## 1. Optimal Corrective Tax on Borrowing to Address Collapse Externality

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### I. Context and policy questions
- 2008 financial crisis prompted reconsideration of tax treatment of the financial sector.
- Recent policy moves cited:
  - United Kingdom: Bank Levy introduced in January 2011, levied at 7.5 basis points on (essentially) banks’ uninsured debt obligations.
  - Ten EU members introduced special charges on financial institutions.
  - United States: related proposals revived in the President’s 2011 budget proposal.
- Central questions:
  - What is the optimal corrective tax to deter inappropriately low capital ratios (base, level, revenue yield relative to social damage)?
  - How does taxation compare with regulation (capital requirements) in effectiveness and implementability?

### II. Nature and magnitude of externalities from financial failures
- Two alternative crisis responses during 2008:
  - Allow collapse and accept large economic disruption.
  - Commit public funds for bailouts, averting damage but creating bailout-expectation externality and equity concerns.
- Empirical estimates and fiscal exposure:
  - BCBS (2010): median cumulative output loss from banking crises of 63 percent of initial GDP, and a mean of over 100 percent.
  - IMF (2010a): cumulative loss already 25 percent of GDP by end-2009 for advanced countries that experienced a systemic crisis.
  - Advanced G-20 economies committed, through guarantees and similar measures, to making an average of 25 percent of GDP available for support operations.
- Fiscal exposures are large; direct fiscal costs imply real resource loss to the extent of distortionary cost of raising public funds and raise fairness concerns.

### III. Modeling framework and key mechanisms
- Bank balance identity: L = K + D.
- Capital ratio k ≡ K/L and failure condition K r < D; critical return R(k) = (k/(1 − k)) ζ.
- Bank owners risk-neutral with limited liability; private failure cost B in addition to equity wipeout.
- Creditors’ required return ρ depends on competition and perceived bailout probability μ.
- Two financial-sector externalities identified:
  - Bailout externality: implicit subsidy from prospect of bailout S(k, μ).
    - Properties: S(k, μ) ≥ 0, S(k, 1) = 0, S(k, μ) = 0 if k = 1; S strictly increasing in μ for Φ < 1; ∂S/∂k ≤ 0 and ∂²S/∂k² ≥ 0 under maintained assumptions.
    - Empirical reduction in borrowing cost due to bailout expectations: range 10–50 basis points, commonly around 20 basis points (discussion around equation (2.13)).
  - Collapse externality: social cost when failure is unmitigated modeled as (1 − μ) Δ B with Δ ≥ 0 (Δ modeled exogenously; systemicness depends on size, interconnectedness, substitutability).

### IV. Private versus social optima and inefficiency
- Bank maximizes private objective j(k; μ); social welfare W(k; μ) defined separately.
- Proposition 1: In the absence of intervention, and whatever the probability of bailout, the capital ratio is inefficiently low and the probability of failure inefficiently high.

### V. Corrective taxation: form, optimal marginal rate, and calibration
- Corrective instrument: tax/subsidy τ(k) on capital ratio (equivalently tax on borrowing or assets).
- Optimal marginal subsidy/tax to decentralize social optimum (Proposition 2, preserved structure):
  - ܶ̇*(k) = (1 − μ) { Φ_R(R(k)) Δ / [ζ(1 + λ)] } ≥ 0
    - Intuition: additive components correcting collapse and bailout externalities, weighted by occurrence probabilities.
- Calibration presented as marginal tax on borrowing:
  - Lower-bound formula for the collapse-externality component preserved (equation (3.13)) with notation for K̄ and Δ̄ as percent of GDP.
  - Numerical illustration (United Kingdom–style parameters: bank capital K̄ ≈ 10 percent of GDP) — Table 1 (optimal corrective tax on borrowing, basis points; probability of crisis in percent; capital ratio k in percent):
    - Capital ratio, k (percent): 6, 8, 10, 12
    - Probability of crisis (annual), Φ(k) (percent): 12.8, 2.6, 0.9, 0
    - Collapse externality (μ = 0):
      - Δ = 63: 32, 12, 6, 0 (basis points)
      - Δ = 100: 51, 16, 9, 0 (basis points)
    - Bailout externality (μ = 1): 38, 8, 2, 0 (basis points)
  - Observations from calibration:
    - Lower-bound optimal tax can be large (e.g., about 50 basis points in extreme case).
    - Very strong nonlinearity: tax negligible at k = 12 percent even with Δ̄ = 100.
  - Approximation for bailout-externality component when μ = 1 (equation (3.14)): τ_D ≈ Φ(k) ζ (preserved approximate form).
- Revenue properties (Proposition 3):
  - For the optimal proportional tax on borrowing:
    - (a) Revenue from bailout-related part more than covers expected fiscal cost of bailout.
    - (b) Revenue from systemic part more than covers expected systemic externality if condition (3.15) holds (inequality involving elasticities and model terms preserved).

### VI. Endogenizing bailout and taxing equity
- Government choice of μ analyzed:
  - Full commitment: first-order condition (4.1) balances Φ Δ versus expected fiscal cost.
    - Proposition 4: If marginal efficiency cost of taxation λ = 0, optimal μ = 1; more generally μ* may be between 0 and 1.
  - No-commitment (government decides after r realized):
    - Government bails out iff R ≤ r̃ satisfies inequality (4.2); creditors’ return and bank bailout valuation adjust accordingly (Γ(k) replaces S(k, μ)).
    - Γ(k) need not be unambiguously decreasing in k; higher k can sometimes raise prospective bailout value to the bank.
- Taxing bank equity:
  - Allowing bank to choose equity K and taxing equity at rate τ_e yields first-order condition (4.5).
  - Proposition 6: Positive optimal tax on equity; equity should be taxed strictly positively when externalities operative (structure of τ_e* preserved).

### VII. Instrument choice under non-convexity and uncertainty
- Non-convexity:
  - Bank payoff non-concavity can produce multiple local optima; linear taxation may fail to implement first-best while regulation (mandated k*) can achieve it.
- Weitzman-style comparison (prices vs quantities):
  - If policy set before distribution Φ realized, tax lets bank adjust k ex post; regulation fixes k ex ante.
  - Proposition 7:
    - (a) If Φ_R = 0 and Δ = 0 then taxation is sufficient to be preferred.
    - (b) Regulation is sufficient to be preferred under condition (5.4) (complex inequality involving model parameters preserved).
  - Intuition: taxation preferred when social marginal benefit curve is relatively flat; regulation preferred when social benefits are steep.

### VIII. Extensions: asymmetric information and nonlinear instruments
- Heterogeneous ability a with private information; private and social first-order conditions vary with a.
- Single minimum capital requirement may fail to achieve first-best due to self-selection constraints.
- Menu and tax approach:
  - Offer menu {k^2, k^1} and impose tax T on choosing the lower capital ratio to deter mimicking (self-selection constraints in equations (5.7)–(5.9)).
  - Proposition 8: First-best implementable via menu {k^2, T} and {k^1, 0} with T satisfying (5.9).
- Implication: nonlinear taxes (marginal rate increasing in borrowing) or differentiated capital requirements can implement efficient outcomes; nonlinear tax schemes can be complex but comparable to differentiated regulatory regimes.

### IX. Policy conclusions, quantitative calibration, and implementation notes
- Corrective tax targets two interrelated externalities: (i) social costs of unmitigated failure of systemic institutions, and (ii) moral hazard from anticipated government bailouts.
- Optimal tax is additive in components, highly nonlinear, rising rapidly as capital ratio falls into risky regions.
- Illustrative calibrations suggest an appropriate tax on borrowing might easily be in the order of 40 basis points in such regions.
- This contrasts with implemented taxes (United Kingdom and elsewhere) and proposed U.S. figures of about 4–8 basis points.
- The bailout component of the corrective tax is optimally set above the insurance-like level that would merely recoup expected fiscal bailout costs.
- Dependence on government commitment:
  - Weaker commitment or limited resources imply a lower optimal tax (banks internalize some bailout-avoidance incentive).
- Practical instrument mix suggested:
  - Binding capital requirement as protection in the most dangerous circumstances.
  - Progressive (nonlinear) tax on borrowing with marginal rate increasing at low capital ratios to reflect heterogeneity and tail risk.
- Implementation considerations:
  - Preexisting tax distortions (bias toward debt finance) suggest corrective tax could partly redress corporate tax non-neutrality; whether bank borrowing should be taxed more heavily remains a question.
  - When capital requirements bind, small taxes have no effect; however, banks typically hold buffers above minima, so a tax can still influence behavior albeit possibly weakened.

*Source: _wp11206 - 1. Optimal Corrective Tax on Borrowing to Address Collapse Externality (IMF Working Paper PDF _wp11206)*

### 1. Optimal Corrective Tax on Borrowing to Address Collapse Externality ............................18

### 1. Optimal Corrective Tax on Borrowing to Address Collapse Externality

### I. Introduction — context and policy questions
- Financial crisis that erupted in 2008 prompted fundamental reconsideration of tax treatment of the financial sector.
- New tax instruments targeted at limiting likelihood and social damage from financial failures have been proposed (IMF (2010a)); implementation has outpaced analysis.
- Examples of recent policy moves:
  - United Kingdom: introduced in January 2011 a Bank Levy, levied at 7.5 basis points on (essentially) banks’ uninsured debt obligations.
  - Ten EU members (including France, Germany, Sweden) have introduced some special charge on financial institutions.
  - United States: related proposals revived in the President’s 2011 budget proposal.
- Traditional approach to externalities from financial failure is regulation (microprudential capital requirements); apparent failings of regulation suggest potential role for other instruments, including taxation.

### II. Nature and magnitude of externalities from financial failures
- Two alternative policy responses to systemic institution distress were common in the crisis:
  - Allow collapse and accept large economic disruption from externalities.
  - Commit sufficient public funds for bailouts, averting damage but creating bailout-expectation externality and equity concerns.
- Literature and empirical estimates cited:
  - BCBS (2010): median cumulative output loss from banking crises of 63 percent of initial GDP, and a mean of over 100 percent.
  - IMF (2010a): cumulative loss at already 25 percent of GDP by end-2009 for advanced countries that experienced a systemic crisis.
  - Advanced G-20 economies committed, through guarantees and similar measures, to making an average of 25 percent of GDP available for support operations.
- Fiscal exposures are large; direct fiscal costs may be transfers but imply real resource loss to extent of distortionary cost of raising public funds and raise fairness concerns.

### III. Two broad policy instruments and central research questions
- Two general ways to address externalities:
  - Regulation: directly regulate behavior (e.g., minimum capital requirements).
  - Taxation: use tax measures to influence behavior indirectly (e.g., taxes on wholesale/unsecured/short-term borrowing).
- Historical dominance of regulation in financial sector contrast with standard prescription of taxes for externalities in other domains (e.g., climate change).
- Key questions addressed in the paper:
  - Characterize optimal tax policies to discourage inappropriately low capital ratios: proper base, likely level, and revenue yield relative to expected social damage.
  - Compare taxation versus regulation: Is regulatory dominance warranted, or is there a purposive role for corrective taxation?
- Notable trade-offs and considerations:
  - Taxation reduces private buffers but increases public-sector buffers; optimal choice depends on correlation of shocks across institutions and between private and public sectors.
  - International coordination historically easier on regulatory side (e.g., Basel standards) than taxation.
  - Microprudential regulation dominance does not preclude a role for macroprudential tools where policy makers have more discretion (Basel III leaves macroprudential strategies unarticulated).

### IV. Modeling approach and paper plan (overview of analytic structure)
- Section II (paper): sets out a model of a representative bank whose decisions determine its own risk of failure, and explores two implied externalities:
  - Externality if failure is allowed to have full social consequences.
  - Externality if government is perceived as committed to prevent those costs by bailing out creditors.
- Section III: characterizes the optimal corrective tax, explores its likely magnitude, and considers whether associated revenue would cover expected social costs of outright failure and of bailout.
- Section IV: extends analysis to endogenize the government’s bailout decision.
- Section V: compares optimal corrective tax with imposing capital requirements, focusing on implementability, effectiveness under uncertainty, and asymmetry of information between policymakers and banks.

### V. Related literature and theoretical positioning
- Paper relates to arguments for corrective taxes on unsecured/short-term borrowing as a means to reduce failure risk or other distortions (citations in source).
- Specific related contributions and motivations cited include:
  - Acharya et al. (2010) as closest in spirit.
  - Shin (2010) and Perotti and Suarez (2009) for maturity-mismatch / short-term debt taxes.
  - Huang and Ratnovski (2009) and Jeanne and Korinek (2010) for alternative justifications for taxing unsecured borrowing.
- Note: regulatory capital requirements are sometimes described as price-based instruments in literature, but may not be equivalent to tax instruments in certain circumstances.

### VI. Key policy implications emphasized in the introduction
- Corrective taxes on banks’ unsecured borrowing are a feasible policy instrument to discourage low capitalization and mitigate collapse externalities.
- Important policy design issues include:
  - Base of the tax (unsecured debt, short-term borrowing, etc.).
  - Appropriate tax level given social costs of failure and bailout probabilities.
  - Revenue yield and whether it offsets expected social costs.
  - Comparative merits versus additional capital requirements for systemically important institutions.
- The practical policy environment features active implementation of bank taxes in several jurisdictions, making an analysis of tax design and comparison with regulation timely and relevant.

*Source: _wp11206 - 1. Optimal Corrective Tax on Borrowing to Address Collapse Externality*

### Section VI concludes.

### _wp11206 - Section VI concludes

### Banking model and failure risk
- Bank balance: L = K + D (equation (2.1)).
- Bank chooses debt D and loans L; equity capital K initially taken as given.
- Return on loans r ≥ 0 is stochastic with density φ and distribution Φ, satisfying:
  - φ(r) ≥ 0 for r ≥ 0; φ(0) = 0 (equation (2.2));
  - ∫(ζ − r) φ(r) dr ≥ 0 (equation (2.3)), where ζ is the risk-free return.
- Failure occurs iff K r < D (equation (2.4)); defining capital ratio k ≡ K/L yields critical return R(k) = (k/(1 − k)) ζ (equation (2.5) structure).
- Bank owners are risk-neutral with limited liability; they incur additional private failure cost B beyond equity wipeout.
- Creditors require return ρ determined by competition and perceived bailout probability μ; equation (2.7) characterizes ρ as a function of μ and k.
- Bank owners maximize expected payoff comprising:
  - private cost of failure (BΦ(R));
  - value of continuing operations absent bailout;
  - expected value of bailout subsidy S(k, μ) (equations (2.6), (2.8), (2.9), (2.10)).

### Financial-sector externalities
- Two externalities when a bank fails:
  - Bailout externality: implicit subsidy from prospect of bailout S(k, μ) (Lemma on properties of S).
    - Properties: (i) S(k, μ) ≥ 0, S(k, 1) = 0, S(k, μ) = 0 if k = 1; (ii) S strictly increasing in μ for Φ < 1; (iii) ∂S/∂k ≤ 0 and ∂²S/∂k² ≥ 0 under maintained assumptions (equation (2.11) and Lemma).
    - S decreases with higher capital ratio k for two reasons: less borrowing and lower bankruptcy probability.
    - Reduction in borrowing rate due to bailout (ρ^μ − ρ) is related to subsidy; empirical estimates of the reduction in borrowing cost are in range 10–50 basis points, commonly around 20 basis points (discussion around equation (2.13)).
  - Collapse externality: wider social cost when failure is unmitigated taken as (1 − μ) Δ B with Δ ≥ 0 (discussion in "The collapse externality").
    - Δ is modeled as exogenous; systemicness considerations (size, interconnectedness, substitutability) noted as relevant but not explicitly modeled.

### Private and social optima; inefficiency
- Bank chooses capital ratio k (with K fixed until Section IV) to maximize private objective j(k; μ) (equation (3.1)).
- First-order condition for the bank: j_k = 0 given by equation (3.2); second-order condition involves Ω and can fail due to non-concavity (equations (3.4), (3.5)).
- Social welfare W(k; μ) defined in equation (2.14) and re-expressed in (3.6); social first-order condition given by equation (3.7).
- Comparison private vs social yields (3.8) and result:
  - Proposition 1: In the absence of intervention, and whatever the probability of bailout, the capital ratio is inefficiently low and the probability of failure inefficiently high (equation (3.9)).

### Corrective taxation: form and calibration
- Corrective instrument: tax/subsidy τ(k) on capital ratio (equivalently tax on borrowing or assets); bank maximand and social welfare with τ appear in (3.10).
- Optimal marginal subsidy to capital ratio (or equivalent marginal tax on borrowing) required to decentralize the social optimum:
  - τ'_*(k) = (1 − μ) [Φ_R(R(k))] Δ / (ζ (1 + ρ)) scaled by (1 + λ)⁻¹ structure — formal expression in Proposition 2 (equation (3.11)):
    - ܶ̇*(k) = (1 − μ) { Φ_R(R(k)) Δ / [ζ(1 + λ)] } ≥ 0 (preserved structure and signs as in (3.11)).
  - Intuition: two components correct collapse and bailout externalities, weighted by probabilities of occurrence.
- Calibration as marginal tax on borrowing (equations (3.12)–(3.14)):
  - Lower-bound formula for the collapse-externality component (equation (3.13)):
    - τ_D(k) ≥ [Φ(k) k^−2 (ζ/(1 + λ)) Δ̄ / K̄ ] (notation preserved as in (3.13)), where K̄ and Δ̄ denote bank capital and collapse costs as percent of GDP.
  - Numerical illustration using United Kingdom–style parameters:
    - Bank capital K̄ ≈ 10 percent of GDP.
    - Two collapse cost scenarios: Δ̄ = 63 and Δ̄ = 100 (percent of GDP).
    - Table 1 (optimal corrective tax on borrowing, basis points; probability of crisis in percent; capital ratio k in percent):
      - Capital ratio, k: 6, 8, 10, 12
      - Probability of crisis (annual), Φ(k): 12.8, 2.6, 0.9, 0
      - Collapse externality (μ = 0):
        - Δ = 63: 32, 12, 6, 0 (basis points)
        - Δ = 100: 51, 16, 9, 0 (basis points)
      - Bailout externality (μ = 1): 38, 8, 2, 0 (basis points)
    - Observations:
      - Lower bound optimal tax can be large (e.g., about 50 basis points in extreme case).
      - Very strong nonlinearity: tax negligible at k = 12 percent even with Δ̄ = 100.
  - Approximation for bailout-externality component when μ = 1 (equation (3.14)):
    - τ_D ≈ Φ(k) ζ(k) — preserved approximate form as in (3.14).
- Revenue properties (Proposition 3):
  - For the optimal proportional tax on borrowing:
    - (a) Revenue from bailout-related part more than covers expected fiscal cost of bailout D S μ (formally B S μ in text).
    - (b) Revenue from systemic part more than covers expected systemic externality (1 − μ) B Φ if condition (3.15) holds (inequality structure preserved):
      - sufficiency condition involves elasticity terms and reads as in equation (3.15).

### Extensions: endogenizing bailout and taxing equity
- Government chooses μ (commitment or no-commitment cases).
- Full commitment case:
  - First-order condition for μ in (4.1): trade-off Φ Δ versus expected fiscal cost; implies:
    - Proposition 4: If marginal efficiency cost of taxation λ = 0, optimal μ = 1; more generally μ* may be between 0 and 1.
- No-commitment (government decides after r realized):
  - Government bails out if and only if R ≤ r̃ satisfies inequality (4.2): R ≥ r̃ ⇔ (1 − μ) Δ B ≤ ζ (residual top-up distortionary cost) — preserved inequality structure in (4.2).
  - Creditors’ required return becomes (4.3); bank’s bailout-value term S(k, μ) replaced by Γ(k) (equation (4.4)).
  - Key difference: Γ(k) need not be unambiguously decreasing in k (unlike S); higher k can make bailout cheaper for government and so raise value of prospective bailout to bank—potentially reversing sign of ∂Γ/∂k and hence the corrective tax component.
- Taxing bank equity:
  - Allowing bank to choose equity K with supply price p(K) (notation χ(K)) and tax on equity rate τ_e leads to first-order condition (4.5).
  - Social optimum implies positive optimal tax on equity (Proposition 6, equation (4.6)):
    - τ_e* = (1/ K ∂W/∂k) {Φ_R(R(k)) Δ + ψ(k)} ≥ 0 structure preserved; bank equity should be taxed strictly positively when externalities operative.

### Instrument choice: taxation vs regulation
- Non-convexity:
  - Bank objective may be non-globally-concave; marginal private cost curve can be non-monotone (equation (5.1) and Figure 2 discussion).
  - Linear taxation may fail to implement first-best when non-convexity causes multiple local optima; regulation that mandates exact capital ratio k* avoids this issue.
  - Minimum capital requirement may still induce over- or under-holding (Figure 2 scenarios).
- Uncertainty (Weitzman-style comparison):
  - If policymakers set policy before distribution Φ is realized, tax allows bank to adjust k ex post; regulation fixes k.
  - Define marginal external benefit S_k (equation (5.2)); assume S'' ≤ 0, private marginal cost curvature ε'' ≥ 0.
  - Weitzman criterion: regulation preferred ex ante iff S'' Σ − ε'' Σ ≥ 0 (translated into model terms).
  - Proposition 7:
    - (a) If Φ_R = 0 and Δ = 0 then taxation is sufficient to be preferred.
    - (b) Regulation is sufficient to be preferred under condition (5.4) (complex inequality involving model parameters preserved as in text).
  - Intuition: taxation better when social marginal benefit curve is flat relative to private marginal cost; regulation better when social benefits are steep (avoiding large ex post welfare losses).

*Source: _wp11206 - Section VI concludes (IMF working paper PDF _wp11206).*

### Appendix 4

### _wp11206 - Appendix 4

### Intuition and comparative advantage: tax versus regulation
- When both Δ and ܵ
௞௞ are (close to) zero, ܥ
௞ remains upward-sloping (by the second order condition (3.4)) while ܤ
௞ is everywhere zero; optimal policy is to allow private decisions to guide the capital ratio, so a (small) tax is preferred to a fixed capital ratio.
- If ߤ = 0 (only the collapse externality applies), (5.3) implies regulation is preferred whenever ߜ ≥ Δ; intuitively, low ߜ makes bank capital choice highly sensitive to taxation, and high Δ makes small capital changes have large social consequences—this condition is plausibly satisfied for systemic institutions.
- If ߤ = 1 (only the bailout externality applies), the relevant magnitude is ܵ
௞௞/Ω rather than the bailout size alone: increasing ܵ
௞௞ flattens ܥ
௞ and steepens ܤ
௞, tilting preference toward regulation. The curvature of S is complex; it is shown in proving part (b) that ܵ
௞௞݇/ܵ൐2(ଶ). Presence of Ω prevents a simple comparison between bailout subsidy size and private costs of failure; nonetheless, larger externalities make regulation more likely to dominate.

### Limitations of the one-shot framework and nonlinear policy implications
- The model treats policy as a one-shot choice; in reality tax rates and regulatory requirements may adjust at different speeds.
- In a nonlinear world, marginal external benefit curves may vary over the cycle (steeper in bad times, flatter in good times), suggesting a hybrid policy:
  - a minimum capital requirement to forestall worst outcomes, and
  - a tax to guard against excessive optimism in good times.
- Nonlinear taxes generally dominate either quantity regulation or proportional tax in this framework (reference: Roberts and Spence, 1976); preceding considerations suggest a higher marginal rate on borrowing at lower capital ratios.

### Asymmetric information: ability heterogeneity and implementation
- Banks have private information a (ability); the bank objective is ݌(a,k), strictly concave in k, with ݌
௔ ≥ 0 and single-crossing ݌
௞௔ ≤ 0. Privately optimal k solves ݌
௞(k*,a) = 0 (equation (5.5)) and is decreasing in a.
- Social welfare is ݌(a,k) − Λ(a,k) with Λ
௞ ≤ 0, Λ
௞௞ ≤ 0, and Λ
௞௔ ≥ 0; the first-best k solves ݌
௞(k^,a) = Λ
௞(k^,a) (equation (5.6)) and is decreasing in a.
- With two ability types a1 < a2, first-best k^1 > k^2. Regulation alone (a single minimum capital requirement) may fail because self-selection constraints bind: setting the minimum at k^2 to place the high-ability firm at its first best leaves the low-ability firm free to choose its private optimum k~2 < k^1, not the social first-best.
- Offering a menu {k^2, k^1} faces mimicking: the low-ability bank may prefer the low capital ratio intended for the high-ability bank.
- A tax T on choosing the lower capital ratio can deter mimicking if it satisfies the self-selection constraints:
  - ݌(k^1,a2) − T ≥ ݌(k^2,a2)  (equation (5.7))
  - ݌(k^2,a1) ≥ ݌(k^1,a1) − T  (equation (5.8))
- PROPOSITION 8: The first-best can be implemented by offering banks the choice between (k^2,T) and (k^1,0), where T satisfies equation (5.9): ݌̃ = max_ε [݌(k^2,a2) − (݌(k^1,a2) − ε), ݌(k^2,a2), 0] ≥ 0 (equation (5.9)); implementation achieved via a menu allowing lower capital conditional on payment of an appropriate tax (equivalently, a subsidy for choosing higher k).
- Conclusion: asymmetric information points to nonlinear taxes (marginal rate increasing in borrowing) or differentiated capital requirements; nonlinear tax schemes can be complex but comparable in complexity to proposed differentiated requirements.

### Policy conclusions and quantitative calibration
- Corrective tax targets two interrelated externalities: (i) social costs of unmitigated failure of systemic institutions, and (ii) moral hazard from anticipated government bailouts. The optimal tax has a simple additive form and is likely highly nonlinear, increasing rapidly as the capital ratio falls into regions implying significant systemic failure risk.
- Illustrative calibrations suggest an appropriate tax on borrowing might easily be in the order of 40 basis points in such regions.
- This contrasts with taxes introduced in practice (United Kingdom and elsewhere in Europe) and those proposed for the United States of about 4–8 basis points.
- The bailout component of the corrective tax is optimally set above the insurance-like level that would merely recoup expected fiscal bailout costs.
- Optimal corrective tax depends on government commitment capacity to bail out: weaker commitment or limited resources imply a lower optimal tax (bank has incentive to limit bailout costs).
- Non-convexities in bank payoffs can favor regulation. When policy must be set under uncertainty ex ante, regulation clearly dominates if social costs of failure exceed private costs (collapse externality). For dominant bailout externality, regulation may still dominate if expected bailout costs are high enough.
- Practical implementation considerations:
  - Preexisting tax distortions (bias toward debt finance) suggest corrective bank tax could redress corporate tax non-neutrality; question remains whether bank borrowing should be taxed more heavily to offset these distortions.
  - Any bank tax will operate in tandem with capital requirements in practice; when capital requirements bind, small taxes have no effect. However, banks typically hold buffers above minima, so a tax on borrowing can still affect behavior, though responsiveness may be weakened by precautionary buffers.
  - Suggested policy mix: a binding capital requirement as protection in the most dangerous circumstances, combined with a progressive tax on borrowing (marginal rate increasing at low capital ratios) to reflect heterogeneity and tail risk.

### Technical proofs and derived expressions (high-level)
- Appendix summarises proofs and derivations used in the main text:
  - Relationships and identities involving Φ, ܵ, ܴ, and derivatives are used to derive comparative-statics (see (A1.2)–(A1.6), (A3.1)–(A3.4), (A4.1)–(A4.2)).
  - With ߤ = 1, optimal tax on k is ܶᇱ = ܵ
௞; derivation of (3.14) involves approximation Φᇱ ≈ Φ and expressions for ߩ (equations (A2.1)–(A2.3)).
  - Proof of Proposition 3 compares revenue from a corrective proportional tax with expected collapse externality cost (equation (A3.1)) and establishes inequalities via identities (A3.2)–(A3.4).
  - Proof of Proposition 7 uses (5.3), (3.4), and second-order condition for part (a); part (b) manipulates (5.3) adding and subtracting Ωߜ and uses A1.6 and (2.11) to obtain (A4.1) and (A4.2).
  - Appendix 5 establishes existence of tax T satisfying self-selection constraints by analyzing the two possibilities for the binding of the low-ability constraint, yielding (5.1)–(5.3).

*Source: _wp11206 - Appendix 4*

### References

### _wp11206 - References

### Systemic risk and macroprudential regulation
- Acharya, Viral, 2009, “A Theory of Systemic Risk and Design of Prudential Regulation,” Journal of Financial Stability, Vol. 5, pp. 224–55.
- Acharya, Viral, Lasse Pedersen, Thomas Philippon, and Matthew Richardson, 2010, “Measuring Systemic Risk,” (mimeo; New York: New York University).
- Adrian, Tobias, and Markus Brunnermeier, 2009, “CoVaR,” Federal Reserve Bank of New York Staff Reports No. 348.
- Bank of England, 2009, The Role of Macroprudential Policy, Discussion Paper.
- Gauthier, Céline, Alfred Lehar, and Moez Souissi, 2010, “Macroprudential Regulation and Systemic Capital Requirements,” Bank of Canada Document de Travail 2010-4 (Ontario: Bank of Canada).
- Haldane, Andrew, 2009. “Banking on the State,” presented at the 12th Annual Federal Reserve of Chicago International Banking Conference, September 25, 2009. Available via the Internet: www.bankofengland.co.uk/publications/speeches/2009/speech409.pdf.
- Korinek, Anton, 2009, “Systemic Risk-Taking: Amplification Effects, Externalities, and Regulatory Responses,” (mimeo; Maryland: University of Maryland).
- Wagner, Wolf, 2010. “In the Quest of Systemic Externalities: a Review of the Literature,” CESifo Economic Studies, Vol. 56, pp. 96–111.
- International Monetary Fund, Financial Stability Board and Basel Committee of Bank Supervisors, 2009, “Guidance to Assess the Systemic Importance of Financial Institutions, Markets and Instruments: Initial Considerations,” available via the Internet: www.imf.org/external/np/g20/pdf/100109.pdf

### Banking regulation, moral hazard, and "too big to fail"
- Baker, Dean, and Travis McArthur, 2009, “The Value of the ‘Too Big to Fail’ Big Bank Subsidy,” Issue Brief (September) (Washington: Center for Economic and Policy Research).
- Demirgüç-Kunt, Asli, and Harry Huizinga, 2010, “Are Banks Too Big to Fail or Too Big to Save? International Evidence from Equity Prices and CDS Spreads,” European Banking Center Discussion Paper No. 2010–15.
- Dewatripont, Matthias, and Jean Tirole, 1993, The Prudential Regulation of Banks (Cambridge: MIT Press).
- Hellman, Thomas F., Kevin C. Murdock, and Joseph E. Stiglitz (2000), “Liberalization, Moral Hazard in Banking and Prudential Regulation: are Capital Requirements Enough?” American Economic Review, Vol. 90, pp. 147–65.
- John, Kose, Teresa John and Lemma Senbet, 1991, “Risk-Shifting Incentives of Depository Institutions: A New Perspective on Federal Deposit Insurance Reform,” Journal of Banking and Finance, Vol. 15, pp. 895–915.
- Sinn, Hans-Werner, 2003, “Risk-Taking, Limited Liability, and the Competition of Bank Regulators,” Finanzarchiv, Vol. 59, pp. 305–29.
- Sinn, Hans-Werner (2010), Casino Capitalism (New York: Oxford University Press).
- Shull, Bernard, 2010, “Too Big To Fail in a Financial Crisis,” Working Paper No. 601, (New York: Hunter College, City University of New York).
- Perotti, Enrico, and Javier Suarez, 2009, “Liquidity Insurance for Systemic Crises,” CEPR Policy Insight No. 31, February.

### Taxation of the financial sector and corrective taxes
- Bianchi, Javier, and Enrique Mendoza, 2010, “Overborrowing, Financial Crises and ‘Macro-Prudential’ Taxes,” NBER Working Paper 16091 (Cambridge: National Bureau of Economic Research).
- Bovenberg, Lans, and Ruud de Mooij, 1994, “Environmental Levies and Distortionary Taxation,” American Economic Review, Vol. 94, pp. 1085–89.
- Christiansen, Vidar, and Stephen Smith, 2009, “Externality-Correcting Taxes and Regulation,” CESifo Working Paper No. 2793 (Munich: Center for Economic Studies Information).
- EEAG, 2011, “Taxation and Regulation of the Financial Sector,” The EEAG Report on the European Economy, CESifo, Munich, pp. 147–69.
- HM Revenue & Customs, 2010, “Bank Levy,” available via the Internet: http://www.hmrc.gov.uk/budget-updates/autumn-tax/tiin1065.htm
- International Monetary Fund, 2010a, A Fair and Substantial Contribution by the Financial Sector: Final Report for the G-20, available via the Internet: www.imf.org/external/np/g20/pdf/062710b.pdf
- Keen, Michael, 2011, “Rethinking the Taxation of the Financial Sector,” CESifo Economic Studies, Vol. 57, pp. 1–24.
- Kocherlakota, Narayana, 2010, “Taxing Risk and the Optimal Regulation of Financial Institutions,” Economic Policy paper, Federal Reserve Bank of Minneapolis.
- Jeanne, Olivier and Anton Korinek, 2010, “Managing Credit Booms and Busts: A Pigouvian Taxation Approach,” (mimeo; Maryland: Johns Hopkins University).
- Korinek, Anton, 2009, “Systemic Risk-Taking: Amplification Effects, Externalities, and Regulatory Responses,” (mimeo; Maryland: University of Maryland).
- Radulescu, Doina Maria, 2010, “The Effects of a Bonus Tax on Manager Compensation and Welfare,” CESifo Working Paper 3030.
- Shackelford, Douglas A., Daniel Shaviro, and Joel Slemrod, 2010, “Taxation and the Financial Sector,” National Tax Journal, Vol. 63, pp. 781–806.
- Shavell, Steven, 2010, “Corrective Taxation versus Liability,” American Economic Review, Paper and Proceedings, Vol. 101, pp. 273–76.
- Shin, Hyun Song, 2010, “Non-Core Liabilities Tax as A Prudential Tool,” Policy Memo, available via the Internet: http://www.princeton.edu/~hsshin/www/NonCoreLiabilitiesTax.pdf
- Weder di Mauro, Beatrice, 2010, “Taxing Systemic Risk: Proposal for a Systemic Risk Charge and a Systemic Risk Fund,” (mimeo; Postfach: University of Mainz).

### Liquidity, funding, and market risk
- Huang, Rocco, and Lev Ratnovski, 2009¸ “The Dark Side of Wholesale Funding,” (mimeo; Washington: International Monetary Fund).
- Hoggarth, Glenn, and Victoria Saporta, 2001, “Costs of Banking Systems Instability: Some Empirical Evidence,” Financial Stability Review, Bank of England, June, pp. 148–65.
- Perotti, Enrico, and Javier Suarez, 2009, “Liquidity Insurance for Systemic Crises,” CEPR Policy Insight No. 31, February.
- Adrian, Tobias, and Markus Brunnermeier, 2009, “CoVaR,” Federal Reserve Bank of New York Staff Reports No. 348.

### Environmental and price-vs-quantity frameworks (theory and policy instruments)
- Pizer, William A., 2002, “Combining Price and Quantity Control to Mitigate Global Climate Change,” Journal of Public Economics, Vol. 85, pp. 409–34.
- Roberts, Marc J., and Michael Spence (1976). “Effluent Charges Under Uncertainty,” Journal of Public Economics, Vol. 5, pp. 193–208.
- Weitzman, Martin, 1974, “Prices vs. Quantities,” Review of Economic Studies ̧ Vol. 41, pp. 477–91.
- Weitzman, Martin, 1978, “Reply to ‘Prices vs. Quantities: A Critical Note on the Use of Approximations,” Review of Economic Studies, Vol. 46, pp. 209–10.

### Other empirical and theoretical contributions
- Levine, Ross, 2005, “Finance and Growth: Theory and Evidence,” (mimeo; (University of Minnesota).
- Radulescu, Doina Maria, 2010, “The Effects of a Bonus Tax on Manager Compensation and Welfare,” CESifo Working Paper 3030.
- Shin, Hyun Song, 2010, “Non-Core Liabilities Tax as A Prudential Tool,” Policy Memo, available via the Internet: http://www.princeton.edu/~hsshin/www/NonCoreLiabilitiesTax.pdf
- International Monetary Fund, 2010b, Global Financial Stability Report, Meeting New Challenges To Stability and Building a Safer System, April, Chapter 2, Systemic Risk and the Redesign of Financial Regulation. April.
- International Monetary Fund, 2010a, A Fair and Substantial Contribution by the Financial Sector: Final Report for the G-20, available via the Internet: www.imf.org/external/np/g20/pdf/062710b.pdf

*References list as provided in the source PDF*

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