设定与时序 / Set-up and timing 共 2 期 \(T=1,2\)。自然每期为代理人抽取两个类型 \(\theta_1,\theta_2\)，二者均只私下揭示给代理人；\(\theta_1\) 服从 \((F,f)\)、\(\theta_2\) 服从 \((G,g)\)。买家线性效用 \(u(q_1,q_2,\theta_1,\theta_2)=\theta_1 q_1+\theta_2 q_2\)。\(\theta_2\) 与 \(\theta_1\) 的关系：\(\theta_2=\lambda\theta_1+(1-\lambda)\varepsilon\)（\(\lambda\in[0,1]\)），\(\varepsilon\) 独立服从 \((H,h)\)；\(\lambda=0\) 时 \(\theta_2\) 独立于 \(\theta_1\)，\(\lambda=1\) 时 \(\theta_2\) 给定 \(\theta_1\) 后确定。时序：1. 自然按 \(F(\theta_1)\) 选 \(\theta_1\)；2. 卖家给出机制 \(\{q_1(\hat\theta_1),q_2(\hat\theta_1,\hat\theta_2),t(\hat\theta_1,\hat\theta_2)\}\)，买家选择接受与否，若接受则报告 \(\hat\theta_1\)；3. 自然按 \(G(\theta_2\mid\theta_1)\) 选 \(\theta_2\)；4. 买家报告 \(\hat\theta_2\) 并支付 \(t(\hat\theta_1,\hat\theta_2)\)。全承诺：假设不允许重新谈判。There are 2 periods \(T=1,2\). Nature draws two types \(\theta_1,\theta_2\) for the agent each period, both only privately revealed to the agent; \(\theta_1\sim(F,f)\), \(\theta_2\sim(G,g)\). The buyer has linear utility \(u(q_1,q_2,\theta_1,\theta_2)=\theta_1 q_1+\theta_2 q_2\). \(\theta_2\) is related to \(\theta_1\) by \(\theta_2=\lambda\theta_1+(1-\lambda)\varepsilon\) (\(\lambda\in[0,1]\)), with \(\varepsilon\) independently \(\sim(H,h)\); \(\lambda=0\) makes \(\theta_2\) independent of \(\theta_1\), \(\lambda=1\) makes \(\theta_2\) deterministic given \(\theta_1\). Timing: 1. nature chooses \(\theta_1\) according to \(F(\theta_1)\); 2. the seller offers a mechanism \(\{q_1(\hat\theta_1),q_2(\hat\theta_1,\hat\theta_2),t(\hat\theta_1,\hat\theta_2)\}\), the buyer chooses to accept or not, and if accepts reports \(\hat\theta_1\); 3. nature chooses \(\theta_2\) according to \(G(\theta_2\mid\theta_1)\); 4. the buyer reports \(\hat\theta_2\) and makes the payment \(t(\hat\theta_1,\hat\theta_2)\). Full commitment: assume no re-negotiation is allowed.

23.3 两特例 / Two special cases \(\theta_1\perp\theta_2\)（\(\lambda=0\)）：第一阶段的任何行动不影响后期，故第一期问题的解与单期筛选模型完全相同；按时序合约在 \(\theta_1\) 揭示后、\(\theta_2\) 揭示前签订，作为垄断者的企业可直接把企业卖给消费者、留给消费者零剩余，利润对企业即第一最优。\(\theta_1=\theta_2\)（\(\lambda=1\)）：\(q_1,q_2\) 总是成对出现（因效用线性，连效用函数里也是），仅成本函数除外；因成本 \(c(\cdot)\) 凸，最优解必有 \(q_1(\theta)=q_2(\theta)=q^d(\theta)\)。\(\theta_1\perp\theta_2\) (\(\lambda=0\)): any action in the first stage won't affect the later period, so the first-period problem's solution is exactly the same as the one-period screening model; by the timing the contract is signed after \(\theta_1\) is revealed but before \(\theta_2\), so the firm, being a monopolist, can sell the firm directly to the consumer leaving the consumer zero surplus, and the profit to the firm is first best. \(\theta_1=\theta_2\) (\(\lambda=1\)): \(q_1,q_2\) always appear together (even in the utility function since utility is linear) except for the cost function; since cost \(c(\cdot)\) is convex, the optimal solution must have \(q_1(\theta)=q_2(\theta)=q^d(\theta)\).

注 23.1 / Remark 23.1 回顾 \(\theta_2=\lambda\theta_1+(1-\lambda)\varepsilon\)。若 \(\lambda=1\)，(23.6) 蕴含 \(q_2\) 与 \(q_1\) 完全相同——合理，因此时 \(\theta_2=\theta_1\) 确定，第一期最优产量也是第二期最优。若 \(\lambda=0\)，\(\theta_2\) 独立于 \(\theta_1\)，第二期最优为第一最优 \(q_2^{FB}\)，由 \(\theta_2=c'(q_2^{FB})\) 刻画——也合理，因第二期部分的合约不影响买家对第一期的认知，故垄断卖家有激励在第二期把买家"榨干"。Recall \(\theta_2=\lambda\theta_1+(1-\lambda)\varepsilon\). If \(\lambda=1\), (23.6) implies \(q_2\) is exactly the same as \(q_1\) — which makes sense, since now \(\theta_2=\theta_1\) for sure, so the optimal quantity for the first period is also optimal for the second. If \(\lambda=0\), \(\theta_2\) is independent of \(\theta_1\), so the optimal second-period result is the first best \(q_2^{FB}\), characterized by \(\theta_2=c'(q_2^{FB})\) — which also makes sense, because the second-period part of the contract won't affect the buyer's perception of the first period, so the monopolist seller has the incentive to take the most out of the buyer from the second-period part.