注记 7.1 / Remark 7.1 注意 \(Y^j\) 的定义域是 \(\mathbb{R}^n\) 而非 \(\mathbb{R}_+^n\)，即 \(Y^j\) 中的向量可有负分量。约定：若 \(y_k^j<0\)，则 \(|y_k^j|\) 为厂商 \(j\) 用作投入的商品 \(k\) 的数量；若 \(y_k^j>0\)，则 \(y_k^j\) 为厂商 \(j\) 作为产出生产的商品 \(k\) 的数量。Note the domain of \(Y^j\) is \(\mathbb{R}^n\) instead of \(\mathbb{R}_+^n\), which means vectors in \(Y^j\) can have negative elements. Our convention: if \(y_k^j<0\), then \(|y_k^j|\) is the amount of good \(k\) used by firm \(j\) as input; if \(y_k^j>0\), then \(y_k^j\) is the amount of good \(k\) produced by firm \(j\) as output.

Assumption 2 & Definition 7.1（strongly convex） 对 \(\forall j\in\mathcal{J}\)：1.（非空）\(\mathbf{0}\in Y^j\)；2.（紧性）\(Y^j\) 闭且有界，故紧；3.（规模报酬递减）\(Y^j\) 强凸。
Def 7.1（强凸）：\(Y^j\) 强凸，若对 \(\forall\mathbf{y}^1,\mathbf{y}^2\in Y^j\)（\(\mathbf{y}^1\ne\mathbf{y}^2\)）及 \(\forall t\in(0,1)\)，\(\exists\bar{\mathbf{y}}\in Y^j\) 使 \(\bar{\mathbf{y}}\gneq(1-t)\mathbf{y}^1+t\mathbf{y}^2\)（即 \(\bar{\mathbf{y}}\) 每个分量不小于 \((1-t)\mathbf{y}^1+t\mathbf{y}^2\) 且至少一处严格大）。For \(\forall j\in\mathcal{J}\): 1. (non-empty) \(\mathbf{0}\in Y^j\); 2. (compactness) \(Y^j\) is closed and bounded, thus compact; 3. (decreasing return to scale) \(Y^j\) is strongly convex.
Def 7.1 (strongly convex): \(Y^j\) is strongly convex if for \(\forall\mathbf{y}^1,\mathbf{y}^2\in Y^j\) (\(\mathbf{y}^1\ne\mathbf{y}^2\)) and \(\forall t\in(0,1)\), \(\exists\bar{\mathbf{y}}\in Y^j\) s.t. \(\bar{\mathbf{y}}\gneq(1-t)\mathbf{y}^1+t\mathbf{y}^2\) (every component in \(\bar{\mathbf{y}}\) no smaller than \((1-t)\mathbf{y}^1+t\mathbf{y}^2\) with at least one strict inequality).

注记 7.2–7.4 / Remarks 7.2–7.4 7.2：条件 3（规模报酬递减）排除了规模报酬不变技术，转而假设规模报酬递减。7.3：由强凸定义，不保证 \((1-t)\mathbf{y}^1+t\mathbf{y}^2\in Y^j\)，\(Y^j\) 甚至可能不凸；此处只需强凸，\(Y^j\) 的凸性无关紧要。7.4：还有一些关于 \(Y^j\) 的合理假设（如 \(y_k^j>0\) 不能对 \(\forall k\) 成立）未被引入，因证明不需要它们。7.2: condition 3 (decreasing return to scale) rules out the constant return to scale technology and instead assumes decreasing returns to scale. 7.3: by definition of strongly convex, it is not guaranteed that \((1-t)\mathbf{y}^1+t\mathbf{y}^2\in Y^j\), so \(Y^j\) might not even be convex; here we only need strongly convex, so the convexity of \(Y^j\) is irrelevant. 7.4: there are other reasonable assumptions about \(Y^j\) (e.g. \(y_k^j>0\) cannot hold for \(\forall k\)) not imposed since they are not used for the proofs.

Definition 7.2（maximized profit） 给定 \(\mathbf{p}\in\mathbb{R}_{++}^n\)，每家厂商 \(j\) 求解 \(\Pi^j(\mathbf{p})\equiv\max_{\mathbf{y}^j\in Y^j}\mathbf{p}\cdot\mathbf{y}^j=\sum_{k=1}^n p_k y_k^j\)，\(\Pi^j(\mathbf{p})\) 为厂商 \(j\) 的最大化利润。注意 \(\mathbf{p}\cdot\mathbf{y}^j=\sum_k p_k y_k^j\) 恰是利润（收入减成本），因 \(y_k^j\) 可取正负分别表示产出与投入。Given \(\mathbf{p}\in\mathbb{R}_{++}^n\), each firm \(j\) solves \(\Pi^j(\mathbf{p})\equiv\max_{\mathbf{y}^j\in Y^j}\mathbf{p}\cdot\mathbf{y}^j=\sum_{k=1}^n p_k y_k^j\), where \(\Pi^j(\mathbf{p})\) is firm \(j\)'s maximized profit. Note \(\mathbf{p}\cdot\mathbf{y}^j=\sum_k p_k y_k^j\) is exactly profit (revenue minus cost) since \(y_k^j\) can take positive and negative values for outputs and inputs.

Theorem 7.1 在 Assumption 2 下，对 \(\forall\mathbf{p}\in\mathbb{R}_{++}^n\)，厂商 \(j\) 的利润最大化问题 \(\mathbf{y}^j(\mathbf{p})=\arg\max_{\mathbf{y}^j\in Y^j}\mathbf{p}\cdot\mathbf{y}^j\) 的解存在且唯一；\(\mathbf{y}^j(\mathbf{p})\) 关于 \(\mathbf{p}\) 在 \(\mathbb{R}_{++}^n\) 上连续；\(\Pi^j(\mathbf{p})=\mathbf{p}\cdot\mathbf{y}^j(\mathbf{p})\) 良定义（存在唯一）且在 \(\mathbb{R}_+^n\) 上连续。Under Assumption 2, for \(\forall\mathbf{p}\in\mathbb{R}_{++}^n\), the solution to firm \(j\)'s profit maximization problem \(\mathbf{y}^j(\mathbf{p})=\arg\max_{\mathbf{y}^j\in Y^j}\mathbf{p}\cdot\mathbf{y}^j\) exists and is unique; \(\mathbf{y}^j(\mathbf{p})\) is continuous in \(\mathbf{p}\) on \(\mathbb{R}_{++}^n\); and \(\Pi^j(\mathbf{p})=\mathbf{p}\cdot\mathbf{y}^j(\mathbf{p})\) is well-defined (existence and uniqueness) and continuous on \(\mathbb{R}_+^n\).

证明 / Proof (Theorem 7.1)

Theorem 7.2 在 Assumption 1 与 Assumption 2 下，消费者 \(i\) 的效用最大化问题 \(\max_{\mathbf{x}^i}u(\mathbf{x}^i)\) s.t. \(\mathbf{p}\cdot\mathbf{x}^i\le\mathbf{p}\cdot\mathbf{e}^i+\sum_{j\in\mathcal{J}}\theta^{ij}\cdot\Pi^j(\mathbf{p})\) 的解对 \(\forall\mathbf{p}\in\mathbb{R}_{++}^n\) 存在且唯一，记为 \(\mathbf{x}^i(\mathbf{p},m^i(\mathbf{p}))\)；它关于 \(\mathbf{p}\) 在 \(\mathbb{R}_{++}^n\) 上连续，\(m^i(\mathbf{p})\) 在 \(\mathbb{R}_+^n\) 上连续。Under Assumption 1 and Assumption 2, the solution to consumer \(i\)'s utility maximization problem \(\max_{\mathbf{x}^i}u(\mathbf{x}^i)\) s.t. \(\mathbf{p}\cdot\mathbf{x}^i\le\mathbf{p}\cdot\mathbf{e}^i+\sum_{j\in\mathcal{J}}\theta^{ij}\cdot\Pi^j(\mathbf{p})\) exists and is unique for \(\forall\mathbf{p}\in\mathbb{R}_{++}^n\), denoted \(\mathbf{x}^i(\mathbf{p},m^i(\mathbf{p}))\); it is continuous in \(\mathbf{p}\) on \(\mathbb{R}_{++}^n\) and \(m^i(\mathbf{p})\) is continuous on \(\mathbb{R}_+^n\).

证明 / Proof (Theorem 7.2)

Definition 7.3（Walrasian equilibrium for production economy） 称 \(\mathbf{p}^\star\in\mathbb{R}_+^n\) 为 \(\mathcal{E}=(u^i,\mathbf{e}^i,\theta^{ij},Y^j)_{i\in\mathcal{I},j\in\mathcal{J}}\) 的瓦尔拉斯均衡，当且仅当 \(\exists\hat{\mathbf{x}}^1,\dots,\hat{\mathbf{x}}^I\in\mathbb{R}_+^n\)、\(\exists\hat{\mathbf{y}}^1\in Y^1,\dots,\hat{\mathbf{y}}^J\in Y^J\) 使：(1) 消费者效用最大化：\(\hat{\mathbf{x}}^i\) 解 \(\max_{\mathbf{x}^i}u(\mathbf{x}^i)\) s.t. \(\mathbf{p}\cdot\mathbf{x}^i\le m^i(\mathbf{p})\)（\(\forall i\)）；(2) 厂商利润最大化：\(\hat{\mathbf{y}}^j\) 解 \(\max_{\mathbf{y}^j\in Y^j}\mathbf{p}\cdot\mathbf{y}^j\)（\(\forall j\)）；(3) 市场出清（可行性）：\(\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j\)。称 \((\hat{\mathbf{x}},\hat{\mathbf{y}})\) 为 WEA。\(\mathbf{p}^\star\in\mathbb{R}_+^n\) is a Walrasian equilibrium of \(\mathcal{E}=(u^i,\mathbf{e}^i,\theta^{ij},Y^j)_{i\in\mathcal{I},j\in\mathcal{J}}\) iff \(\exists\hat{\mathbf{x}}^1,\dots,\hat{\mathbf{x}}^I\in\mathbb{R}_+^n\), \(\exists\hat{\mathbf{y}}^1\in Y^1,\dots,\hat{\mathbf{y}}^J\in Y^J\) such that: (1) consumer's utility maximization: \(\hat{\mathbf{x}}^i\) solves \(\max_{\mathbf{x}^i}u(\mathbf{x}^i)\) s.t. \(\mathbf{p}\cdot\mathbf{x}^i\le m^i(\mathbf{p})\) for \(\forall i\); (2) firm's profit maximization: \(\hat{\mathbf{y}}^j\) solves \(\max_{\mathbf{y}^j\in Y^j}\mathbf{p}\cdot\mathbf{y}^j\) for \(\forall j\); (3) market clearing (feasibility): \(\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j\). We call \((\hat{\mathbf{x}},\hat{\mathbf{y}})\) a WEA.

注记 7.5 / Remark 7.5 市场出清条件 \(\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j\) 假设了无生产外部性，即总生产恰为个体生产计划之和 \(\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j\)。The market clearing condition \(\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j\) assumes no production externalities, i.e. aggregate production is just the sum of individual production plans \(\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j\).

Definition 7.4（Excess demand for production economy） 价格 \(\mathbf{p}\) 处的超额需求 \(\mathbf{z}(\mathbf{p})=\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i(\mathbf{p},m^i(\mathbf{p}))-\sum_{i\in\mathcal{I}}\mathbf{e}^i-\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j(\mathbf{p})\)。Excess demand at price \(\mathbf{p}\): \(\mathbf{z}(\mathbf{p})=\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i(\mathbf{p},m^i(\mathbf{p}))-\sum_{i\in\mathcal{I}}\mathbf{e}^i-\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j(\mathbf{p})\).

Theorem 7.3（Existence） 设 Assumption 1 与 Assumption 2 成立，若 \(\exists\bar{\mathbf{y}}_1\in Y^1,\dots,\bar{\mathbf{y}}_J\in Y^J\) 使 \(\sum_{j\in\mathcal{J}}\bar{\mathbf{y}}^j+\sum_{i\in\mathcal{I}}\mathbf{e}^i\gg\mathbf{0}\)，则存在瓦尔拉斯均衡价格 \(\mathbf{p}^\star\in\mathbb{R}_{++}^n\)。Suppose Assumption 1 and Assumption 2 hold; if \(\exists\bar{\mathbf{y}}_1\in Y^1,\dots,\bar{\mathbf{y}}_J\in Y^J\) such that \(\sum_{j\in\mathcal{J}}\bar{\mathbf{y}}^j+\sum_{i\in\mathcal{I}}\mathbf{e}^i\gg\mathbf{0}\), then there exists a Walrasian equilibrium price \(\mathbf{p}^\star\in\mathbb{R}_{++}^n\).

证明 / Proof (Theorem 7.3)

$$ > \begin{aligned} > \mathbf{p}\cdot\mathbf{z}(\mathbf{p})&=\mathbf{p}\cdot\left(\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i(\mathbf{p},m^i(\mathbf{p}))-\sum_{i\in\mathcal{I}}\mathbf{e}^i-\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j(\mathbf{p})\right)\\ > &=\sum_{i\in\mathcal{I}}\mathbf{p}\cdot\hat{\mathbf{x}}^i(\mathbf{p},m^i(\mathbf{p}))-\sum_{i\in\mathcal{I}}\mathbf{p}\cdot\mathbf{e}^i-\sum_{j\in\mathcal{J}}\mathbf{p}\cdot\hat{\mathbf{y}}^j(\mathbf{p})\\ > &=\sum_{i\in\mathcal{I}}\mathbf{p}\cdot\hat{\mathbf{x}}^i(\mathbf{p},m^i(\mathbf{p}))-\sum_{i\in\mathcal{I}}\mathbf{p}\cdot\mathbf{e}^i-\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}\theta^{ij}\Pi^j(\mathbf{p})\\ > &=\sum_{i\in\mathcal{I}}\underbrace{\left(\mathbf{p}\cdot\hat{\mathbf{x}}^i(\mathbf{p},m^i(\mathbf{p}))-\mathbf{p}\cdot\mathbf{e}^i-\sum_{j\in\mathcal{J}}\theta^{ij}\Pi^j(\mathbf{p})\right)}_{=0}\\ > &=0 > \end{aligned} > $$

$$ > \begin{aligned} > \sum_{i\in\mathcal{I}}m^i(\bar{\mathbf{p}})&=\sum_{i\in\mathcal{I}}\left(\bar{\mathbf{p}}\cdot\mathbf{e}^i+\sum_{j\in\mathcal{J}}\theta^{ij}\Pi^j(\bar{\mathbf{p}})\right)\\ > &=\bar{\mathbf{p}}\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\Pi^j(\bar{\mathbf{p}})\\ > &\ge\bar{\mathbf{p}}\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\bar{\mathbf{p}}\cdot\bar{\mathbf{y}}^j\\ > &=\bar{\mathbf{p}}\left(\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\bar{\mathbf{y}}^j\right)>0 > \end{aligned} > $$

Definition 7.5（feasibility）& Definition 7.6（Pareto efficiency） Def 7.5：在生产经济中，\((\mathbf{x},\mathbf{y})\) 可行，当且仅当 \(\mathbf{x}^i\in\mathbb{R}_+^n\)（\(\forall i\)），\(\mathbf{y}^j\in Y^j\)（\(\forall j\)），\(\sum_{i\in\mathcal{I}}\mathbf{x}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\mathbf{y}^j\)，其中 \(\mathbf{x}=(\mathbf{x}^1,\dots,\mathbf{x}^I)\)、\(\mathbf{y}=(\mathbf{y}^1,\dots,\mathbf{y}^J)\)。
Def 7.6：\((\mathbf{x},\mathbf{y})\) Pareto 有效，当且仅当可行且 \(\nexists(\tilde{\mathbf{x}},\tilde{\mathbf{y}})\) 使 \(u^i(\tilde{\mathbf{x}}^i)\ge u^i(\mathbf{x}^i)\)（\(\forall i\)）且至少一处严格。Def 7.5: in a production economy, \((\mathbf{x},\mathbf{y})\) is feasible iff \(\mathbf{x}^i\in\mathbb{R}_+^n\) (\(\forall i\)), \(\mathbf{y}^j\in Y^j\) (\(\forall j\)), \(\sum_{i\in\mathcal{I}}\mathbf{x}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\mathbf{y}^j\), where \(\mathbf{x}=(\mathbf{x}^1,\dots,\mathbf{x}^I)\), \(\mathbf{y}=(\mathbf{y}^1,\dots,\mathbf{y}^J)\).
Def 7.6: \((\mathbf{x},\mathbf{y})\) is Pareto efficient iff it is feasible and \(\nexists(\tilde{\mathbf{x}},\tilde{\mathbf{y}})\) s.t. \(u^i(\tilde{\mathbf{x}}^i)\ge u^i(\mathbf{x}^i)\) for \(\forall i\) with at least one strict inequality.

Theorem 7.4（First Welfare Theorem） 若每个 \(u^i\) 严格单调，则每个 WEA 都是 Pareto 有效的。If each \(u^i\) is strictly increasing, then every WEA is Pareto efficient.

证明 / Proof (Theorem 7.4)

$$\mathbf{p}^\star\tilde{\mathbf{x}}^i\ge\mathbf{p}^\star\hat{\mathbf{x}}^i\tag{7.1}$$

$$\mathbf{p}^\star\sum_{i\in\mathcal{I}}\tilde{\mathbf{x}}^i>\mathbf{p}^\star\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i\tag{7.2}$$

$$\sum_{i\in\mathcal{I}}\tilde{\mathbf{x}}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\tilde{\mathbf{y}}^j\tag{7.3}$$

$$\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i=\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j\tag{7.4}$$

注记 7.6 / Remark 7.6 此证明依赖于消费者与厂商面对相同价格 \(\mathbf{p}^\star\)。This proof depends on the fact that consumers and firms face the same price \(\mathbf{p}^\star\).

Definition 7.7（Core） 考虑私有制生产经济 \(\mathcal{E}\)，其所有权份额皆为 0 或 1，即 \(\theta^{ij}\in\{0,1\}\)（\(\forall i,j\)）。\(\mathcal{E}\) 的核是所有可行配置 \((\mathbf{x},\mathbf{y})\) 之集，满足不存在非空子集 \(S\subseteq\mathcal{I}\) 与配置 \((\tilde{\mathbf{x}},\tilde{\mathbf{y}})\)（\(\tilde{\mathbf{y}}^j\in Y^j\) 对每个 \(j\)）使：(1) \(\sum_{i\in S}\tilde{\mathbf{x}}^i=\sum_{i\in S}\mathbf{e}^i+\sum_{i\in S}\sum_{j\in\mathcal{J}:\theta^{ij}=1}\tilde{\mathbf{y}}^j\)；(2) \(u^i(\tilde{\mathbf{x}}^i)\ge u^i(\mathbf{x}^i)\)（\(\forall i\in S\)）至少一处严格。Consider a production economy \(\mathcal{E}\) with private ownership in which all ownership shares are 0 or 1, i.e. \(\theta^{ij}\in\{0,1\}\) (\(\forall i,j\)). The core of \(\mathcal{E}\) is the set of all feasible allocations \((\mathbf{x},\mathbf{y})\) such that there does not exist a nonempty subset \(S\subseteq\mathcal{I}\) and an allocation \((\tilde{\mathbf{x}},\tilde{\mathbf{y}})\) (\(\tilde{\mathbf{y}}^j\in Y^j\) for every \(j\)) such that: (1) \(\sum_{i\in S}\tilde{\mathbf{x}}^i=\sum_{i\in S}\mathbf{e}^i+\sum_{i\in S}\sum_{j\in\mathcal{J}:\theta^{ij}=1}\tilde{\mathbf{y}}^j\); (2) \(u^i(\tilde{\mathbf{x}}^i)\ge u^i(\mathbf{x}^i)\) (\(\forall i\in S\)) with at least one strict inequality.

Theorem 7.5 若每个 \(u^i\) 严格单调，则 \(\mathcal{E}\) 的每个 WEA 都在 Def 7.7 定义的核中。If each \(u^i\) is strictly increasing, then every WEA of \(\mathcal{E}\) is in the core defined in Def 7.7.

证明 / Proof (Theorem 7.5)

$$\mathbf{p}\sum_{i\in S}\tilde{\mathbf{x}}^i>\mathbf{p}\sum_{i\in S}\hat{\mathbf{x}}^i\tag{7.5}$$

$$ > \begin{aligned} > \mathbf{p}\sum_{i\in S}\sum_{j:\theta^{ij}=1}\hat{\mathbf{y}}^j&\ge\mathbf{p}\sum_{i\in S}\sum_{j:\theta^{ij}=1}\tilde{\mathbf{y}}^j\\ > \Rightarrow\mathbf{p}\sum_{i\in S}\mathbf{e}^i+\mathbf{p}\sum_{i\in S}\sum_{j:\theta^{ij}=1}\hat{\mathbf{y}}^j&\ge\mathbf{p}\sum_{i\in S}\mathbf{e}^i+\mathbf{p}\sum_{i\in S}\sum_{j:\theta^{ij}=1}\tilde{\mathbf{y}}^j\\ > \Rightarrow\mathbf{p}\sum_{i\in S}\hat{\mathbf{x}}^i&\ge\mathbf{p}\sum_{i\in S}\tilde{\mathbf{x}}^i > \end{aligned}\tag{7.6} > $$

注记 7.7–7.8 / Remarks 7.7–7.8 7.7：因经济中有生产，可行性现涉及生产；任一联盟可考虑的总资源 = 其禀赋 + 其拥有的厂商生产出的全部商品。为何 \(\theta^{ij}\) 只能取 0 或 1？因这意味着对任一联盟，厂商要么被联盟完全拥有与控制，要么完全不被拥有、不影响联盟。0-1 所有权使任一联盟能独立地为其拥有的厂商安排生产、且不受其不拥有厂商的生产计划影响，从而像交换经济一样与整个经济完全分离。若份额可为分数，联盟对其厂商的生产计划可能被联盟外、亦持有正份额的agent否决，使配置可行与否变得不清楚。7.8：已证若每个 \(u^i\) 严格单调则 WEA 在核中；由核的定义，WEA 不被大联盟阻塞，故 WEA Pareto 有效。但此 Pareto 效率证明不如 Thm 7.4 一般，因对 \(\mathcal{E}\) 的所有权结构加了限制。7.7: since there is production, feasibility now involves production; the aggregate resources for any coalition = their endowment plus all goods produced by the firms they own. Why \(\theta^{ij}\in\{0,1\}\)? Because then for any coalition, a firm is either completely owned and controlled by the coalition, or not owned at all and unaffecting the coalition. 0-1 ownership lets any coalition make production independently for the firms it owns, unaffected by the production plans of firms it doesn't own, achieving perfect separateness from the whole economy like the exchange economy. If shares were fractional, a coalition's production plan for its firms could be rejected by agents outside the coalition who also hold positive shares, making feasibility unclear. 7.8: we proved that if each \(u^i\) is strictly increasing then WEA is in the core; by the definition of the core, WEA is unblocked by the grand coalition, hence Pareto efficient. But this Pareto-efficiency proof is less general than Thm 7.4 because it imposes restrictions on the ownership structures of \(\mathcal{E}\).

Theorem 7.6（Second Welfare Theorem） 设 Assumption 1 与 Assumption 2 成立，且 \(\exists\bar{\mathbf{y}}_1\in Y^1,\dots,\bar{\mathbf{y}}_J\in Y^J\) 使 \(\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j+\sum_{i\in\mathcal{I}}\mathbf{e}^i\gg\mathbf{0}\)。若 \((\hat{\mathbf{x}},\hat{\mathbf{y}})\) Pareto 有效，则存在一组（一次性）收入转移 \(T_1,\dots,T_I\) 使 \(\sum_{i\in\mathcal{I}}T_i=0\)，且 \(\exists\mathbf{p}^\star\in\mathbb{R}_{++}^n\) 使：(1) \(\hat{\mathbf{x}}^i\) 在 \(\mathbf{p}^\star\mathbf{x}^i\le m^i(\mathbf{p}^\star)+T_i\) 下最大化 \(u^i(\mathbf{x}^i)\)（\(\forall i\)）；(2) \(\hat{\mathbf{y}}^j\) 在 \(\mathbf{y}^j\in Y^j\) 下最大化 \(\mathbf{p}^\star\mathbf{y}^j\)（\(\forall j\)）。二者合起来即 \((\hat{\mathbf{x}},\hat{\mathbf{y}})\) 为 WEA。Suppose Assumption 1 and Assumption 2 hold, and \(\exists\bar{\mathbf{y}}_1\in Y^1,\dots,\bar{\mathbf{y}}_J\in Y^J\) such that \(\sum_{j\in\mathcal{J}}\hat{\mathbf{y}}^j+\sum_{i\in\mathcal{I}}\mathbf{e}^i\gg\mathbf{0}\). If \((\hat{\mathbf{x}},\hat{\mathbf{y}})\) is Pareto efficient, then there exists a set of (lump-sum) income transfers \(T_1,\dots,T_I\) with \(\sum_{i\in\mathcal{I}}T_i=0\), and \(\exists\mathbf{p}^\star\in\mathbb{R}_{++}^n\) such that: (1) \(\hat{\mathbf{x}}^i\) maximizes \(u^i(\mathbf{x}^i)\) s.t. \(\mathbf{p}^\star\mathbf{x}^i\le m^i(\mathbf{p}^\star)+T_i\) (\(\forall i\)); (2) \(\hat{\mathbf{y}}^j\) maximizes \(\mathbf{p}^\star\mathbf{y}^j\) s.t. \(\mathbf{y}^j\in Y^j\) (\(\forall j\)). Combined, \((\hat{\mathbf{x}},\hat{\mathbf{y}})\) is a WEA.

证明 / Proof (Theorem 7.6)

$$\nabla u^i(\hat{\mathbf{x}}^i)=\begin{bmatrix}\frac{\partial u^i}{\partial x_1^i}\\\frac{\partial u^i}{\partial x_2^i}\\\vdots\\\frac{\partial u^i}{\partial x_n^i}\end{bmatrix}$$

$$\mathbf{p}^\star=\frac{1}{\lambda_i}\nabla u^i(\hat{\mathbf{x}}^i)\gg\mathbf{0}\quad\text{for }\forall i\in\mathcal{I},\qquad\nabla u^i(\hat{\mathbf{x}}^i)=\lambda_i\mathbf{p}^\star\quad\text{for }\forall i\in\mathcal{I}\tag{7.7}$$

$$\nabla u^i(\hat{\mathbf{x}}^i)\hat{\mathbf{y}}^j\ge\nabla u^i(\hat{\mathbf{x}}^i)\mathbf{y}^j\quad\text{for }\forall\mathbf{y}^j\in Y^j\tag{7.8}$$

$$ > \begin{aligned} > u^i(\mathbf{x}_m)&=u^i(\hat{\mathbf{x}}^i)+\nabla u^i(\hat{\mathbf{x}}^i)(\mathbf{x}_m-\hat{\mathbf{x}}^i)\\ > &=u^i(\hat{\mathbf{x}}^i)+\nabla u^i(\hat{\mathbf{x}}^i)t(\tilde{\mathbf{y}}^j-\hat{\mathbf{y}}^j)\\ > &>u^i(\hat{\mathbf{x}}^i) > \end{aligned} > $$

$$\mathbf{p}^\star\hat{\mathbf{x}}^i=\mathbf{p}^\star\mathbf{e}^i+\sum_{j\in\mathcal{J}}\theta^{ij}\mathbf{p}^\star\hat{\mathbf{y}}^j+T_i\tag{7.9}$$

$$ > \begin{aligned} > \mathbf{p}^\star\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i&=\mathbf{p}^\star\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}}\theta^{ij}\mathbf{p}^\star\hat{\mathbf{y}}^j+\sum_{i\in\mathcal{I}}T_i\\ > &=\mathbf{p}^\star\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\mathbf{p}^\star\hat{\mathbf{y}}^j+\sum_{i\in\mathcal{I}}T_i\\ > \text{by feasibility }\mathbf{p}^\star\sum_{i\in\mathcal{I}}\hat{\mathbf{x}}^i&=\mathbf{p}^\star\sum_{i\in\mathcal{I}}\mathbf{e}^i+\sum_{j\in\mathcal{J}}\mathbf{p}^\star\hat{\mathbf{y}}^j\\ > \Rightarrow\sum_{i\in\mathcal{I}}T_i&=0.\quad\blacksquare > \end{aligned} > $$