本章导读 上一章 [[bilateral-trade-impossibility]] 证明双边私人信息下事后有效的交易机制无法兼顾激励相容 (IC) 与个人理性 (IR)。本章在放松事后有效后讨论两个第二好问题，沿用 §28 的全部记号。§29.1 社会福利最大化问题：在 IC + IR 约束下最大化期望社会剩余 \(\mathbb{E}[\varphi(\theta_B-\theta_S)]\)。由定理 28.3，IC+IR 可行性等价于虚拟剩余约束 (29.3)；由不可能性定理该约束在最优处必然取等(29.6)。拉格朗日法给出阈值解 \(\varphi_\lambda\) (29.7)：只有当 \(\theta_B-\theta_S\ge\frac{\lambda}{1+\lambda}(\frac{1-F_B}{f_B}+\frac{F_S}{f_S})\) 时才交易，\(\lambda>0\) 由约束取等定出；注 29.1 交易的最小价差严格为正（该交易的有时不交易），注 29.2 解只由 \(\varphi\) 刻画（\(t\) 决定剩余分配、不唯一）；例 29.1 均匀 $[0,1]$ 给 \(\lambda=\frac12\)、价差 \(\ge\frac14\) 才交易（图 30）。§29.2 平台利润最大化问题：平台向买家收 \(t_B\)、付卖家 \(t_S\)，最大化 \(\mathbb{E}[t_B-t_S]\) (29.9)。由会计恒等式利润 = 总剩余 − 双方剩余 (29.10/注 29.3)；代入 IC 刻画与命题 28.1，并令边界效用为零满足 IR，得阈值解 \(\varphi\) (29.14)：\(\theta_B-\theta_S\ge\frac{1-F_B}{f_B}+\frac{F_S}{f_S}\) 才交易——等价于在福利解中令 \(\lambda\to\infty\)，交易价差门槛更大；例 29.1 给价差 \(\ge\frac12\)（图 31）。图 30、31 已转述。

约束必然取等 / The constraint must bind 由不可能性定理 28.4，对有效且 IC、IR 的机制 \(\{\varphi(\cdot,\cdot),t(\cdot,\cdot)\}\)，(29.1) 与 (29.3) 的解必使 (29.3) 取等。反证：若最优 \(\hat\varphi\) 使 (29.3) 严格大于零（\(\mathbb{E}[\hat\varphi(\dots)]>0\)，即 (29.4)），则由连续性必存在更靠近事后有效 \(\varphi^\star(\cdot,\cdot)=\mathbf 1\{\theta_B\ge\theta_S\}\) 的机制 \(\tilde\varphi\)（在更多点 \((\theta_B,\theta_S)\) 上 \(\tilde\varphi=\varphi^\star\)）仍满足 (29.4)、仍 IC 且 IR；但 \(\tilde\varphi\) 更靠近事后有效机制就产生更高社会剩余，与 \(\hat\varphi\) 解 (29.1)&(29.3) 矛盾。故 (29.3) 在最优处取等。By the impossibility theorem 28.4, for an efficient IC and IR mechanism \(\{\varphi(\cdot,\cdot),t(\cdot,\cdot)\}\), the solution to (29.1) and (29.3) must make (29.3) bind. By contradiction: if the optimal \(\hat\varphi\) made (29.3) strictly positive (\(\mathbb{E}[\hat\varphi(\dots)]>0\), i.e. (29.4)), then by continuity there exists a mechanism \(\tilde\varphi\) closer to the ex-post efficient \(\varphi^\star(\cdot,\cdot)=\mathbf 1\{\theta_B\ge\theta_S\}\) (with \(\tilde\varphi=\varphi^\star\) at more points \((\theta_B,\theta_S)\)) that still satisfies (29.4) and is still IC and IR; but being closer to the ex-post efficient mechanism, \(\tilde\varphi\) generates higher social surplus, contradicting that \(\hat\varphi\) solves (29.1)&(29.3). Hence (29.3) binds at the optimum.

注 29.1、注 29.2 / Remarks 29.1, 29.2 注 29.1：交易发生所需的最小价差 \(\theta_B-\theta_S\) 严格为正：\(0<\theta_B-\theta_S<\frac{\lambda}{1+\lambda}(\frac{1-F_B(\theta_B)}{f_B(\theta_B)}+\frac{F_S(\theta_S)}{f_S(\theta_S)})\) 这一带状区域内该交易却不交易——可行（IC+IR）的福利最大化机制有时不能实现本应发生的交易。注 29.2：解仅由 \(\varphi(\cdot,\cdot)\) 刻画，因 \(t(\cdot,\cdot)\) 只决定剩余在买卖双方间的分配份额，并不唯一。Remark 29.1: the minimum gap \(\theta_B-\theta_S\) required for trade is strictly positive: in the band \(0<\theta_B-\theta_S<\frac{\lambda}{1+\lambda}(\frac{1-F_B(\theta_B)}{f_B(\theta_B)}+\frac{F_S(\theta_S)}{f_S(\theta_S)})\) trade should happen but doesn't — the feasible (IC+IR) welfare-maximizing mechanism sometimes fails to realize trades that should occur. Remark 29.2: the solution is characterized only by \(\varphi(\cdot,\cdot)\), since \(t(\cdot,\cdot)\) merely determines the share of surplus distributed between the two sides and is not unique.

例 29.1（均匀分布 / Uniform distribution） 设 \(\theta_B,\theta_S\) 独立同服从 $[0,1]$ 上的均匀分布。则 \(\frac{1-F_B(\theta_B)}{f_B(\theta_B)}=1-\theta_B\)、\(\frac{F_S(\theta_S)}{f_S(\theta_S)}=\theta_S\)，故 (29.7) 的门槛 \(\frac{\lambda}{1+\lambda}(1-\theta_B+\theta_S)\)。化简交易条件 \(\theta_B-\theta_S\ge\frac{\lambda}{1+\lambda}(1-(\theta_B-\theta_S))\) 得Let \(\theta_B,\theta_S\) be i.i.d. uniform on $[0,1]\(. Then \)\frac{1-F_B(\theta_B)}{f_B(\theta_B)}=1-\theta_B$ and \(\frac{F_S(\theta_S)}{f_S(\theta_S)}=\theta_S\), so the threshold in (29.7) is \(\frac{\lambda}{1+\lambda}(1-\theta_B+\theta_S)\). Simplifying the trade condition \(\theta_B-\theta_S\ge\frac{\lambda}{1+\lambda}(1-(\theta_B-\theta_S))\) gives

$$\varphi_\lambda(\theta_B,\theta_S)=\begin{cases}1 & \text{if } (\theta_B-\theta_S)\ge\frac{\lambda}{1+2\lambda}\\[4pt]0 & \text{if } (\theta_B-\theta_S)<\frac{\lambda}{1+2\lambda}\end{cases}\tag{29.8}$$

把 (29.8) 代回 (29.6) 求 \(\lambda\)：经多步积分化简（\(\int_{\frac{\lambda}{1+2\lambda}}^1\dots d\theta_B=0\)）最终得 \(\frac{\lambda}{1+2\lambda}=\frac14\)，即 \(\lambda=\frac12\)。于是本例中只有当 \(\theta_B-\theta_S\ge\frac14\) 时才交易。Substituting (29.8) into (29.6) to solve for \(\lambda\): after several integration steps (\(\int_{\frac{\lambda}{1+2\lambda}}^1\dots d\theta_B=0\)) one obtains \(\frac{\lambda}{1+2\lambda}=\frac14\), i.e. \(\lambda=\frac12\). So in this example trade happens only when \(\theta_B-\theta_S\ge\frac14\).

图 30（已转述 / Figure 30, paraphrased） 在 \((\theta_B,\theta_S)\in[0,1]^2\) 的单位正方形中画两条平行线：对角线 \(\theta_S=\theta_B\) 与下移直线 \(\theta_S=\theta_B-\frac14\)。区域 \(A\)+区域 \(B\)（对角线下方整片 \(\{\theta_S\le\theta_B\}\)）是事后有效机制下该交易的点集；区域 \(B\) 单独（\(\theta_S\le\theta_B-\frac14\)）是福利最大化机制下实际交易的点集。两者之差（\(\theta_B-\frac14<\theta_S\le\theta_B\) 的窄带 \(A\)）即被牺牲掉的本应发生的交易。In the unit square \((\theta_B,\theta_S)\in[0,1]^2\), draw two parallel lines: the diagonal \(\theta_S=\theta_B\) and the shifted line \(\theta_S=\theta_B-\frac14\). Region \(A\) + region \(B\) (the whole area below the diagonal, \(\{\theta_S\le\theta_B\}\)) is the set where trade should happen under the ex-post efficient mechanism; region \(B\) alone (\(\theta_S\le\theta_B-\frac14\)) is the set where trade actually happens under the welfare-maximizing mechanism. The difference (the narrow band \(A\) with \(\theta_B-\frac14<\theta_S\le\theta_B\)) is the socially desirable trade that is sacrificed.

注 29.3（会计恒等式 / Remark 29.3: accounting identity） \(\mathbb{E}[t_B(\theta_B)-t_S(\theta_S)]=\mathbb{E}[\varphi(\theta_B,\theta_S)(\theta_B-\theta_S)-U_B(\theta_B)-U_S(\theta_S)]\) 恒成立：平台利润等于全部产生的剩余减去买卖双方各自的剩余，是一个会计恒等式。\(\mathbb{E}[t_B(\theta_B)-t_S(\theta_S)]=\mathbb{E}[\varphi(\theta_B,\theta_S)(\theta_B-\theta_S)-U_B(\theta_B)-U_S(\theta_S)]\) always holds: the platform's profit equals the total generated surplus minus the surplus of both seller and buyer — an accounting identity.

平台解 = 福利解令 \(\lambda\to\infty\) / Platform solution = welfare solution with \(\lambda\to\infty\) 平台利润最大化解 (29.14) 等价于在福利最大化解 (29.7) 中令 \(\lambda=\infty\)（此时 \(\frac{\lambda}{1+\lambda}=1\)）。故平台利润最大化机制要求更大的价差 \(\theta_B-\theta_S\) 才交易——比福利最大化机制牺牲掉更多本应发生的交易。直觉：平台为攫取自身利润压低买家、抬高卖家的有效报价，把交易门槛进一步抬高。The platform-profit-maximizing solution (29.14) is equivalent to setting \(\lambda=\infty\) in the welfare-maximizing solution (29.7) (so that \(\frac{\lambda}{1+\lambda}=1\)). Hence the platform mechanism requires an even larger gap \(\theta_B-\theta_S\) before trade occurs — sacrificing even more socially desirable trade than the welfare mechanism. Intuition: to extract its own profit the platform marks the buyer down and the seller up, pushing the trade threshold still higher.

例 29.1（续，均匀分布 / continued, uniform） 仍设 \(\theta_B,\theta_S\) 独立同服从 $[0,1]$。则 \(\frac{1-F_B(\theta_B)}{f_B(\theta_B)}+\frac{F_S(\theta_S)}{f_S(\theta_S)}=(1-\theta_B)+\theta_S=1-(\theta_B-\theta_S)\)。由 (29.14)，交易条件 \(\theta_B-\theta_S\ge1-(\theta_B-\theta_S)\) 即 \(\theta_B-\theta_S\ge\frac12\)：Again let \(\theta_B,\theta_S\) be i.i.d. uniform on $[0,1]\(. Then \)\frac{1-F_B(\theta_B)}{f_B(\theta_B)}+\frac{F_S(\theta_S)}{f_S(\theta_S)}=(1-\theta_B)+\theta_S=1-(\theta_B-\theta_S)\(. By (29.14), the trade condition \)\theta_B-\theta_S\ge1-(\theta_B-\theta_S)$ becomes \(\theta_B-\theta_S\ge\frac12\):

$$\varphi(\theta_B,\theta_S)=\begin{cases}1 & \text{if } (\theta_B-\theta_S)\ge\frac12\\[4pt]0 & \text{if } (\theta_B-\theta_S)<\frac12\end{cases}$$

图 31（已转述 / Figure 31, paraphrased） 在单位正方形中画三条平行线：对角线 \(\theta_S=\theta_B\)、\(\theta_S=\theta_B-\frac14\)、\(\theta_S=\theta_B-\frac12\)。区域 \(A\)+\(B\)+\(C\)（整片 \(\{\theta_S\le\theta_B\}\)）是事后有效机制下该交易的点集；区域 \(B\)+\(C\)（\(\theta_S\le\theta_B-\frac14\)）是福利最大化机制下实际交易的点集；区域 \(C\) 单独（\(\theta_S\le\theta_B-\frac12\)）是平台利润最大化机制下实际交易的点集。门槛随"有效 → 福利最大 → 平台利润最大"逐级抬高，交易越来越少。In the unit square draw three parallel lines: the diagonal \(\theta_S=\theta_B\), \(\theta_S=\theta_B-\frac14\), and \(\theta_S=\theta_B-\frac12\). Region \(A\)+\(B\)+\(C\) (the whole area \(\{\theta_S\le\theta_B\}\)) is the set where trade should happen under the ex-post efficient mechanism; region \(B\)+\(C\) (\(\theta_S\le\theta_B-\frac14\)) is where trade actually happens under the welfare-maximizing mechanism; region \(C\) alone (\(\theta_S\le\theta_B-\frac12\)) is where trade actually happens under the platform-profit-maximizing mechanism. The threshold rises step by step as "efficient → welfare-max → platform-profit-max", with less and less trade.