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The first welfare theorem from the point of view of partial analysis|Econ经济作业代写Economics代考

We attack this claim by way of partial analysis first and turn to total analysis later on. Our partial analysis (we concentrate on one or two markets leaving the repercussions on and from other markets aside) concerns

• exchange optimality (is it possible to make a consumer better off without making another one worse off?),
• production optimality (is it possible to produce more of one good without producing less of any other good?), and
• the optimal product mix (is it better to produce more of one good and less of another one?).

In the next three subsections, we consider each type of optimality separately. Within each type, we go through three steps:
(1) We first characterize Pareto optimality (concerning exchange, production, and product mix). Clearly, for this specification prices are unimportant.
(2) We then consider how households and firms adapt to given good and factor prices.
(3) We finally show that the adaption according to the second point (price-taking assumption) is done in a way compatible with the first (Pareto optimality).
4.1.3. Exchange optimality. We know the exchange Edgeworth box from chapter XIV (pp. 385). Assume two households $A$ and $B$ and two goods 1 and 2. Along the contract curve or exchange curve,
$$\left|\frac{d x_{2}^{A}}{d x_{1}^{A}}\right|=M R S^{A} \stackrel{!}{=} M R S^{B}=\left|\frac{d x_{2}^{B}}{d x_{1}^{B}}\right|$$
holds. This is the first step.
The second step consists of determining the decisions taken by the households at the given prices. Since at each of the two household optima, the

The first welfare theorem from the point of view of general equilibrium analysis|ECON经济作业代写ECONOMICS代考

A coalition $S$ can block an allocation if it can present an allocation that improves the lot of its members and that can be afforded by $S$ :
Definition XIX.11 (BLOCKABLE ALLOCATION, CORE). Let
$$\mathcal{E}=\left(N, G,\left(\omega^{i}\right){i \in N},\left(\precsim^{i}\right){i \in N}\right)$$
be an exchange economy. A coalition $S \subseteq N$ is said to block an allocation $\left(y^{i}\right){i \in N}$, if an allocation $\left(z^{i}\right){i \in N}$ exists such that

• $z^{i} \succsim^{i} y^{i}$ for all $i \in S, z^{i} \succ^{i} y^{i}$ for some $i \in S$ and
• $\sum_{i \in S} z^{i} \leq \sum_{i \in S} \omega^{i}$
hold.
An allocation is not blockable if there is no coalition that can block it. The set of all feasible and non-blockable allocations is called the core of an exchange economy.

Within the Edgeworth box, the core can be depicted graphically. We see the endowment point and the associated exchange lens in figure XIX.7. Every household (considered a one-man coalition) blocks any allocation that lies below the indifference curve cutting his endowment point. Therefore, the core is contained inside the exchange lens. Both households together block any allocation that is not Pareto efficient. Thus, the core is the intersection of the exchange lens and the contract curve.

The second welfare theorem|Econ经济作业代写Economics代考

The reader is invited to consider figure XIX.9 and the Pareto opimum highlighted. If point $E$ is given as endowment point, the associated Walras allocation is indeed the Pareto optimum. If, however, the original endowment is $D$ instead of $E$, we can redistribute endowments by transfering some units of good 1 from $B$ to $A$.

Figure XIX.10 illustrates why we assume convexity in the above theorem. Agent $B$ does not have convex preferences. At the prices given by the price line, he does not demand his part of the Pareto optimum but some point $C$.FIGURE XIX.9. The second welfare theorem
FIGURE XIX.10. Convexity is a necessary condition for the second welfare theorem.

THE FIRST WELFARE THEOREM FROM THE POINT OF VIEW OF PARTIAL ANALYSIS|ECON经济作业代写ECONOMICS代考

• 交换最优性（是否有可能在不让另一个消费者变得更糟的情况下让消费者变得更好？），
• 生产最优性（是否有可能生产更多的一种商品而不生产更少的任何其他商品？），以及
• 最佳产品组合（生产更多一种商品而减少另一种商品是否更好？）。

（1）我们首先描述帕累托最优（关于交换、生产和产品组合）。显然，对于这个规格，价格并不重要。
(2) 然后我们考虑家庭和企业如何适应给定的商品和要素价格。
(3) 我们最终表明，根据第二点（定价假设）的适应是以与第一点（帕累托最优）兼容的方式进行的。
4.1.3。交换最优性。我们从第十四章（第 385 页）中知道交换御剑盒。假设有两个家庭一种和乙以及两种商品 1 和 2。沿着合约曲线或交换曲线，
|dX2一种dX1一种|=米R小号一种=!米R小号乙=|dX2乙dX1乙|

THE FIRST WELFARE THEOREM FROM THE POINT OF VIEW OF GENERAL EQUILIBRIUM ANALYSIS|ECON经济作业代写ECONOMICS代考

$$\mathcal{E}=\left(N, G,\left(\omega^{i}\right) {i \in N},\left(\precsim^{i}\right) {i \在 N}\right)$$

• 和一世≿一世和一世对所有人一世∈小号,和一世≻一世和一世对于一些一世∈小号和
• ∑一世∈小号和一世≤∑一世∈小号ω一世
抓住。
如果没有联盟可以阻止分配，则分配是不可阻止的。所有可行且不可阻塞的分配集合称为交换经济的核心。

THE SECOND WELFARE THEOREM|ECON经济作业代写ECONOMICS代考

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