设定 / Set-up 政府是委托人，目标是最大化消费者剩余减转移支付 \(CS(q)-t\)：\(q\) 为电力产量；\(CS(\cdot)\) 是关于 \(q\) 递增且凹的消费者剩余；\(t\) 为消费者向电力企业的转移支付。企业是代理人，成本 \(c(q)=\theta q+K\)，其中 \(\theta\) 是企业私人已知的类型，累积分布 \(F(\theta)\)、支撑 \([\underline\theta,\overline\theta]\)；企业利润 \(\Pi(q,t)=R(q)-c(q)+t=R(q)-\theta q-K+t\)，其中 \(R(q)\) 是企业生产 \(q\) 电力（除转移 \(t\) 外）的收入。政府设计的直接显示机制为 \(\{q(\cdot),t(\cdot)\}\)。The government is the principal, whose target is to maximize consumer surplus minus transfer \(CS(q)-t\): \(q\) is the amount of produced electricity; \(CS(\cdot)\) is the consumer surplus, increasing and concave in \(q\); \(t\) is the transfer payment from consumers to the electricity firm. The firm is the agent, with cost \(c(q)=\theta q+K\), where \(\theta\) is the firm's privately known type with c.d.f. \(F(\theta)\) on support \([\underline\theta,\overline\theta]\); the firm's profit is \(\Pi(q,t)=R(q)-c(q)+t=R(q)-\theta q-K+t\), where \(R(q)\) is the firm's revenue from producing \(q\) amount of electricity other than from the transfer \(t\). The direct revelation mechanism designed by the government is \(\{q(\cdot),t(\cdot)\}\).

证明（必要性）/ Proof (necessity) 可实施显示机制须满足 \(\Pi(\theta)=\max_{\hat\theta}\underbrace{R(q(\hat\theta))-\theta q(\hat\theta)-K+t(\hat\theta)}_{\equiv\Pi(\hat\theta\mid\theta)}=\max_{\hat\theta}\Pi(\hat\theta\mid\theta)\)。条件 2：由包络条件 \(\Pi'(\theta)=\Pi_2(\hat\theta\mid\theta)|_{\hat\theta=\theta}=-q(\hat\theta)|_{\hat\theta=\theta}=-q(\theta)\)，故 \(\Pi(\theta)=\Pi(\underline\theta)+\int_{\underline\theta}^\theta\Pi'(\theta)\,d\theta=\Pi(\underline\theta)-\int_{\underline\theta}^\theta q(\theta)\,d\theta\)。条件 1：由一阶条件 \(\Pi_1(\hat\theta\mid\theta)|_{\hat\theta=\theta}=0\) 与局部二阶条件 \(\Pi_{11}(\hat\theta\mid\theta)|_{\hat\theta=\theta}\le0\)，全微分 \(\Pi_1=0\) 得 \(\underbrace{\Pi_{11}(\theta\mid\theta)}_{\le0}d\theta+\Pi_{12}(\theta\mid\theta)\,d\theta=0\Rightarrow\Pi_{12}(\theta\mid\theta)\ge0\)；而 \(\Pi_{12}(\theta\mid\theta)=\Pi_{21}(\theta\mid\theta)=-q'(\theta)\ge0\Leftrightarrow q'(\theta)\le0\)，证毕。条件 1、2 也充分（同 §20.4.3）。\(\blacksquare\)An implementable revelation mechanism must satisfy \(\Pi(\theta)=\max_{\hat\theta}\underbrace{R(q(\hat\theta))-\theta q(\hat\theta)-K+t(\hat\theta)}_{\equiv\Pi(\hat\theta\mid\theta)}=\max_{\hat\theta}\Pi(\hat\theta\mid\theta)\). Condition 2: by the envelope condition \(\Pi'(\theta)=\Pi_2(\hat\theta\mid\theta)|_{\hat\theta=\theta}=-q(\hat\theta)|_{\hat\theta=\theta}=-q(\theta)\), so \(\Pi(\theta)=\Pi(\underline\theta)+\int_{\underline\theta}^\theta\Pi'(\theta)\,d\theta=\Pi(\underline\theta)-\int_{\underline\theta}^\theta q(\theta)\,d\theta\). Condition 1: by the first-order condition \(\Pi_1(\hat\theta\mid\theta)|_{\hat\theta=\theta}=0\) and the local second-order condition \(\Pi_{11}(\hat\theta\mid\theta)|_{\hat\theta=\theta}\le0\), totally differentiating \(\Pi_1=0\) gives \(\underbrace{\Pi_{11}(\theta\mid\theta)}_{\le0}d\theta+\Pi_{12}(\theta\mid\theta)\,d\theta=0\Rightarrow\Pi_{12}(\theta\mid\theta)\ge0\); and \(\Pi_{12}(\theta\mid\theta)=\Pi_{21}(\theta\mid\theta)=-q'(\theta)\ge0\Leftrightarrow q'(\theta)\le0\), which proves it. Conditions 1, 2 are also sufficient (same as §20.4.3). \(\blacksquare\)

(21.1) 的推导（分部积分）/ Derivation of (21.1) (integration by parts) 先用 \(\Pi(\underline\theta)-\int_{\underline\theta}^\theta q\,ds=\Pi(\overline\theta)+\int_\theta^{\overline\theta}q(s)\,ds\) 改写，再分部积分（取常数使 \(F(\underline\theta)+K=0\)）：First rewrite \(\Pi(\underline\theta)-\int_{\underline\theta}^\theta q\,ds=\Pi(\overline\theta)+\int_\theta^{\overline\theta}q(s)\,ds\), then integrate by parts (choosing a constant so that \(F(\underline\theta)+K=0\)):

$$ > \begin{aligned} > \mathbb{E}_\theta[\Pi(\theta)]&=\Pi(\overline\theta)+\int_{\underline\theta}^{\overline\theta}\left(\int_\theta^{\overline\theta}q(s)\,ds\right)f(\theta)\,d\theta\\[2pt] > &=\Pi(\overline\theta)+\int_{\underline\theta}^{\overline\theta}q(\theta)F(\theta)\,d\theta\\[2pt] > &=\Pi(\overline\theta)+\int_{\underline\theta}^{\overline\theta}q(\theta)\frac{F(\theta)}{f(\theta)}f(\theta)\,d\theta=\Pi(\overline\theta)+\mathbb{E}_\theta\!\left[q(\theta)\frac{F(\theta)}{f(\theta)}\right] > \end{aligned} > $$

保证 \(q'(\cdot)\le0\) / Ensuring \(q'(\cdot)\le0\) 全微分 (21.3)：\(\Lambda_{qq}\,dq+\Lambda_{q\theta}\,d\theta=0\Rightarrow q'(\theta)=-\dfrac{\Lambda_{q\theta}(q,\theta)}{\Lambda_{qq}(q,\theta)}\le0\)。故 \(q'(\cdot)\le0\) 等价于 \(\Lambda_{q\theta}(q,\theta)\le0\)，由施加 \(\dfrac{F(\theta)}{f(\theta)}\) 关于 \(\theta\) 递增（充分非必要）保证。结论：施加 \(\dfrac{F(\theta)}{f(\theta)}\) 递增，则 \(\Lambda_q=0\) 的点态解给出满足 \(q'(\theta)\le0\) 的最优 \(q(\theta)\)。Totally differentiate (21.3): \(\Lambda_{qq}\,dq+\Lambda_{q\theta}\,d\theta=0\Rightarrow q'(\theta)=-\dfrac{\Lambda_{q\theta}(q,\theta)}{\Lambda_{qq}(q,\theta)}\le0\). So \(q'(\cdot)\le0\) is equivalent to \(\Lambda_{q\theta}(q,\theta)\le0\), guaranteed by imposing \(\dfrac{F(\theta)}{f(\theta)}\) increasing in \(\theta\) (sufficient but not necessary). In conclusion, imposing \(\dfrac{F(\theta)}{f(\theta)}\) increasing, the point-wise solution to \(\Lambda_q=0\) gives the optimal \(q(\theta)\) with \(q'(\theta)\le0\).

21.2.5 步骤 5：价格菜单 / Step 5: price menu 找实施直接机制 \(\{q(\theta),\Pi(\theta)\}_{\theta\in\Theta}\) 的 \(P(q)\)。(a) \(P(q)\) 为买 \(q\) 的总价：\(P(q)=\{t(\theta)+R(q(\theta))\text{ if }q=q(\theta)\text{ for any }\theta;\ \infty\text{ otherwise}\}\)。(b) 用 \(q(\cdot)\) 之逆写 \(P(q)=t(q^{-1}(q))+R(q)\)；因 \(q(\cdot)\) 未必严格单调，定义 \(\overline\theta(q)=\sup_\theta\{\theta:q(\theta)=q\}\)，则 \(P(q)=t(\overline\theta(q))+R(q)\) 即所求价格菜单。Find \(P(q)\) implementing the direct mechanism \(\{q(\theta),\Pi(\theta)\}_{\theta\in\Theta}\). (a) \(P(q)\) is the total price paid for \(q\): \(P(q)=\{t(\theta)+R(q(\theta))\text{ if }q=q(\theta)\text{ for any }\theta;\ \infty\text{ otherwise}\}\). (b) Using the inverse of \(q(\cdot)\), \(P(q)=t(q^{-1}(q))+R(q)\); since \(q(\cdot)\) is not necessarily strictly monotone, define \(\overline\theta(q)=\sup_\theta\{\theta:q(\theta)=q\}\), then \(P(q)=t(\overline\theta(q))+R(q)\) is the desired price menu.