设定 / Set-up 创业者有一个项目创意，可付出努力 \(e\) 来推进，但没有钱。努力 \(e\) 产生成本 \(\Phi(e)\)，关于 \(e\) 递增且凸，\(\Phi'(e)>0\)、\(\Phi''(e)>0\)。创业者风险中性，最大化期望支付；产出 \(x\in\mathcal{X}=[0,\overline x]\) 的分布由累积分布 \(F(x\mid e)\) 刻画，密度 \(f(x\mid e)\)。创业者需向债权人融资，并设计回款方案 \(r(x)\)——即产出 \(x\) 实现后须偿还给债权人的金额，故需设计函数 \(r(\cdot)\)；创业者有有限责任，\(0\le r(x)\le x\)。假设债权人也风险中性，记其本金加利息为 \(I\)；创业者在期望意义上至少偿还 \(I\) 才能使该项目对债权人理性，这构成债权人的 IR 约束。The entrepreneur has an idea of a project and can put in effort \(e\) to carry it out, but has no money. Effort \(e\) incurs a cost \(\Phi(e)\), increasing and convex in \(e\), \(\Phi'(e)>0\), \(\Phi''(e)>0\). The entrepreneur is risk-neutral and maximizes the expected payoff; the outcome \(x\in\mathcal{X}=[0,\overline x]\) has a distribution characterized by the c.d.f. \(F(x\mid e)\) with density \(f(x\mid e)\). The entrepreneur needs to raise money from a creditor and design a payback scheme \(r(x)\) — the amount to pay back once outcome \(x\) is realized, so he must design the function \(r(\cdot)\); the entrepreneur has limited liability, \(0\le r(x)\le x\). Assume the creditor is also risk-neutral, with initial outlay plus interest denoted \(I\); the entrepreneur must pay back at least \(I\) in expectation to make the project rational for the creditor, forming the creditor's IR constraint.

三个假设 / Three assumptions 1. 问题 (18.1) 有一个对创业者有吸引力的解。(a) 即可能存在期望意义上 \(x-r(x)>0\) 的解；(b) 有此假设便无需再为创业者显式添加 IR 约束。2. 最优解的努力水平 \(e\) 低于第一最优水平。(a) 第一最优努力即创业者若用自有资金做项目时会付出的努力；(b) 这使问题有趣，否则就成了无隐藏行动、无道德风险的博弈。3. 假设函数 \(\int_z^{\overline x}(x-r(x))f(x\mid e)\,dx-\Phi(e)\) 对任意 \(z\in(0,\overline x)\) 关于 \(e\) 严格凹，并在某个 \(e\in(0,\overline e)\) 取唯一最大值。(a) 此条件弱于 CDFC；(b) 它蕴含 FOA 问题与原问题等价。1. Problem (18.1) has a solution attractive to the entrepreneur. (a) This means it is possible to have a solution with positive \(x-r(x)\) in expectation; (b) with this assumption we don't need to explicitly add another IR constraint for the entrepreneur. 2. The optimal solution has an effort level \(e\) below the first-best level. (a) The first-best effort is the effort the entrepreneur would put in if he used his own money; (b) this makes the problem interesting, otherwise we'd have a game without hidden action and moral hazard. 3. Assume the function \(\int_z^{\overline x}(x-r(x))f(x\mid e)\,dx-\Phi(e)\) is strictly concave in \(e\) for any \(z\in(0,\overline x)\), attaining a unique maximum for some \(e\in(0,\overline e)\). (a) This condition is weaker than CDFC; (b) it implies the equivalence of the FOA problem and the original problem.

\(\mu>0\) 与 MLRP / \(\mu>0\) and MLRP 由 MLRP（定义 17.1），\(\dfrac{f_e(x\mid e)}{f(x\mid e)}\) 随 \(x\) 递增。又 \(\mu>0\)，因为：(i) 由拉格朗日的设置方式 \(\mu\ge0\)（脚注 18.1：否则 \(\mu<0\) 意味创业者付出过多努力使 \(\int(x-r(x))f_e\,dx<\Phi'(e)\)，不可能最优）；(ii) 若 \(\mu=0\)，IC 一阶条件不紧约束，创业者会选最低努力，这与其从 \(x-r(x)\) 中获得的激励不一致；故绝大多数情形 \(\mu=0\) 不成立，即便有 \(f(x\mid e)\) 使 \(\mu=0\) 成立，那也是努力恒为最低的无趣情形。所以 \(\mu>0\)。By MLRP (Definition 17.1), \(\dfrac{f_e(x\mid e)}{f(x\mid e)}\) increases in \(x\). Also \(\mu>0\), because: (i) \(\mu\ge0\) by the way the Lagrangian is set up (footnote 18.1: otherwise \(\mu<0\) means the entrepreneur puts in too much effort such that \(\int(x-r(x))f_e\,dx<\Phi'(e)\), which cannot be optimal); (ii) if \(\mu=0\), the IC-f.o.c. is not binding and the entrepreneur chooses the lowest effort, inconsistent with the incentive coming from \(x-r(x)\); so in most cases \(\mu=0\) cannot be true, and even where \(f(x\mid e)\) makes \(\mu=0\) true, it is the uninteresting case where effort is always the lowest. So \(\mu>0\).

图 29 / Figure 29（已转述 / paraphrased） 图 29a（债权人支付 \(r(x)\)）：在 \(x\le\hat x\) 段，\(r(x)=x\) 沿 45 度线上升至高度 \(\hat x\)；在 \(x>\hat x\) 段 \(r(x)=0\)（债权人在高产出区一分钱都拿不到）。图 29b（创业者支付 \(x-r(x)\)）：在 \(x\le\hat x\) 段恒为 0（创业者把全部产出还给债权人）；在 \(x>\hat x\) 段 \(x-r(x)=x\)，以斜率 1 上升。这是一种"活/死"式的极端方案——低产出全归债权人、高产出全归创业者。Figure 29a (creditor's payoff \(r(x)\)): on the \(x\le\hat x\) segment, \(r(x)=x\) rises along the 45-degree line to height \(\hat x\); on the \(x>\hat x\) segment \(r(x)=0\) (the creditor gets nothing in the high-output region). Figure 29b (entrepreneur's payoff \(x-r(x)\)): on \(x\le\hat x\) it is 0 (the entrepreneur pays all output back to the creditor); on \(x>\hat x\), \(x-r(x)=x\) rises with slope 1. This is an extreme "live/die" scheme — low output all to the creditor, high output all to the entrepreneur.

命题 18.1 与定义 18.1 / Proposition 18.1 and Definition 18.1 命题 18.1：若 \(r(x)\) 关于 \(x\) 弱递增且 \(f(x\mid e)\) 满足 MLRP，则对风险中性的创业者与债权人，最优回款方案必为债务合约。定义 18.1（债务合约）：债务合约 \(r^D(\cdot)\) 定义为 \(r^D(x)=\min\{x,D\}\)，其中 \(D\) 为本金加预定利息。Proposition 18.1: If \(r(x)\) is weakly increasing in \(x\) and \(f(x\mid e)\) satisfies MLRP, then for a risk-neutral entrepreneur and creditor, the optimal payback scheme must be a debt contract. Definition 18.1 (Debt contract): a debt contract \(r^D(\cdot)\) is defined by \(r^D(x)=\min\{x,D\}\), where \(D\) is the initial outlay plus the pre-determined interest.

命题 18.1 证明 / Proof of Proposition 18.1 反设不然。则带单调性的最优合约记为 \(r^\star(x)\)，它不是债务合约，对应最优努力 \(e^\star\)。我们分三步证明 \((r^\star(\cdot),e^\star)\) 并非最优。Suppose not. Then the optimal contract with monotonicity is denoted \(r^\star(x)\), which is not a debt contract, with corresponding optimal effort \(e^\star\). We prove \((r^\star(\cdot),e^\star)\) is not optimal in three steps.

第一步：因 \(r^D(\cdot)\) 弱递增，可找到 \(D\) 使下式成立（\(I\) 为本金加预定利息）：First, since \(r^D(\cdot)\) is weakly increasing, we can find \(D\) such that (\(I\) is initial outlay plus pre-determined interest):

$$I=\int_{\mathcal{X}}r^\star(x)f(x\mid e^\star)\,dx=\int_{\mathcal{X}}r^D(x)f(x\mid e^\star)\,dx$$

记 \(r^D(\cdot)\) 对应的最优努力为 \(e^D\)。可论证 \(e^D>e^\star\)（细节见 Innes 1990）：债务合约 \(r^D(\cdot)\) 在 \(x\le D\) 处最大斜率为 1、\(x>D\) 处斜率为 0；非债务合约 \(r^\star(\cdot)\)（脚注 18.2：非债务＋单调性意味 \(r^\star(\cdot)\) 在 \(D\) 之后某处有严格正斜率）要有同样的期望值 \(I\)，必存在 \(\tilde x\) 使 \(r^D(x)\ge r^\star(x)\)（\(x\le\tilde x\)）、\(r^D(x)\tilde x\)）（脚注 18.3：因有限责任，\(r^\star(x)\) 在 \(D\) 之前位于 \(r^D(x)\) 下方且不会与之相交）。Denote the optimal effort with \(r^D(\cdot)\) by \(e^D\). We can show \(e^D>e^\star\) (see Innes 1990 for details): the debt contract \(r^D(\cdot)\) has maximum slope 1 for \(x\le D\) and slope 0 for \(x>D\); for a non-debt contract \(r^\star(\cdot)\) (footnote 18.2: non-debt plus monotonicity imply \(r^\star(\cdot)\) has strictly positive slope somewhere after \(D\)) to have the same expectation \(I\), there must be \(\tilde x\) such that \(r^D(x)\ge r^\star(x)\) for \(x\le\tilde x\) and \(r^D(x)\tilde x\) (footnote 18.3: by limited liability, \(r^\star(x)\) is below and cannot cross \(r^D(x)\) before \(D\)).

（比较图，已转述：\(r^D(x)\) 红线先沿 45 度升至 \(D\) 再水平；\(r^\star(x)\) 蓝线呈 S 形，在 \(\tilde x>D\) 处自下方穿过 \(r^D\)。）相比 \(r^\star(\cdot)\)，债务合约 \(r^D(\cdot)\) 把回款"从右向左"移动，也即把对创业者的支付"从左向右"移动。因 \(f(x\mid e)\) 满足 MLRP，最优努力 \(e^D\) 上升，即 \(e^D>e^\star\)。(Comparison figure, paraphrased: the red line \(r^D(x)\) rises along the 45-degree line to \(D\) then flattens; the blue line \(r^\star(x)\) is S-shaped, crossing \(r^D\) from below at \(\tilde x>D\).) Compared with \(r^\star(\cdot)\), the debt contract \(r^D(\cdot)\) shifts payback from right to left, which shifts the payoff to the entrepreneur from left to right. Since \(f(x\mid e)\) satisfies MLRP, the optimal effort \(e^D\) increases, i.e. \(e^D>e^\star\).

第二步：在债务合约下做分部积分：Second, under debt contracts, do integration by parts:

$$\int_{\mathcal{X}}r^D(x)f(x\mid e^D)\,dx=\int_{\mathcal{X}}\min\{x,D\}f(x\mid e^D)\,dx=\overline x-\int_0^D F(x\mid e^D)\,dx$$

同理 \(\int_{\mathcal{X}}r^D(x)f(x\mid e^\star)\,dx=\overline x-\int_0^D F(x\mid e^\star)\,dx\)。由 MLRP，\(e^D>e^\star\) 蕴含对任意 \(x\) 有 \(F(x\mid e^D)\le F(x\mid e^\star)\)，故Similarly \(\int_{\mathcal{X}}r^D(x)f(x\mid e^\star)\,dx=\overline x-\int_0^D F(x\mid e^\star)\,dx\). By MLRP, \(e^D>e^\star\) implies \(F(x\mid e^D)\le F(x\mid e^\star)\) for any \(x\), so

$$\int_{\mathcal{X}}r^D(x)f(x\mid e^D)\,dx>\int_{\mathcal{X}}r^D(x)f(x\mid e^\star)\,dx=I$$

这意味 \((r^D(\cdot),e^D)\) 使风险中性的债权人更好。which implies \((r^D(\cdot),e^D)\) makes the risk-neutral creditor better off.

第三步：因 \(\int_{\mathcal{X}}r^\star(x)f(x\mid e^\star)\,dx=\int_{\mathcal{X}}r^D(x)f(x\mid e^\star)\,dx\)，故 \(\int_{\mathcal{X}}(x-r^\star(x))f(x\mid e^\star)\,dx=\int_{\mathcal{X}}(x-r^D(x))f(x\mid e^\star)\,dx\)，即创业者本可在 \(r^D(\cdot)\) 下选同样努力 \(e^\star\) 得到同样支付，但他却选了更高的 \(e^D\)，故其在 \((r^D(\cdot),e^D)\) 下支付更高：Third, since \(\int_{\mathcal{X}}r^\star(x)f(x\mid e^\star)\,dx=\int_{\mathcal{X}}r^D(x)f(x\mid e^\star)\,dx\), we have \(\int_{\mathcal{X}}(x-r^\star(x))f(x\mid e^\star)\,dx=\int_{\mathcal{X}}(x-r^D(x))f(x\mid e^\star)\,dx\), i.e. the entrepreneur could choose the same effort \(e^\star\) under \(r^D(\cdot)\) to obtain the same payoff, but instead chooses a higher \(e^D\), so his payoff is higher under \((r^D(\cdot),e^D)\):

$$\int_{\mathcal{X}}(x-r^\star(x))f(x\mid e^\star)\,dx<\int_{\mathcal{X}}(x-r^D(x))f(x\mid e^D)\,dx$$

故创业者在债务合约 \(r^D(\cdot)\) 加努力 \(e^D\) 下也更好。综上，创业者与债权人在债务合约下都更好，与 \((r^\star(\cdot),e^\star)\) 的最优性矛盾。\(\blacksquare\)So the entrepreneur is also better off under the debt contract \(r^D(\cdot)\) with effort \(e^D\). Therefore both entrepreneur and creditor are better off under the debt contract, contradicting the optimality of \((r^\star(\cdot),e^\star)\). \(\blacksquare\)