Bail-Ins, Optimal Regulation, and Crisis Resolution: Clayton & Schaab (2025)

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TL;DR

Clayton and Schaab build a tractable three-period dynamic contracting model of bank liability structure in the tradition of Innes (1990). Banks face an initial and a continuation monitoring incentive problem. In the presence of fire sales from liquidations, the privately optimal contract combines short-term standard debt (which forces liquidation in bad states and provides strong incentives) and long-term bail-in debt (which avoids resource costs of liquidation by writing down to pledgeable income). A social planner that internalizes the fire-sale externality intervenes in both the level and composition of debt: it favors less standard debt and less total debt on the margin. The policy combines a maximum leverage requirement to limit total debt with a TLAC requirement that limits standard debt and can be met with bail-in debt. In the linear-private-benefit illustration, both socially optimal debt thresholds are weakly below their private-optimum counterparts. The model shows that bail-ins replace bailouts as a recapitalization tool and that statutory provisions increasing the cost of bailouts improve welfare, providing a unified rationalization of postcrisis regulation. The framework connects to demand-based explanations of standard debt (Bolton and Oehmke (2019); Walther and White (2020)) and extends the macroprudential literature on pecuniary externalities from Davila and Korinek (2018) to the bank contracting setting.

Core results

#	Result	Locator	Magnitude as reported
R1	Privately optimal contract uses both standard and bail-in debt	Proposition 1, Corollary 1, pp. 2822-2825	Liability structure has three regions: liquidation ($R_1 \le R_\ell^p$), bail-in write-down ($R_\ell^p < R_1 \le R_u^p$), and no write-down ($R_1 > R_u^p$); implemented with short-term standard debt face value $(1-b)R_\ell^p Y_0$ and long-term bail-in debt face value $(1-b)(R_u^p - R_\ell^p)Y_0$
R2	Social optimum has the private optimum’s structure, with wedges that discourage standard debt and total debt on the margin	Proposition 3, Equations (24)-(28), pp. 2827-2829	Wedges $\tau_\ell^s \ge 0$ and $\tau_u^s \ge 0$ reflect the social cost of liquidations $\lambda^s \sigma \gamma^s$. In the linear-private-benefit special case, $R_\ell^s \le R_\ell^p$ and $R_u^s \le R_u^p$
R3	All three model ingredients (initial incentive problem, continuation incentive problem, costly liquidation) are needed for bail-in debt to be part of the optimal contract	Proposition 2, p. 2826	If $B_0(e_0) = 0$: bail-in debt alone suffices. If $B_1 = 0$: long-term debt alone suffices. If $\gamma = 1$: standard debt alone suffices. All three ingredients are needed for the combined structure
R4	With planner commitment over bailouts, bail-ins dominate bailouts: the socially optimal contract with no bailouts is Pareto efficient	Proposition 4, p. 2832	No bailouts ($T_0 = T_1 = 0$) is Pareto efficient; bailouts are redundant recapitalization when bail-ins are available; both instruments can achieve same state contingencies in bank debt contracts
R5	Without planner commitment, Pareto-efficient debt levels still eliminate bailouts entirely; welfare is increasing in the cost of bailouts F	Proposition 5, p. 2834	For any $F \ge 0$, Pareto-efficient debt levels $(R_\ell^s, R_u^s)$ result in no banks being bailed out; welfare is strictly increasing in F because higher F relaxes the no-bailout constraint (Equation 29)
R6	Macroprudential policy and an ex post resolution authority can implement the social optimum	Corollary 2, pp. 2829-2830	Require $R_\ell^s$ in non-bail-in-able senior debt and $R_u^s-R_\ell^s$ in bail-in-able junior debt; the authority writes down junior debt to $R_1-R_\ell^s$ when $R_\ell^s \le R_1 \le R_u^s$, yielding the Proposition 3 allocation

Overall (paper’s conclusion). The paper provides a single contracting framework that rationalizes both a maximum leverage requirement and a TLAC requirement as jointly optimal responses to fire sale externalities. Bail-in debt is the instrument of choice because it combines the incentive properties of standard debt with the loss-absorbing properties of equity, making it superior to outside equity as a loss-absorbing instrument. Bail-ins replace bailouts in the regulatory toolkit; statutory provisions that increase the cost of engaging in bailouts complement bail-in regulation by allowing looser regulatory constraints while still preventing bailouts in equilibrium.

Theory / model

The model has three periods ($t = 0, 1, 2$), a unit continuum of banks, penniless investors, and arbitrageurs. Banks invest in a firm of variable scale $Y_0 = A_0 + I_0 > 0$ using their own equity $A_0 > 0$ and investor funds $I_0 \ge 0$ (p. 2815).

Project quality shocks. At each of dates 1 and 2, the bank experiences a stochastic quality shock $R_t \in [\underline{R}, \overline{R}]$ that adjusts project scale to $Y_t = R_t Y_{t-1}$, so final scale is $Y_2 = R_1 R_2 Y_0$. The project pays one unit of the consumption good per unit of final scale at date 2 but yields no dividend on date 1. Shocks $R_t$ are independent and idiosyncratic with densities (p. 2815):

$f_t(R_t | e_{t-1}) = e_{t-1} f_{tH}(R_t) + (1 - e_{t-1}) f_{tL}(R_t).$

Effort and MLRP. Date 0 effort is continuous, $e_0 \in [0,1]$; date 1 effort is binary, $e_1 \in \{0,1\}$. Higher effort increases the quality of the return distribution. The paper assumes the monotone likelihood ratio property (MLRP): the likelihood ratio $\Lambda_t(R_t) \equiv f_{tH}(R_t) / f_{tL}(R_t)$ is increasing in $R_t$.

Private benefits. The banker’s date 0 private benefit is $B_0(e_0) Y_0$, where $B_0$ is decreasing and concave with $B_0(1) = 0$. The banker’s date 1 private benefit is $(1 - e_1) B_1 Y_1$ for $0 < B_1 < 1$ (p. 2816).

Resource constraints. Along a history $(R_1, R_2)$, investor repayment $x_1(R_1)$, $x_2(R_1, R_2)$ and bank consumption $c_1(R_1)$, $c_2(R_1, R_2)$ satisfy (p. 2817):

c_1(R_1) + x_1(R_1) = \alpha(R_1) \gamma R_1 Y_0, \tag{1}

c_2(R_1, R_2) + x_2(R_1, R_2) = (1 - \alpha(R_1)) R_1 R_2 Y_0. \tag{2}

Limited liability requires $c_1(R_1), c_2(R_1, R_2) \ge 0$.

Investor participation constraint. Investors must at least break even (p. 2818):

Y_0 - A_0 \le \mathbb{E}\bigl[x(R_1) \mid e_0 = e_0^*\bigr]. \tag{6}

Fire sale and liquidation price. A representative arbitrageur with borrowing constraints generates a fire sale externality. The equilibrium liquidation price $\gamma$ satisfies the market-clearing condition (pp. 2820-2821):

\gamma(\Omega) = \frac{\partial \mathcal{F}(\Omega)}{\partial \Omega}, \quad \Omega = \int_{\underline{R}}^{\overline{R}} \alpha(R_1) R_1 f_1(R_1 | e_0^*) \, dR_1. \tag{16}

When $\partial^2 \mathcal{F} / \partial \Omega^2 < 0$, more liquidations reduce the price, creating the fire sale. The liquidation price elasticity is $\sigma = -(\Omega / \gamma)(\partial \gamma / \partial \Omega)$.

Date 1 incentive compatibility. High effort $e_1^*(R_1) = 1$ is incentive compatible if (p. 2818):

\mathbb{E}[c_2(R_1, R_2)(\Lambda_2(R_2) - 1) \mid e_1 = 0] \ge B_1 R_1 Y_0. \tag{9}

Date 0 optimal effort. The bank’s optimal date 0 effort $e_0^*$ satisfies (p. 2819):

-B_0'(e_0^*) Y_0 = \mathbb{E}_0[c(R_1)(\Lambda_1(R_1) - 1) \mid e_0 = 0]. \tag{12}

Other numbered model equations (pp. 2817-2820). The page’s resource, participation, date-1 incentive, and date-0 effort equations above are part of the full model system. The remaining payoff, monotonicity, choice, and market clearing equations are (locators identify the printed equation pages):

Bank limited liability and total bank payoff (pp. 2817-2818):

c_1(R_1), c_2(R_1,R_2) \ge 0. \tag{3}

c_1(R_1)+c_2(R_1,R_2)+B_0(e_0^*)Y_0+(1-\alpha(R_1))(1-e_1^*(R_1))B_1R_1Y_0. \tag{4}

Investor expected payoff (p. 2817):

x(R_1\mid e_1^*,\alpha,x_1,x_2)=\begin{cases}x_1(R_1)+\mathbb{E}[x_2(R_1,R_2)\mid e_1=e_1^*(R_1)],&\alpha(R_1)=0,\\x_1(R_1),&\alpha(R_1)=1.\end{cases} \tag{5}

Repayment monotonicity (p. 2818):

R_1\ge R_1'\Rightarrow x(R_1)\ge x(R_1'). \tag{7}

R_2\ge R_2'\Rightarrow x_2(R_1,R_2)\ge x_2(R_1,R_2'). \tag{8}

Date-1 expected bank payoff and date-0 effort choice (p. 2819):

c(R_1\mid e_1^*,\alpha,c_1,c_2,Y_0)=\begin{cases}c_1(R_1)+\mathbb{E}[c_2(R_1,R_2)+(1-e_1^*(R_1))B_1R_1Y_0\mid e_1=e_1^*(R_1)],&\alpha(R_1)=0,\\c_1(R_1),&\alpha(R_1)=1.\end{cases} \tag{10}

\max_{e\in[0,1]}\mathbb{E}[c(R_1)\mid e_0=e]+B_0(e)Y_0. \tag{11}

The bank chooses the feasible contract to maximize (p. 2819):

\mathbb{E}[c(R_1)\mid e_0=e_0^*]+B_0(e_0^*)Y_0. \tag{13}

The arbitrageur’s demand and market clearing (pp. 2820-2821):

\frac{\partial F(\Omega)}{\partial\Omega}=\gamma. \tag{14}

\Omega(\alpha,e_0^*)=\int_{\underline R}^{\overline R}\alpha(R_1)R_1f_1(R_1\mid e_0^*)\,dR_1. \tag{15}

Remaining numbered contract and policy equations. The date-2 repayment schedule, private liability schedule, optimal date-0 effort condition, social liability schedule, planner’s first-order conditions, and planner’s effort condition complete the main-text numbered equations:

Lemma 1’s repayment schedule and the private bank’s liability schedule (pp. 2821-2823):

x_1(R_1)=0,\quad x_2(R_1,R_2)=\begin{cases}R_2R_1Y_0,&R_2\le R_2^u(R_1),\\R_2^u(R_1)R_1Y_0,&R_2>R_2^u(R_1),\end{cases}\quad R_2^u(R_1)\le\overline R_2. \tag{17}

L(R_1)=\begin{cases}(1-b)R_\ell^pY_0,&R_1\le R_\ell^p,\\(1-b)R_1Y_0,&R_\ell^p\le R_1\le R_u^p,\\(1-b)R_u^pY_0,&R_u^p\le R_1.\end{cases} \tag{19}

The private optimum’s effort condition (p. 2823):

-B_0'(e_0^*)=\int_{R_\ell^p}^{\overline R}\max\{bR_1,R_1-(1-b)R_u^p\}(f_{1H}(R_1)-f_{1L}(R_1))\,dR_1. \tag{22}

The planner’s liability schedule, first-order conditions, and effort condition (pp. 2827-2828):

L(R_1)=\begin{cases}(1-b)R_\ell^sY_0^s,&R_1\le R_\ell^s,\\(1-b)R_1Y_0^s,&R_\ell^s\le R_1\le R_u^s,\\(1-b)R_u^sY_0^s,&R_u^s\le R_1.\end{cases} \tag{23}

\frac{1-\Lambda_1(R_\ell^s)}{(1-e_0^s)+e_0^s\Lambda_1(R_\ell^s)}\frac{1}{|B_0''(e_0^s)|}b\lambda^sG^s=b+\lambda^s(1-b-\gamma^s)+\tau_\ell^s. \tag{24}

\frac{F_{1L}(R_u^s)-F_{1H}(R_u^s)}{|B_0''(e_0^s)|}\lambda^sG^s=(\lambda^s-1)(1-F_1(R_u^s\mid e_0^s))-\tau_u^s. \tag{25}

-B_0'(e_0^s)=\int_{R_\ell^s}^{\overline R}\max\{bR_1,R_1-(1-b)R_u^s\}(f_{1H}(R_1)-f_{1L}(R_1))\,dR_1. \tag{28}

Method

This is a pure theory paper. The model is solved by characterizing the set of feasible contracts (satisfying limited liability, resource constraints, investor participation, repayment monotonicity, and incentive compatibility at both dates), then finding the contract that maximizes bank expected utility subject to those constraints.

Pledgeability reduction (Lemma 1, p. 2822). The binary date 1 effort problem reduces to a Holmstrom and Tirole (1997) style pledgeability constraint. The optimal contract sets $x_1(R_1) = 0$ and repays investors on date 2 at a threshold $R_2^u(R_1)$. Bank high effort on date 1 is incentive compatible if and only if (p. 2822):

c(R_1) \ge b R_1 Y_0. \tag{18}

Here $b = \int_{\overline{R}_2^u}^{\overline{R}} [R_2 - \overline{R}_2^u] f_{2H}(R_2) dR_2$ is a constant (Equation 18).

Mapping to promised liabilities. The paper maps actual-repayment contracts to promised “face value” liabilities $L(R_1)$: if $L(R_1) \le (1-b)R_1 Y_0$ the bank avoids liquidation; if $L(R_1) > (1-b)R_1 Y_0$ the bank is liquidated with actual repayment $\gamma R_1 Y_0$ (p. 2822).

First-order conditions for the private optimum (Proposition 1, pp. 2822-2823). The privately optimal liquidation threshold $R_\ell^p$ and total debt threshold $R_u^p$ satisfy:

\underbrace{\frac{1 - \Lambda_1(R_\ell^p)}{(1 - e_0^*) + e_0^* \Lambda_1(R_\ell^p)} \frac{1}{|B_0''(e_0^*)|}}_{\text{Incentive Provision}} b \lambda G = \underbrace{b + \lambda(1 - b - \gamma)}_{\text{Liquidation Costs}}, \tag{20}

\underbrace{\frac{F_{1L}(R_u^p) - F_{1H}(R_u^p)}{|B_0''(e_0^*)|}}_{\text{Incentive Provision}} \lambda G = \underbrace{(\lambda - 1)(1 - F_1(R_u^p | e_0^*))}_{\text{Investor Repayment}}, \tag{21}

where $\lambda > 1$ is the Lagrange multiplier on the investor participation constraint and $G = \int_{\underline{R}}^{R_\ell^p} \gamma R_1 (f_{1H}(R_1) - f_{1L}(R_1)) dR_1 + \int_{R_\ell^p}^{\overline{R}} (1-b) \min\{R_1, R_u^p\} (f_{1H}(R_1) - f_{1L}(R_1)) dR_1$.

Social planner’s first-order conditions (Proposition 3, pp. 2827-2828). The planner’s optimum satisfies the same structural equations as the private optimum but with wedge terms $+\tau_\ell^s$ and $-\tau_u^s$ on the right-hand sides of Equations (24) and (25):

\tau_\ell^s = \left(1 - \frac{1 - \Lambda_1(R_\ell^s)}{(1 - e_0^s) + e_0^s \Lambda_1(R_\ell^s)} \frac{1}{|B_0''(e_0^s)|} b L^s\right) \lambda^s \sigma \gamma^s \ge 0, \tag{26}

\tau_u^s = \frac{F_{1L}(R_u^s) - F_{1H}(R_u^s)}{|B_0''(e_0^s)|} L^s \sigma \gamma^s \ge 0, \tag{27}

where $\sigma$ is the liquidation price elasticity and $\gamma^s$ is the equilibrium liquidation price, and $L^s = \int_{R_\ell^s}^{\overline R} R_1 (f_{1L}(R_1) - f_{1H}(R_1))\,dR_1 \ge 0$. The wedge $\tau_\ell^s$ reflects the social cost of additional liquidations induced by standard debt; $\tau_u^s$ reflects the social cost of more total debt through the indirect effort channel.

No-bailout constraint (Proposition 5, p. 2833). In the case without planner commitment, the planner does not bail out banks if the total losses from liquidations are high enough:

(1 - \gamma(\Omega)) \Omega Y_0 \le F. \tag{29}

Empirical specifications

This is a pure theory paper with no empirical component. There are no regressions, no estimation equations, and no data. The paper characterizes optimal contracts and regulatory instruments analytically. The special case of linear private benefits ($B_0(e_0) = b_0(1-e_0)$) is worked out in Internet Appendix B.3 as a closed-form tractable illustration.

Datasets used

Dataset	Role in paper	Wiki page
None	Pure theory paper; no data used	N/A

When to read the full paper

Read Clayton and Schaab (2025) if you are designing or evaluating:

• Optimal bank capital structure rules combining leverage caps and TLAC requirements
• The comparison between bail-in debt and outside equity as loss-absorbing instruments (Proposition 2 and Section 2.2.1), extending Dewatripont and Tirole (1994)
• The rationale for statutory provisions that increase the cost of bailouts, such as Dodd-Frank Act restrictions (Section 4.2); connects to Farhi and Tirole (2012) and Chari and Kehoe (2016)
• The good bank / bad bank approach to resolving large banks with partial liquidations (Section 5.2)
• The connection between the incentive-based explanation and the result of Keister and Mitkov (2023) that anticipated bailouts suppress bail-in debt issuance
• The connection between the incentive-based and demand-based (safety premium) explanations for the coexistence of standard and bail-in debt (Section 5.3)

Attribution and rights

Clayton, C., and A. Schaab. 2025. “Bail-Ins, Optimal Regulation, and Crisis Resolution.” The Review of Financial Studies 38(9): 2810-2843. https://doi.org/10.1093/rfs/hhaf002.

Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For commercial re-use contact reprints@oup.com. This page is an LLM-distilled summary (extract-only); it is not human-verified and has not been reproduced. Read the original at the DOI above.