank internalizes more risk → lowers loan rates).
  - Proposition 2: Increasing collateralization of wholesale funding expands credit rationing:
    - ∂Ψ / ∂ λ > 0.
- Channels (three):
  - 1) Direct supply reduction: higher collateral requirement uses balance sheet for risk-free assets → fewer loanable funds.
  - 2) Bank risk-taking channel: loss of collateral on default increases bank exposure → lower br_b to mitigate adverse selection.
  - 3) Financier channel: collateral reduces financier risk → lower br_d → higher charter value → further reduces bank risk-taking and br_b.
- Historical note (model interpretation):
  - Pre-crisis: securitized loans functioned as collateral without reducing loan supply (λ ≈ 0).
  - Post-crisis: collateral backed by public debt increased λ and thus predicted increase in credit rationing.

### VII. Mixed funding (retail + unsecured wholesale)
- Setup:
  - Representative mixed financier holds fraction φ ∈ (0,1) in insured deposits and 1−φ in unsecured wholesale funding.
  - Market funding condition expressed in equation (16).
- Comparative statics:
  - Proposition 3: The larger the share of insured deposits, the greater the extent of credit rationing:
    - ∂Ψ / ∂ φ > 0.
  - Proposition 4: The larger the share of insured deposits, the smaller the impact of capital requirements on the degree of credit rationing:
    - ∂^2 Ψ / ∂(Q/X) ∂ φ < 0.
- Interpretation:
  - More insured deposits → cheaper funding → higher charter value → bank becomes more risk-averse → lower br_b → more rationing.
  - But larger φ weakens the feedback between br_b and br_d, so capital requirement changes have a dampened effect on Ψ when φ is higher.

### VIII. Conclusions — summary of substantive findings
- Core findings:
  - Higher capital requirements increase credit rationing (Lemma 1).
  - Unsecured wholesale funding amplifies the impact of capital requirements on credit rationing relative to retail (Proposition 1).
  - Collateralization of wholesale funding expands credit rationing via supply, bank-incentive, and financier channels (Proposition 2).
  - Greater shares of insured retail deposits increase baseline credit rationing but reduce the sensitivity of rationing to capital requirements (Propositions 3 and 4).
- Scope and limits:
  - Analysis is positive (descriptive) — does not assess welfare implications.
  - Model abstracts from monitoring/moral hazard decisions by depositors; results on deposit insurance are model-specific.
- Policy relevance:
  - Highlights that bank funding modes materially affect how capital requirements translate into credit supply outcomes; policymakers should account for funding composition and collateral practices when assessing capital regulation effects.

*Source: _wp1330 - References . . . . . . . . . . . . . . . . . . . . . . . . . .   20*

### References

### References

### Bank capital regulation and lending
- Berger, Allen N., and Gregory F. Udell, 1994, "Did Risk-Based Capital Allocate Bank Credit and Cause a "Credit Crunch" in the United States?" Journal of Money, Credit and Banking, Vol. 26, pp. 585-628.
- Furlong, Frederick T., 1992, "Capital Regulation and Bank Lending," Federal Reserve Bank of San Fransisco Economic Review, Vol. 3, pp. 23-33.
- Hancock, Diana, and James A. Wilcox, 1994, "Bank Capital and Credit Crunch: the Roles of Risk-Weighted and Unweighted Capital Regulations," Journal of the American Real Estate Urban Economic Association, Vol. 22, pp. 59-94.
- Hancock, Diana, Andrew J. Laing, and James A. Wilcox, 1995, "Bank Capital Shocks: Dynamic E§ects on Securities, Loans, and Capital," Journal of Banking and Finance, Vol. 19, pp. 661-677.
- Haubrich, Joseph G., and Paul Wachtel, 1993, "Capital Requirements and Shifts in Commercial Bank Portfolios," Federal Reserve Bank of Cleveland Economic Review, Vol. 29, pp. 2-15.
- Jacklin, Charles J., 1993, "Bank Capital Requirements and Incentives for Lending," Federal Reserve Bank of San Fransisco Working Paper 93-07 (San Fransisco: Federal Reserve Bank).
- Thakor, Anjan V., 1996, "Capital Requirements, Monetary Policy, and Aggregate Bank Lending: Theory and Empirical Evidence," Journal of Finance, Vol. 51, pp. 279-324.
- BIS Macroeconomic Assessment Group Report, 2010, "Assessing the Macroeconomic Impact of the Transition to Stronger Capital and Liquidity Requirements," (Basel).

### Monetary policy, risk-taking, and transmission mechanisms
- Adrian, Tobias and Hyun Song Shin, 2010, "Financial Intermediaries and Monetary Economics," Handbook of Monetary Economics 3A, Ch. 12, eds. Benjamin Friedman and Michael Woodford, pp. 601-650.
- Altunbas, Yener, Leonardo Gambacorta, and David Marquez-Ibanez, 2010, "Bank Risk and Monetary Policy," Journal of Financial Stability, Vol. 6, pp. 121-129.
- Agur, Itai and Maria Demertzis, 2012, "Excessive Bank Risk-Taking and Monetary Policy," ECB Working Paper No. 1457 (Frankfurt: ECB).
- Borio, Claudio, and Haibin Zhu, 2012, "Capital Regulation, Risk-Taking and Monetary Policy: A Missing Link in the Transmission Mechanism?" Journal of Financial Stability, Vol. 8, pp. 236-251.
- Maddaloni, Angela, and JosÈ-Luis Peydro, 2011, "Bank Risk-Taking, Securitization, Supervision, and Low Interest-Rates: Evidence from the Euro Area and US Lending Standards," Review of Financial Studies, Vol. 24, pp. 2121-65.

### Credit crunch, borrower costs, and lending spreads
- Bernanke, Ben S., Cara S. Lown, and Benjamin N. Friedman, 1991, "Credit Crunch," Brookings Papers on Economic Activity, Vol. 2, pp. 205-247.
- Peek, Joe, and Eric S. Rosengren, 1995a, "Bank Regulation and the Credit Crunch," Journal of Banking and Finance, Vol. 19, pp. 679ñ692.
- Peek, Joe, and Eric S. Rosengren, 1995b, "The Capital Crunch: Neither a Borrower Nor a Lender Be," Journal of Money, Credit and Banking, Vol. 27, pp. 625-638.
- Hubbard, R. Glenn, Kenneth N. Kuttner, and Darius N. Palia, 2002, "Are There Bank E§ects in BorrowersíCosts of Funds? Evidence From a Matched Sample of Borrowers and Banks," Journal of Business, Vol. 75, pp. 559-581.
- Santos, Jo„o A.C., and Andrew Winton, 2010, "Bank Capital, Borrower Power, and Loan Rates," Working Paper.
- Lown, Cara S., and Stavros Peristiani, 1996, "The Behavior of Consumer Loan Rates During the 1990 Credit Slowdown," Journal of Banking and Finance, Vol. 20, pp. 1673-1694.
- Fisher, Markus, Julian Mattes, Sascha Ste§en, 2010, "Bank Capital Ratios, Competition and Loan Spreads," Working Paper.

### Causes, remedies, and principles of regulation
- Diamond, Douglas W., and Raghuram G. Rajan, 2009, "The Credit Crisis: Conjectures About Causes and Remedies," American Economic Review, Papers and Proceedings, pp. 606-610.
- Brunnermeier, Markus, Andrew Crockett, Charles Goodhart, Avinash D. Persaud, and Hyun Song Shin, 2009, "The Fundamental Principles of Regulation," Geneva Reports on the World Economy 11 (Geneva).
- Institute of International Finance Research Note, 2011, "Bank Funding and Regulatory Reform." (Washington).

### Theory, information frictions, and tools
- Stiglitz, Joseph E., and Andrew Weiss, 1981, "Credit Rationing in Markets with Imperfect Information," American Economic Review, Vol. 71, pp. 393ñ410.
- Hull, John C., 2002, Options, Futures, and Other Derivatives (5th ed.) (New Jersey: Prentice Hall).

*Source: _wp1330 - References*

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