时序 / Timing 第一：委托人 \(P\) 给代理人 \(A\) 提供合约 \(w(x):X\to[\underline w,\overline w]\subseteq\mathbb{R}\)，即从产出空间 \(X\) 到工资的映射。离散设定 \(X=\{x_1,\dots,x_n\}\)；连续设定 \(X=[\underline x,\overline x]\subseteq\mathbb{R}\)。第二：代理人接受/拒绝；若接受则选努力水平 \(e\)。离散设定 \(e\in\mathcal{E}=\{e_1,\dots,e_m\}\)；连续设定 \(e\in\mathcal{E}=[0,\overline e]\)。最后：产出 \(x\in X\) 实现，其分布依赖于 \(A\) 的努力 \(e\)，由累积分布 \(F(x\mid e)\) 刻画，代理人获得工资 \(w(x)\)。\(x\) 连续时有密度 \(f(x\mid e)\)，离散时有离散概率 \(\{\phi_1,\dots,\phi_n\}\)。First, principal \(P\) offers agent \(A\) a contract \(w(x):X\to[\underline w,\overline w]\subseteq\mathbb{R}\), a mapping from the output space \(X\) to a wage. Discrete setting \(X=\{x_1,\dots,x_n\}\); continuous setting \(X=[\underline x,\overline x]\subseteq\mathbb{R}\). Second, the agent accepts/rejects; if accepting, chooses an effort level \(e\). Discrete \(e\in\mathcal{E}=\{e_1,\dots,e_m\}\); continuous \(e\in\mathcal{E}=[0,\overline e]\). Finally, output \(x\in X\) is realized, its distribution depending on \(A\)'s effort \(e\), characterized by the c.d.f. \(F(x\mid e)\), and the agent gets wage \(w(x)\). If \(x\) is continuous there is a density \(f(x\mid e)\); if discrete, a set of discrete probabilities \(\{\phi_1,\dots,\phi_n\}\).

注 17.1 / Remark 17.1 离散设定更容易：在有限个点上定义的工资函数 \(w(\cdot)\) 之解的存在性容易证明。而连续设定即便一阶条件能给出最优性，我们也无法真正确定那个涉及不可数无穷多点的解是否存在。The discrete setting is easier: the existence of a solution (the wage function \(w(\cdot)\)) defined over finitely many points can easily be proved. In the continuous setting, even though the f.o.c. can give optimality, we don't really know whether the solution, which involves uncountably infinite points, exists.

双方支付 / Payoffs of both sides 委托人支付 \(V=v(x-w(x))\)，风险中性（\(v''(\cdot)=0\)），故 \(V=x-w(x)\)，是期望效用最大化者。代理人支付 \(U=u(w(x))-\Psi(e)\)，其中 \(u'(\cdot)>0\)、\(u''(\cdot)<0\)、\(\Psi'(\cdot)>0\)、\(\Psi''(\cdot)>0\)。我们不假设代理人风险中性，而在后续讨论中假设其风险厌恶。代理人选行动时会对不同 \(e\) 下的 \(x\) 取期望，并选 \(e\) 最大化净于努力负效用的期望支付——这是其事前理性的最优做法，即便对风险厌恶的代理人也如此。The principal's payoff is \(V=v(x-w(x))\), risk-neutral (\(v''(\cdot)=0\)), so \(V=x-w(x)\), an expected-utility maximizer. The agent's payoff is \(U=u(w(x))-\Psi(e)\), with \(u'(\cdot)>0\), \(u''(\cdot)<0\), \(\Psi'(\cdot)>0\), \(\Psi''(\cdot)>0\). We do not assume the agent is risk-neutral, and assume the agent is risk-averse in the remaining discussion. When choosing an action, the agent takes an expectation over \(x\) with different \(e\)'s and chooses \(e\) to maximize the expected payoff net of the disutility of effort, because this is the best rational thing to do ex ante, even for a risk-averse agent.

两个约束 / The two constraints 个人理性 (IR)：\(\mathbb{E}[u(w(x))\mid e]-\Psi(e)\ge\underline U\)——接受合约的期望支付不低于外部选项 \(\underline U\)，否则代理人不接受。激励相容 (IC)：\(e\in\arg\max_{\tilde e\in\mathcal{E}}\mathbb{E}[u(w(x))\mid\tilde e]-\Psi(\tilde e)\)——委托人希望的努力 \(e\) 也是代理人在给定 \(w(\cdot)\) 下效用最大化的努力，故代理人无激励偏离 \(e\)。因此，对该最大化问题（关于 \(w(\cdot)\) 与 \(e\)）满足 IR、IC 的解，不仅解了委托人的问题，也解了代理人的问题，是整个问题的解。Individual Rationality (IR): \(\mathbb{E}[u(w(x))\mid e]-\Psi(e)\ge\underline U\) — the expected payoff from accepting the contract is no less than the outside option \(\underline U\), else the agent won't accept. Incentive Compatibility (IC): \(e\in\arg\max_{\tilde e\in\mathcal{E}}\mathbb{E}[u(w(x))\mid\tilde e]-\Psi(\tilde e)\) — the effort \(e\) desired by the principal is also the utility-maximizing effort for the agent given \(w(\cdot)\), so the agent has no incentive to deviate from \(e\). Therefore a solution \((w(\cdot),e)\) to this maximization problem subject to IR and IC not only solves the principal's problem but also solves the agent's, so it is a solution to the whole problem.

第一最优 / First best (17.1) 与 (17.2) 共同钉死最优的工资 \(w\) 与努力 \(e\)。我们称之为该问题的第一最优（first best），因为不存在隐藏行动、努力完全可签约，委托人获得最佳可能结果。后续把第一最优努力水平记为 \(e^{fb}\)。(17.1) and (17.2) together pin down the optimal wage \(w\) and effort \(e\). We call this the first best for the problem, because there is no hidden action and effort is perfectly contractable, so the principal obtains the best possible outcome. In later discussion we refer to the first-best effort level as \(e^{fb}\).

注 17.2 与最优合约 / Remark 17.2 and the optimal contract 注 17.2：无不确定性情形与"努力可签约"情形同构，因为努力与产出之间是一一映射；即便努力不可直接观测、不可签约，产出也完美揭示努力水平，故对产出签约等价于对努力签约。记解为 \(x=x^\star\)，则最优工资合约为 \(w(x)=w^\star\)（当 \(x\ge x^\star\)），否则 \(-\infty\)；委托人实现 \(e^\star=G^{-1}(x^\star)\)，即确定性下的第一最优 \(e^{fb}\)，努力被完全揭示、零隐藏行动。Remark 17.2: the no-uncertainty case is isomorphic to the effort-contractable case, since there is a one-to-one mapping between effort and output; even if effort is not directly observable or contractable, output perfectly reveals the effort level, so contracting on output is equivalent to contracting on effort. Denote the solution \(x=x^\star\); then the optimal wage contract is \(w(x)=w^\star\) (when \(x\ge x^\star\)), otherwise \(-\infty\); the principal implements \(e^\star=G^{-1}(x^\star)\), the first-best \(e^{fb}\) under certainty, with effort fully revealed and zero hidden action.

注 17.3 / Remark 17.3 即便存在隐藏行动，也可通过把所有权给到能隐藏行动的代理人来获得第一最优努力——让他们为自己工作。理性的代理人对自己没有隐藏行动，故不完全信息问题被解决。主要启示：股票与期权激励是有正当理由的。Even when hidden action does exist, we can still obtain the first-best effort level by giving the ownership to the agent who can hide their actions — letting them work for themselves. A rational agent has no hidden action to himself, so the incomplete-information problem is solved. The main takeaway: stock and option incentives are justifiable.

\(\lambda,\mu\) 皆为正 / Both \(\lambda\) and \(\mu\) are positive \(\lambda>0\)：\(\lambda\) 是 IR 约束的影子价值；若 \(\lambda=0\)，委托人会想支付更低工资，因 IR 松弛。\(\mu>0\)：\(\mu=0\) 蕴含 \(\frac{1}{u'(w(x))}=\lambda\)，即 \(w(x)\perp x\)（工资与产出无关），而这对理性代理人意味着 \(e=e_L\)；但此处 \(e=e_H\) 最优，故 \(\mu>0\)。\(\lambda>0\): \(\lambda\) is the shadow value of the IR constraint; if \(\lambda=0\), the principal would want a lower wage as IR is slack. \(\mu>0\): \(\mu=0\) implies \(\frac{1}{u'(w(x))}=\lambda\), i.e. \(w(x)\perp x\) (wage independent of output), which for a rational agent implies \(e=e_L\); but here \(e=e_H\) is optimal, so \(\mu>0\).

定义 17.1（单调似然比性质 MLRP）/ Definition 17.1 (MLRP) 若 \(f(x\mid e)\) 关于 \(e\) 可微，则称该分布满足单调似然比性质 (MLRP)，当且仅当对任意 \(e\)，\(\dfrac{f_e(x\mid e)}{f(x\mid e)}\) 随 \(x\) 递增。在两努力情形下，\(f\) 满足 MLRP 当且仅当 \(\dfrac{f_H(x)-f_L(x)}{f_H(x)}\) 随 \(x\) 递增。故当 \(f\) 满足 MLRP 时，\(w(x)\) 随 \(x\) 递增。If \(f(x\mid e)\) is differentiable in \(e\), the distribution satisfies the monotone likelihood ratio property (MLRP) if and only if for any \(e\), \(\dfrac{f_e(x\mid e)}{f(x\mid e)}\) increases in \(x\). In the two-effort case, \(f\) satisfies MLRP iff \(\dfrac{f_H(x)-f_L(x)}{f_H(x)}\) increases in \(x\). So when \(f\) satisfies MLRP, \(w(x)\) increases in \(x\).

注 17.4：MLRP 蕴含 FOSD / Remark 17.4: MLRP implies FOSD 如下面将证明的，MLRP 蕴含一阶随机占优 (FOSD)。设条件分布有累积分布 \(F(x\mid e)\)，FOSD 指对所有 \(x\)，\(F(x\mid e)\ge F(x\mid\tilde e)\) 当且仅当 \(e\le\tilde e\)。换言之，更高的 \(e\) 提供更好的产出分布（更偏向高值）。As will be shown below, MLRP implies First Order Stochastic Dominance (FOSD). Suppose the conditional distribution has c.d.f. \(F(x\mid e)\); FOSD means \(F(x\mid e)\ge F(x\mid\tilde e)\) for all \(x\) if and only if \(e\le\tilde e\). In other words, higher \(e\) provides a better distribution of output (leaning towards higher value).

断言 17.1：MLRP 蕴含 FOSD（证明）/ Claim 17.1: MLRP implies FOSD (Proof) 注意 \(\dfrac{f_e(x\mid e)}{f(x\mid e)}\) 的期望为零，因为Note that \(\dfrac{f_e(x\mid e)}{f(x\mid e)}\) has an expectation of zero, because

$$\int_{\underline x}^{\overline x}\frac{f_e(x\mid e)}{f(x\mid e)}f(x\mid e)\,dx=\int_{\underline x}^{\overline x}f_e(x\mid e)\,dx=\frac{\partial}{\partial e}\int_{\underline x}^{\overline x}f(x\mid e)\,dx=\frac{\partial}{\partial e}1=0$$

故 \(\dfrac{f_e}{f}\) 必先负后正（MLRP 要求其递增）。于是对 \(X=[\underline x,\overline x]\) 及任意 \(x<\overline x\)，有 \(\int_{\underline x}^x\left(\dfrac{f_e}{f}\right)f\,dx<0\Rightarrow\int_{\underline x}^x f_e(x\mid e)\,dx<0\Rightarrow F_e(x\mid e)<0\)，即 \(e\le\tilde e\) 时 \(F(x\mid e)\ge F(x\mid\tilde e)\) 对所有 \(x\) 成立，正是 FOSD。注意一般而言 FOSD 不蕴含 MLRP。\(\blacksquare\)So \(\dfrac{f_e}{f}\) must start negative and end positive (MLRP requires it increasing). Thus for \(X=[\underline x,\overline x]\) and any \(x<\overline x\), \(\int_{\underline x}^x\left(\dfrac{f_e}{f}\right)f\,dx<0\Rightarrow\int_{\underline x}^x f_e(x\mid e)\,dx<0\Rightarrow F_e(x\mid e)<0\), i.e. for \(e\le\tilde e\), \(F(x\mid e)\ge F(x\mid\tilde e)\) for all \(x\), which is exactly FOSD. Note that in general FOSD does not imply MLRP. \(\blacksquare\)

注 17.5 / Remark 17.5 当 \(f\) 满足 MLRP 时，它也满足 FOSD，从而最优工资 \(w(x)\) 随 \(x\) 递增。这很合理：FOSD 意味着更高的 \(e\) 蕴含更高的期望 \(x\)，故委托人对更高的产出 \(x\) 给出更高的工资以鼓励更高的努力 \(e\)。When \(f\) satisfies MLRP, it also satisfies FOSD, so the optimal wage \(w(x)\) increases in \(x\). This makes sense: FOSD means higher \(e\) implies higher expected \(x\), so the principal offers a higher wage for a higher outcome \(x\) to encourage higher effort \(e\).

趋于第一最优 / Approaching the first best 若委托人令 \(k\to-\infty\)，第一项 \(\dfrac{F(k\mid e_L)}{F(k\mid e_H)}\) 爆炸式增大，故第二项必须趋于零，即 \(u(w_1)\to\Delta\)。这意味着委托人通过令 \(k\to-\infty\) 可任意逼近第一最优结果，也就说明最初那个内部解并不是真正的解（最优合约不存在）。If the principal sets \(k\to-\infty\), the first term \(\dfrac{F(k\mid e_L)}{F(k\mid e_H)}\) blows up, so the second term must go to zero, i.e. \(u(w_1)\to\Delta\). This means the principal can get arbitrarily close to the first-best outcome by setting \(k\to-\infty\), which means the initial interior solution was not a solution (the optimal contract does not exist).

17.4.4 信息原理 / Information Principle 定义 17.2：随机变量 \(x\) 是 \(y\) 关于 \(e\) 的充分统计量，当且仅当存在函数 \(g,h\) 使得 \(f(x,y\mid e)=g(x\mid e)\,h(y\mid x)\)，对所有 \(x\in X\)、\(y\in Y\)、\(e\in\mathcal{E}\) 成立。注 17.6：信息原理是说，一个额外信号 \(y\) 对激励有价值，当且仅当它携带了关于代理人努力 \(e\) 的、不包含在 \(x\) 中的额外信息（即 \(x\) 不是 \(y\) 关于 \(e\) 的充分统计量）。Definition 17.2: a random variable \(x\) is a sufficient statistic for \(y\) with respect to \(e\) if and only if there exist functions \(g,h\) such that \(f(x,y\mid e)=g(x\mid e)\,h(y\mid x)\), for all \(x\in X\), \(y\in Y\), \(e\in\mathcal{E}\). Remark 17.6: the information principle says that an additional signal \(y\) is valuable for incentives if and only if it carries additional information about the agent's effort \(e\) not contained in \(x\) (i.e. \(x\) is not sufficient for \(y\) with respect to \(e\)).

IC 难以验证 / IC is hard to check IC 约束 (17.8) 非常难以核验，因为现在有连续的 \(e\) 要代入 (17.6) 与 IR (17.7) 求各 \(e\) 下的解 \(w(\cdot)\)，再把所有这些解代回 IC 检验是否存在满足 IC 的 \(e\)——这是不可能完成的。The IC constraint (17.8) is very difficult to check, because now there is a continuum of \(e\) to plug into program (17.6) and IR (17.7) for a solution \(w(\cdot)\) w.r.t. each \(e\), and then plug all those solutions back into IC to obtain the \(e\) such that IC holds, which is impossible to do.

FOA 只是必要而非充分 / FOA is only necessary, not sufficient 解 FOA 问题的解只是原问题的必要条件而非充分条件。换言之，FOA 这一松弛可能生成对原问题而言虚假的解。见下面图 28 的说明。A solution that solves the FOA problem is only necessary but not sufficient to solve the original problem. In other words, the FOA relaxation might generate a false solution to the original problem. See the illustration of Figure 28 below.

图 28 / Figure 28（IC 一阶条件可能给出错误解，已转述 / IC-f.o.c. could give wrong solution, paraphrased） 横轴为委托人的选择、纵轴为代理人的选择。黑色曲线描述基于 IC 一阶条件（IC 的松弛版本）委托人与代理人共同选出的解；红线是原始 IC 约束给出的真实选择集。若用 IC 一阶条件求解，很可能得到点 \(p\) 处的解——它显然不是原问题的有效解，因为点 \(p\) 违反 IC 约束。不过我们仍可解 FOA 问题以至少得到一个结果，下一步是把解 \((w(\cdot),e)\) 代入原始 IC (17.8) 检验是否满足：若满足，则我们足够幸运地得到了有效解；若不满足，则 FOA 问题的解失败。The horizontal axis is the principal's choices, the vertical axis the agent's choices. The black curve describes the solution that both principal and agent select based on the IC-f.o.c. (the relaxed version of IC); the red line is the actual choice set given by the original IC constraint. If we solve with the IC-f.o.c., we likely get the solution at point \(p\) — clearly not a valid solution to the original problem since \(p\) violates IC. We can still solve the FOA problem to get at least a result; the next step is to plug the solution \((w(\cdot),e)\) into the original IC (17.8) to check: if satisfied, we are lucky enough to have a valid solution; if not, the solution to the FOA problem fails.

命题 17.1 / Proposition 17.1 \(\mu\) 与 \(\lambda\) 皆为正。\(\mu\) and \(\lambda\) are both positive.

证明 / Proof 首先需 \(\mu\ne0\)：若 \(\mu=0\)，则 IC 一阶条件松弛，代理人会选最高或最低努力，是不有趣的角点解；对内部解，\(\mu\ne0\)。整理 (17.12)：First, we need \(\mu\ne0\): if \(\mu=0\), the IC-f.o.c. is slack and the agent chooses the highest or lowest effort, an uninteresting corner solution; for an interior solution, \(\mu\ne0\). Rearrange (17.12):

$$\frac{1}{u'(w(x))}=\lambda+\mu\left(\frac{f_e(x\mid e)}{f(x\mid e)}\right) \;\Rightarrow\; \left(\frac{1}{u'(w(x))}-\lambda\right)\frac{f(x\mid e)}{\mu}=f_e(x\mid e) \tag{17.13}$$

又知 \(\int_{\underline x}^{\overline x}f_e(x\mid e)\,dx=\dfrac{\partial}{\partial e}\int f(x\mid e)\,dx=\dfrac{\partial}{\partial e}1=0\)，代入 (17.13)：We also know \(\int_{\underline x}^{\overline x}f_e(x\mid e)\,dx=\dfrac{\partial}{\partial e}\int f(x\mid e)\,dx=\dfrac{\partial}{\partial e}1=0\); plug in (17.13):

$$\int_{\underline x}^{\overline x}\left(\frac{1}{u'(w(x))}-\lambda\right)\frac{f(x\mid e)}{\mu}\,dx=0 \;\Rightarrow\; \int_{\underline x}^{\overline x}\frac{1}{u'(w(x))}f(x\mid e)\,dx=\lambda \;\Rightarrow\; \mathbb{E}\!\left[\frac{1}{u'(w(x))}\,\Big|\,e\right]=\lambda$$

因 \(u'(w(x))>0\) 恒成立，故 \(\lambda>0\)。再考虑代理人一阶条件 \(\int u(w(x))f_e(x\mid e)\,dx=\Psi'(e)\)，代入 (17.13) 并用 \(\lambda=\mathbb{E}[1/u'(w(x))\mid e]\)，得Since \(u'(w(x))>0\) always, \(\lambda>0\). Next consider the agent's f.o.c. \(\int u(w(x))f_e(x\mid e)\,dx=\Psi'(e)\); plug in (17.13) and use \(\lambda=\mathbb{E}[1/u'(w(x))\mid e]\) to get

$$\text{Cov}\!\left(u(w(x)),\,\frac{1}{u'(w(x))}\,\Big|\,e\right)=\mu\Psi'(e) \tag{17.14}$$

因更高的 \(u(w(x))\) 蕴含更高的 \(w(x)\) 从而更高的 \(\dfrac{1}{u'(w(x))}\)，故 (17.14) 左端严格为正；又 \(\Psi(\cdot)\) 凸、\(\Psi'(e)>0\)，二者共同蕴含 \(\mu>0\)。\(\blacksquare\)Since higher \(u(w(x))\) implies higher \(w(x)\) and thus higher \(\dfrac{1}{u'(w(x))}\), the LHS of (17.14) is strictly positive; and \(\Psi(\cdot)\) is convex with \(\Psi'(e)>0\), which together imply \(\mu>0\). \(\blacksquare\)

推论 / Corollary 命题 17.1 蕴含：在 FOA 提供有效解的前提下，\(w(x)\) 随 \(x\) 递增。Proposition 17.1 implies that \(w(x)\) increases in \(x\), given that the FOA problem provides a valid solution.

定义 17.3（凸分布函数条件 CDFC）/ Definition 17.3 (Convex distribution function condition, CDFC) 称分布满足凸分布函数条件 (CDFC)，当且仅当其累积分布满足 \(F_{ee}(x\mid e)\ge0\)。A distribution satisfies the convex distribution function condition (CDFC) if and only if its c.d.f. satisfies \(F_{ee}(x\mid e)\ge0\).

命题 17.2 / Proposition 17.2 若分布同时满足 MLRP 与 CDFC，则 IC 约束是凹的，从而 FOA 问题的解也解原问题。If a distribution satisfies both MLRP and CDFC, then the IC constraint is concave, and thus the solution to the FOA problem also solves the original problem.

证明 / Proof 考虑代理人的问题，即在给定 \(w(\cdot)\) 下对 \(e\) 最大化期望效用 \(\max_e\int_{\underline x}^{\overline x}u(w(x))f(x\mid e)\,dx-\Psi(e)\) (17.15)。分部积分改写其目标函数（用 \(F(\overline x\mid e)=1\)、\(F(\underline x\mid e)=0\)）：Consider the agent's problem, maximizing expected utility over \(e\) at given \(w(\cdot)\), \(\max_e\int_{\underline x}^{\overline x}u(w(x))f(x\mid e)\,dx-\Psi(e)\) (17.15). Integrate by parts to rewrite the objective (using \(F(\overline x\mid e)=1\), \(F(\underline x\mid e)=0\)):

$$\int_{\underline x}^{\overline x}u(w(x))f(x\mid e)\,dx-\Psi(e)=u(w(\overline x))-\int_{\underline x}^{\overline x}u'(w(x))w'(x)F(x\mid e)\,dx-\Psi(e) \tag{17.16}$$

因 \(F\) 满足 MLRP，知 \(w'(x)>0\)，又 \(u'(w(x))>0\)。对 (17.16) 关于 \(e\) 二次求导：Since \(F\) satisfies MLRP, \(w'(x)>0\), and \(u'(w(x))>0\). Twice differentiate (17.16) w.r.t. \(e\):

$$-\underbrace{\int_{\underline x}^{\overline x}\underbrace{u'(w(x))w'(x)}_{\ge0}\,\underbrace{F_{ee}(x\mid e)}_{\ge0:\text{ CDFC}}dx}_{\ge0}-\underbrace{\Psi''(e)}_{>0}<0 \tag{17.17}$$

这意味着代理人问题的目标函数 (17.15) 关于 \(e\) 严格凹，故一阶条件是 IC (17.8) 最优解的充分必要条件，即 FOA 问题的解也解原问题。\(\blacksquare\)This implies the objective (17.15) in the agent's problem is strictly concave in \(e\), so the f.o.c. is a sufficient and necessary condition for the optimal solution in IC (17.8), i.e. the solution to FOA also solves the original problem. \(\blacksquare\)

注 17.7、注 17.8 / Remarks 17.7, 17.8 注 17.7：CDFC 是很强的假设，可被放松。可设想一种情形：即便没有 CDFC，只要 \(-\Psi''(e)\) 足够负、使 (17.17) 左端整体为负，仍可保持 FOA 问题与原问题的等价。注 17.8：一些常用分布并不满足 CDFC，例如以 \(e\) 为均值的正态分布 \(f(x\mid e)=\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12(\frac{x-e}{\sigma})^2}\)；可计算其 \(F_{ee}(x\mid e)=\int_{-\infty}^x\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12(\frac{x-e}{\sigma})^2}\dfrac{(x-e)^2-\sigma^2}{\sigma^4}\,dx\)，它对任意 \(x\) 并不总 \(\ge0\)。Remark 17.7: CDFC is a very strong assumption that can be relaxed. We can imagine a case where, even without CDFC, as long as \(-\Psi''(e)\) is negative enough to make the whole LHS of (17.17) negative, we still have the equivalence of the FOA and the original problem. Remark 17.8: some commonly used distributions don't satisfy CDFC, such as the normal with \(e\) in the mean \(f(x\mid e)=\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12(\frac{x-e}{\sigma})^2}\); one computes \(F_{ee}(x\mid e)=\int_{-\infty}^x\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12(\frac{x-e}{\sigma})^2}\dfrac{(x-e)^2-\sigma^2}{\sigma^4}\,dx\), which is not always \(\ge0\) for any \(x\).