11. Hard vs. Soft Budget Constraint

Jun He May 31, 2026

公司金融Corporate Finance 预算硬约束Hard Budget Constraint 完全合约Complete Contract 低效清算Inefficient Liquidation 重新谈判Renegotiation 债权人数量Number of Creditors 战略违约Strategic Default 学习笔记Study Note

Note

本章主题：预算硬约束 vs 软约束。 两个模型对照「能否重新谈判」。Bolton-Scharfstein (1990)（§11.1，完全合约 = 硬约束）：合约完全 → 低效清算不可重新谈判，故清算威胁可信（掠夺），用以阻止经理虚报现金流。三期、经理零财富、$t{=}1$ 现金流 $x$（概率 $\theta$）或 0，报告 $\hat x$ 决定还款 $R$ 与清算概率 $\beta$。最优完全合约（揭示原理→讲真话直接机制）解 (11.1)–(11.3)：可证真报状态零清算 $\beta_x^\star=0$、零报状态须正清算 $\beta_0^\star>0$（否则无法维持讲真话），化简得 $\beta_0^\star=\frac{I}{(1-\theta)\alpha y+\theta y}>0$ (11.6)。Remark 11.1：$\beta_0^\star>0$（低效清算）是无效率之源，源于维持讲真话的需要；Remark 11.2：这是完全合约保证的硬约束，机制类似 Townsend (1979) 的 CSV（都用低效威慑工具逼真话）；Remark 11.3：刻画风投与年轻公司——中期现金流好则继续、坏则终止。Bolton-Scharfstein (1996)（§11.2，不完全合约 = 软约束）：低效清算可重新谈判，经理得 $S(N)>0$、债权人得 $L_0(N)>\alpha y$，$N$ 为竞争债权人数量（假设 $S'(N)<0$、$L_0'(N)<0$）。新 IC (11.7)、IR (11.8) 给出 $\beta_0^\star=\frac{I}{(1-\theta)L_0(N)+\theta(y-S(N))}$。$L_0$↑→无效率↓（债权人补偿更好），$S$↑→无效率↑（虚报更诱人）。$N$ 的权衡：更多债权人使 $L_0$↓（更难保本）但 $S$↓（扭曲更小）。两类违约：战略违约（$\hat x(x)=0$，虚报激励太强→$N$ 宜大以降 $S$）、流动性违约（$\hat x(0)=0$，还不上→清算前谈判宜易→$N$ 宜小）。

Note

Chapter theme: hard vs. soft budget constraint. Two models contrast on "whether renegotiation is possible." Bolton-Scharfstein (1990) (§11.1, complete contract = hard constraint): contract completeness → inefficient liquidation is non-renegotiable, so the liquidation threat is credible (predation), used to prevent the manager from misreporting cash flow. Three periods, manager has zero wealth, $t{=}1$ cash flow $x$ (prob $\theta$) or 0, the report $\hat x$ determines repayment $R$ and liquidation probability $\beta$. The optimal complete contract (a truth-telling direct mechanism by the revelation principle) solves (11.1)–(11.3): one can show the truthful state has zero liquidation $\beta_x^\star=0$ while the zero-report state needs positive liquidation $\beta_0^\star>0$ (otherwise truth-telling fails), giving $\beta_0^\star=\frac{I}{(1-\theta)\alpha y+\theta y}>0$ (11.6). Remark 11.1: $\beta_0^\star>0$ (inefficient liquidation) is the source of inefficiency, arising from the need to maintain truth-telling; Remark 11.2: this is a hard constraint guaranteed by the complete contract, the mechanism resembling Townsend's (1979) CSV (both use an inefficient deterrent to enforce truth-telling); Remark 11.3: it depicts venture capital and young firms — continue if interim cash flow is good, terminate if bad. Bolton-Scharfstein (1996) (§11.2, incomplete contract = soft constraint): inefficient liquidation can be renegotiated, the manager getting $S(N)>0$ and creditors $L_0(N)>\alpha y$, where $N$ is the number of competing creditors (assume $S'(N)<0$, $L_0'(N)<0$). The new IC (11.7) and IR (11.8) give $\beta_0^\star=\frac{I}{(1-\theta)L_0(N)+\theta(y-S(N))}$. Higher $L_0$ → less inefficiency (better creditor compensation); higher $S$ → more inefficiency (misreporting more attractive). The trade-off in $N$: more creditors make $L_0$ smaller (harder break-even) but also $S$ smaller (less distortion). Two default types: strategic default ($\hat x(x)=0$, misreporting incentive too strong → larger $N$ to lower $S$) and liquidity default ($\hat x(0)=0$, cannot repay → easier renegotiation before liquidation → smaller $N$).

11.1 Complete Contract and Hard Budget Constraint: Bolton and Scharfstein (1990)

Bolton and Scharfstein (1990) 讨论一个完全合约模型：因合约完全，低效清算前无法重新谈判，故低效清算成为一个可信威胁（掠夺 predation），用以阻止经理虚报。

11.1.1 设定.

所有主体风险中性，贴现率零，三期 $t=0,1,2$。
只有一个项目：
$t=0$ 需初始投资 $I$；
$t=1$ 产生现金流 $\tilde x$：

$$\tilde x=\begin{cases}x>0 & \text{w.p. }\theta\\[2pt] 0 & \text{w.p. }1-\theta\end{cases}$$

$t=2$ 产生现金流 $\tilde y$（取 $\varepsilon\to0$）：

$$\tilde y=\begin{cases}y>0 & \text{w.p. }1-\varepsilon\\[2pt] 0 & \text{w.p. }\varepsilon\end{cases}$$

经理零财富，故向债权人借 $I$。
$t=1$ 经理报告现金流 $\hat x(\tilde x)$：
若报告 $x$（$\hat x=x$），则 $t=1$ 付 $R_x$ 给债权人，并以概率 $\beta_x$ 被清算；
若报告 0（$\hat x=0$），则 $t=1$ 付 $R_0$，并以概率 $\beta_0$ 被清算；
假设 $x>R_x>R_0$ 且 $\beta_x<\beta_0$；
清算技术低效，给出回报 $\alpha y$（$\alpha\in(0,1)$）。

Bolton and Scharfstein (1990) discuss a model of complete contract, in which no renegotiation before inefficient liquidation is possible due to contract completeness. So inefficient liquidation becomes a credible threat (predation) to prevent the manager from misreporting.

11.1.1 Setup.

All agents risk neutral, discount rate zero, three periods $t=0,1,2$.
There is only one project:
at $t=0$ it requires an initial investment of $I$;
at $t=1$ it generates cash flow $\tilde x$:

$$\tilde x=\begin{cases}x>0 & \text{w.p. }\theta\\[2pt] 0 & \text{w.p. }1-\theta\end{cases}$$

at $t=2$ it generates cash flow $\tilde y$ (with $\varepsilon\to0$):

$$\tilde y=\begin{cases}y>0 & \text{w.p. }1-\varepsilon\\[2pt] 0 & \text{w.p. }\varepsilon\end{cases}$$

The manager has zero wealth, so he borrows $I$ from the creditor.
At $t=1$ the manager reports cash flow $\hat x(\tilde x)$:
if he reports $x$ ($\hat x=x$), then at $t=1$ he pays $R_x$ to the creditor and suffers liquidation with probability $\beta_x$;
if he reports 0 ($\hat x=0$), then at $t=1$ he pays $R_0$ and suffers liquidation with probability $\beta_0$;
assume $x>R_x>R_0$ and $\beta_x<\beta_0$;
the liquidation technology is inefficient, giving a payoff $\alpha y$ ($\alpha\in(0,1)$).

11.1.2 Optimal Complete Contract

合约的完全性来自不可重新谈判的清算：以概率 $\beta_x$ 或 $\beta_0$ 公司必被清算。由于清算只在 $t=1$ 发生，经理在 $t=2$ 总会报告 0 并享有 $\tilde y=y$（$\varepsilon\to0$）。由揭示原理，$t=0$ 的最优合约对应一个讲真话的直接机制。

经理 $t=1$ 讲真话的 IC 约束：

$$\underbrace{x-R_x+(1-\beta_x)y}_{\text{truth-telling if }\tilde x=x}\ge\underbrace{x-R_0+(1-\beta_0)y}_{\text{misreporting if }\tilde x=x}$$

债权人保本的 IR 约束：

$$\theta(R_x+\beta_x\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)=I$$

有限责任 LL：$R_0=0$ 且 $R_x\le x$。

故 $t=0$ 的最优合约求解：

$$\max_{R_x,R_0,\beta_x,\beta_0}\theta[x-R_x+(1-\beta_x)y]+(1-\theta)[-R_0+(1-\beta_0)y] \tag{11.1}$$

$$\text{s.t.}\quad x-R_x+(1-\beta_x)y\ge x-R_0+(1-\beta_0)y \quad[\text{IC}] \tag{11.2}$$

$$\theta(R_x+\beta_x\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)=I \quad[\text{IR}] \tag{11.3}$$

$$R_0=0,\ R_x\le x \quad[\text{LL}];\qquad \beta_x\in[0,1],\ \beta_0\in[0,1]$$

可证最优时 $\beta_x=0$、$\beta_0>0$（证明见下方）。代入 $\beta_x=0$、$\beta_0>0$、$R_0=0$，问题化简为

$$\min_{\beta_0}\ I+(1-\alpha)(1-\theta)\beta_0 y \tag{11.4}$$

$$\text{s.t.}\quad R_x\le\beta_0 y \quad[\text{IC}] \tag{11.5}$$

目标随 $\beta_0$ 递增，故取 IC 绑定 $R_x=\beta_0 y$，代入 IR (11.3) 得

$$\theta y\beta_0^\star+(1-\theta)\beta_0^\star\alpha y=I\ \Rightarrow\ \beta_0^\star=\frac{I}{(1-\theta)\alpha y+\theta y}>0 \tag{11.6}$$

Note

证明 / 最优合约的 $\beta_x=0$ 与 $\beta_0>0$（点击展开） 断言一：最优时 $\beta_x=0$。 反设 $\beta_x>0$。对 $0<\varepsilon<\beta_x$，令 $\beta_x'=\beta_x-\varepsilon>0$、$R_x'=R_x+\alpha y\varepsilon$。代入 IR (11.3) 的左端： $$\theta(R_x'+\beta_x'\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)=\theta\big[R_x+\alpha y\varepsilon+(\beta_x-\varepsilon)\alpha y\big]+(1-\theta)(R_0+\beta_0\alpha y)=\theta(R_x+\beta_x\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)$$ 与原 IR 左端相等，故 IR 不受影响。再代入 IC (11.2) 左端： $$x-R_x'+(1-\beta_x')y=x-(R_x+\alpha y\varepsilon)+[1-(\beta_x-\varepsilon)]y=x-R_x+(1-\beta_x)y+\underbrace{(1-\alpha)\varepsilon y}_{>0}>x-R_x+(1-\beta_x)y$$ 即 IC 被放松，从而可通过降低 $R_0$（使 (11.2) 仍成立）来提高目标函数。这与「$\beta_x>0$ 是最优」矛盾。故 $\beta_x=0$。 断言二：最优时 $\beta_0>0$。 反设 $\beta_0=0$。则 IC (11.2)（已 $\beta_x=0$）给出 $$x-R_x+(1-0)y\ge x-R_0+(1-0)y\ \Rightarrow\ -R_x-\beta_x y\ge -R_0\ \Rightarrow\ R_x\le R_0-\beta_x y\le 0$$ 即 $R_x\le 0$。结合 $R_0=0$、$\beta_0=0$，债权人收入 $\theta R_x\le 00$。$\blacksquare$

The completeness of the contract comes from the non-renegotiable liquidation: with probability $\beta_x$ or $\beta_0$, the firm must be liquidated. Since liquidation only happens at $t=1$, the manager always reports 0 at $t=2$ and enjoys $\tilde y=y$ ($\varepsilon\to0$). By the revelation principle, the optimal contract at $t=0$ corresponds to a truth-telling direct mechanism.

The manager's truth-telling IC constraint at $t=1$:

$$\underbrace{x-R_x+(1-\beta_x)y}_{\text{truth-telling if }\tilde x=x}\ge\underbrace{x-R_0+(1-\beta_0)y}_{\text{misreporting if }\tilde x=x}$$

The creditor's break-even IR constraint:

$$\theta(R_x+\beta_x\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)=I$$

Limited liability LL: $R_0=0$ and $R_x\le x$.

So the optimal contract at $t=0$ solves:

$$\max_{R_x,R_0,\beta_x,\beta_0}\theta[x-R_x+(1-\beta_x)y]+(1-\theta)[-R_0+(1-\beta_0)y] \tag{11.1}$$

$$\text{s.t.}\quad x-R_x+(1-\beta_x)y\ge x-R_0+(1-\beta_0)y \quad[\text{IC}] \tag{11.2}$$

$$\theta(R_x+\beta_x\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)=I \quad[\text{IR}] \tag{11.3}$$

$$R_0=0,\ R_x\le x \quad[\text{LL}];\qquad \beta_x\in[0,1],\ \beta_0\in[0,1]$$

One can show that optimally $\beta_x=0$ and $\beta_0>0$ (proof below). Plugging $\beta_x=0$, $\beta_0>0$, $R_0=0$, the problem reduces to

$$\min_{\beta_0}\ I+(1-\alpha)(1-\theta)\beta_0 y \tag{11.4}$$

$$\text{s.t.}\quad R_x\le\beta_0 y \quad[\text{IC}] \tag{11.5}$$

The objective increases in $\beta_0$, so the IC binds ($R_x=\beta_0 y$); plugging into IR (11.3) gives

$$\theta y\beta_0^\star+(1-\theta)\beta_0^\star\alpha y=I\ \Rightarrow\ \beta_0^\star=\frac{I}{(1-\theta)\alpha y+\theta y}>0 \tag{11.6}$$

Note

Proof / optimal contract has $\beta_x=0$ and $\beta_0>0$ (click to expand) Claim 1: optimally $\beta_x=0$. Suppose not, so $\beta_x>0$. For $0<\varepsilon<\beta_x$, let $\beta_x'=\beta_x-\varepsilon>0$ and $R_x'=R_x+\alpha y\varepsilon$. Plug into the LHS of IR (11.3): $$\theta(R_x'+\beta_x'\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)=\theta\big[R_x+\alpha y\varepsilon+(\beta_x-\varepsilon)\alpha y\big]+(1-\theta)(R_0+\beta_0\alpha y)=\theta(R_x+\beta_x\alpha y)+(1-\theta)(R_0+\beta_0\alpha y)$$ which equals the original IR LHS, so IR is not affected. Plug into the LHS of IC (11.2): $$x-R_x'+(1-\beta_x')y=x-(R_x+\alpha y\varepsilon)+[1-(\beta_x-\varepsilon)]y=x-R_x+(1-\beta_x)y+\underbrace{(1-\alpha)\varepsilon y}_{>0}>x-R_x+(1-\beta_x)y$$ so IC is relaxed, and the objective can be increased by reducing $R_0$ (keeping (11.2) holding). This contradicts $\beta_x>0$ being optimal. Hence $\beta_x=0$. Claim 2: optimally $\beta_0>0$. Suppose not, so $\beta_0=0$. Then IC (11.2) (with $\beta_x=0$) gives $$x-R_x+(1-0)y\ge x-R_0+(1-0)y\ \Rightarrow\ -R_x-\beta_x y\ge -R_0\ \Rightarrow\ R_x\le R_0-\beta_x y\le 0$$ i.e. $R_x\le 0$. With $R_0=0$ and $\beta_0=0$, the creditor's revenue $\theta R_x\le 00$. $\blacksquare$

Remarks (11.1)

Tip

Remark 11.1（无效率之源）由 (11.6)，$\beta_0^\star>0$，即当 $\hat x=0$（流动性违约）时必须进行低效清算。这是出于维持讲真话的需要。所以 $\beta_0^\star>0$（经理的扭曲激励）正是无效率之源。

Tip

Remark 11.2（硬约束与 CSV 的类比）这种低效清算的威胁是由完全合约保证的预算硬约束。其机制与 Townsend (1979) 讨论的 CSV 相似：两种设定都用一个低效的威慑工具来强制经理讲真话。

Tip

Remark 11.3（风投与年轻公司）该模型刻画了风险投资与年轻公司的关系：若中期现金流好，则继续；若坏，则终止。

Tip

Remark 11.1 (the source of inefficiency) From (11.6), $\beta_0^\star>0$, which means inefficient liquidation must be conducted when $\hat x=0$ (liquidity default). This is because of the need to maintain truth-telling. So $\beta_0^\star>0$ (the manager's distorted incentive) is the source of inefficiency.

Tip

Remark 11.2 (hard constraint and the CSV analogy) This threat of inefficient liquidation is a hard budget constraint guaranteed by the complete contract. The mechanism looks similar to the CSV discussed by Townsend (1979), in the sense that both setups involve an inefficient deterring tool to enforce truth-telling of the manager.

Tip

Remark 11.3 (venture capital and young firms) This model depicts the relation between venture capital and young firms: if interim cash flow is good, continue; if bad, terminate.

11.2 Incomplete Contract and Number of Creditors: Bolton and Scharfstein (1996)

Bolton and Scharfstein (1996) 用与 1990 文相同的设定，唯一不同是低效清算不再被强制执行、可以重新谈判。

11.2.1 设定.

保留 §11.1.1 的所有条件。
低效清算可重新谈判。谈判后，债权人得 $L_0>\alpha y$、经理得 $S>0$。
设有 $N$ 个相互竞争的债权人：
谈判回报 $L_0$ 与 $S$ 都受 $N$ 影响，即 $L_0(N)$、$S(N)$；
自然假设 $S'(N)<0$——债权人越多谈判越难，经理从谈判中得到越少；
对 $L_0(N)$ 较复杂，涉及两个相反效应：
- $N$ 越大，对任意固定的蛋糕，债权人作为整体能榨取更高比例，使 $L_0$ 上升；
- $N$ 越大，谈判越难，低效清算更可能发生，缩小蛋糕从而使 $L_0$ 下降；
- Bolton-Scharfstein (1996) 讨论了第二种效应占优的条件；
- 故我们假设 $L_0'(N)<0$。

Bolton and Scharfstein (1996) use the same setup as Bolton and Scharfstein (1990) except that the inefficient liquidation is not enforced and can be renegotiated.

11.2.1 Setup.

Keep all conditions in subsection 11.1.1.
The inefficient liquidation can be renegotiated. After renegotiation, the payoff to the creditor is $L_0>\alpha y$ and the payoff to the manager is $S>0$.
Suppose there are $N$ creditors competing with each other:
the renegotiation payoffs $L_0$ and $S$ are affected by $N$, i.e. $L_0(N)$, $S(N)$;
naturally assume $S'(N)<0$ — the manager gets less from renegotiation since negotiating with more creditors is harder;
for $L_0(N)$ it's complicated, involving two competing effects:
- higher $N$ means that for any fixed size of pie, creditors as a group can squeeze a higher fraction, which increases $L_0$;
- higher $N$ makes renegotiation harder, so inefficient liquidation is more likely, which reduces the size of the pie and thus reduces $L_0$;
- Bolton-Scharfstein (1996) discuss certain conditions under which the second effect is dominant;
- so we assume $L_0'(N)<0$.

11.2.2 Optimal Incomplete Contract

讲真话的 IC 约束. 在完全合约下，IC 由 (11.2) 给出（代入 $\beta_x=0$、$R_0=0$）：

$$x-R_x+y\ge x+\underbrace{\beta_0\cdot 0}_{\text{liquidation gives manager }0}+(1-\beta_0)y$$

在本不完全合约下，IC 变为

$$x-R_x+y\ge x+\beta_0\cdot S(N)+(1-\beta_0)y \tag{11.7}$$

与完全合约唯一的差别是：经过重新谈判，清算给经理的回报从 $0$ 变为 $S(N)$。

债权人保本的 IR 约束. 完全合约下 IR 由 (11.3) 给出（代入 $\beta_x=0$、$R_0=0$）为 $\theta R_x+(1-\theta)\beta_0\alpha y=I$。本不完全合约下 IR 变为

$$\theta R_x+(1-\theta)\beta_0 L_0(N)=I \tag{11.8}$$

唯一差别：经过重新谈判，清算给债权人的回报从 $\alpha y$ 变为 $L_0(N)$。

求最优 $\beta_0^\star$. 按与 §11.1.2 完全相同的步骤（用新 IC (11.7) 与新 IR (11.8)，IC 绑定 $R_x=\beta_0(y-S(N))$ 代入 IR），得与 (11.6) 类似的表达式：

$$\beta_0^\star=\frac{I}{(1-\theta)L_0(N)+\theta(y-S(N))}$$

$\beta_0^\star$ 是无效率的指标。
$L_0$ 越高 → 无效率越少：债权人补偿更好，更易保本。
$S$ 越高 → 无效率越多：经理虚报回报更好，扭曲讲真话激励、提高低效清算频率。

由假设 $L_0'(N)<0$、$S'(N)<0$，存在权衡：更高的 $N$（更多债权人）使 $L_0$ 变小（债权人更难保本），但也使 $S$ 变小（经理激励扭曲更小）。

两类违约：

战略违约 (strategic default)：$\hat x(x)=0$。经理虚报激励太强，故 $N$ 宜大以降低 $S(N)$、即降低虚报激励。
流动性违约 (liquidity default)：$\hat x(0)=0$。经理无力偿还 $R_x$，故清算前的重新谈判宜更易，即 $N$ 宜小。

Truth-telling IC constraint. Under the complete contract, IC is given by (11.2) (with $\beta_x=0$, $R_0=0$):

$$x-R_x+y\ge x+\underbrace{\beta_0\cdot 0}_{\text{liquidation gives manager }0}+(1-\beta_0)y$$

In this incomplete contract setting, IC becomes

$$x-R_x+y\ge x+\beta_0\cdot S(N)+(1-\beta_0)y \tag{11.7}$$

where the only difference with the complete contract is that the liquidation payoff to the manager changes from $0$ to $S(N)$ through renegotiation.

Creditor's break-even IR constraint. Under the complete contract, IR (11.3) (with $\beta_x=0$, $R_0=0$) is $\theta R_x+(1-\theta)\beta_0\alpha y=I$. In this incomplete contract setting, IR becomes

$$\theta R_x+(1-\theta)\beta_0 L_0(N)=I \tag{11.8}$$

where the only difference is that the liquidation payoff to the creditors changes from $\alpha y$ to $L_0(N)$ through renegotiation.

Solving for the optimal $\beta_0^\star$. Following exactly the same procedure as in §11.1.2 (with the new IC (11.7) and IR (11.8), the IC binding at $R_x=\beta_0(y-S(N))$ plugged into IR), we get an expression similar to (11.6):

$$\beta_0^\star=\frac{I}{(1-\theta)L_0(N)+\theta(y-S(N))}$$

$\beta_0^\star$ is the indicator of inefficiency.
Higher $L_0$ → less inefficiency: creditors get better compensation, so break even more easily.
Higher $S$ → more inefficiency: the manager has a better payoff from misreporting, distorting the truth-telling incentive and increasing the frequency of inefficient liquidation.

By the assumptions $L_0'(N)<0$ and $S'(N)<0$, there is a trade-off: higher $N$ (more creditors) makes $L_0$ smaller (harder for creditors to break even), but also makes $S$ smaller (smaller manager incentive distortion).

Two types of defaults:

Strategic default: $\hat x(x)=0$. The manager has too strong an incentive to misreport, so $N$ is better to be large to reduce $S(N)$, i.e. reduce the misreporting incentive.
Liquidity default: $\hat x(0)=0$. The manager is not able to repay $R_x$, so renegotiation before liquidation is better to be easier, i.e. $N$ is better to be small.

References

Bolton, P. and D. S. Scharfstein (1990). A theory of predation based on agency problems in financial contracting. The American Economic Review, 93–106.
Bolton, P. and D. S. Scharfstein (1996). Optimal debt structure and the number of creditors. Journal of Political Economy 104(1), 1–25.
Townsend, R. M. (1979). Optimal contracts and competitive markets with costly state verification. Journal of Economic Theory 21(2), 265–293.