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经济代写|微观经济学代写Microeconomics代考|WACm; Kuhn–Tucker Conditions and Cost Minimization

Cost minimization has an implication analogous to the WAPM. If for a given output $q^{\circ}$ and given input rentals $v$ the firm finds it optimal to utilize an input vector $\mathbf{x}^{\circ}=h\left(\mathbf{v}, q^{\circ}\right)$, it must mean that any other input vector capable of producing $q^{\circ}$ (or more) must cost at least $\mathbf{v x}^{\circ}$, in other words, $\mathbf{v x}^{\circ} \leq \mathbf{v x}$ for all $\mathbf{x}$ such that $f(\mathbf{x}) \geq q^{\circ}$, a result sometimes called Weak Axiom of Cost Minimization, WACm for short. If at input prices $v^{\circ}$ the firm chooses input vector $\mathbf{x}^{\circ}$ and at input prices $\mathbf{v}^$ the firm chooses input vector $\mathbf{x}^$ to produce the same output, proceeding in the same way as for the WAPM one reaches the conclusion
$$
\left(\mathbf{v}^0-\mathbf{v}^\right)\left(\mathbf{x}^0-\mathbf{x}^\right) \leq 0,
$$
more often expressed as
$$
\Delta \mathbf{v} \cdot \Delta \mathbf{x} \leq 0 .
$$
This implies, for example, that if only one input price changes, the demand for that input cannot change in the same direction (and if it changes at all it must change in the opposite direction). (Note that if inputs were measured as negative quantities the inequality sign would be reversed, still this is not the same result as was derived from the WAPM, because here output is kept fixed.)

In general, not all possible inputs will be used in positive amounts by a firm; the condition $v / v_j=\mathrm{MP}i / \mathrm{MP}{\mathrm{j}}$ must hold for inputs both used in positive quantities; the more general necessary first-order conditions for cost minimization are derivable from the Kuhn-Tucker theorem ( Sect. 4.8). In the present case, with $q^{\circ}$ the given output, the function to be maximized is $-\mathbf{v x}$ and the constraints are $f(\mathbf{x})-q^{\circ} \geq 0$, and $x_i \geq 0, \mathrm{i}=1, \ldots, \mathrm{n}$. The Lagrangian is $-\mathbf{v} \mathbf{x}+i^{\circ}\left(f(\mathbf{x})-q^{\circ}\right)+\sum_{\mathrm{i}} \lambda_i x_i$ indicating $\partial f f \partial x_i$ as $f_i$, the first-order conditions are therefore:
Kuhn-Tucker first-order necessary conditions for cost minimization:
$$
-v_i+\lambda_i+\lambda_i=0, \quad i=1, \ldots, n,
$$
with complementary slackness conditions $\lambda^{\circ}\left(f(\mathbf{x})-q^{\circ}\right)=0$ and $\lambda_r x_i=0$, that is, if $\left(f(x)-q^{\circ}\right)>0$, then $\lambda^{\circ}=0$, and if $\lambda_i>0$, i.e. if $v_i>\lambda f_i$, then $x_i=0$.

经济代写|微观经济学代写Microeconomics代考|Supply Curves: Short-Period Marshallian Analysis, Quasi-Rents

The traditional, Marshallian approach distinguishes short-period from long-period profit maximization and short-period from long-period supply curves of firms. As Marshall puts it, the long-period or normal value of a commodity “is that which economic forces tend to bring about in the long run. It is the average value which economic forces would bring about if the general conditions of life were stationary for a run of time long enough to enable them all to work out their full effect’ (1972, V, iii, 6; p. 289). At the level of the single firm, the distinction between short-period and long-period decisions is traditionally described as follows: in the short period a firm cannot alter the quantities of some of the inputs, in the long period all inputs can be varied.

This is often put in terms of whether the more durable part of the capital goods used by the firm, its fixed plant, is given or variable. However, it takes very little time for a firm to sell or rent out its fixed plant to other firms, or to buy or rent the fixed plants of other firms (a firm is a legal entity, not to be confused with its plants): for example, a firm can rent a building or flat if needed for a particular production or period. So the idea that the short period is characterized by each firm having a given fixed plant is difficult to defend. If one re-reads the inventor of the short-period/long-period distinction, Alfred Marshall, one finds that the fixity he had in mind referred not to the individual firm but to the industry: the short period was the analytical period within which there was not enough time significantly to change the amounts available to the entire industry of specialized durable inputs necessary to the industry and requiring considerable time for their production. Thus Marshall’s short-period analysis of the fishing industry takes as given the number of fishing ships (and of experienced fishermen), not the number of firms nor how the fishing ships are divided among different firms (1972, V, v, 4; p. 307). In the language of plants, one can say that in the short period the total number of fixed plants available to an industry is given, how they are divided among firms is secondary: a well-run firm owning two separate plants will behave in the same way as two firms each owning one of the two plants because it will want to maximize the profit from each plant.

However, once the point is grasped, then precisely because how the plants are divided among firms is secondary if all firms are efficient, to assume that the distribution of plants among firms is given will make no difference for the determination of the industry’s supply decisions (which are what is relevant to determine the equilibrium of the market one is studying.). Therefore we can follow the standard textbook approach after all.

经济代写|微观经济学代写Microeconomics代考|BEA470

经济代写|微观经济学代写Microeconomics代考|WACm; Kuhn–Tucker Conditions and Cost Minimization

最优的 $\mathbf{x}^{\circ}=h\left(\mathbf{v}, q^{\circ}\right)$ ,它必须意味着任何其他能够产生 $q^{\circ}$ (或更多) 必须至少花费 $\mathbf{v x}{ }^{\circ}$ , 换句话说, $\mathbf{v} \mathbf{x}^{\circ} \leq \mathbf{v} \mathbf{x}$ 对所有人 $\mathbf{x}$ 这样 $f(\mathbf{x}) \geq q^{\circ}$ ,这个结果有时被称为成本最小化的弱公理,简称 WACm。如果按 出,以与 WAPM 相同的方式进行,得出结论
更常表示为
$$
\Delta \mathbf{v} \cdot \Delta \mathbf{x} \leq 0
$$
这意味着,例如,如果只有一个投入品价格发生变化,则对该投入品的需求不能沿相同方向变化 (如果它 发生变化,它必须沿相反方向变化)。(请注意,如果输入被测量为负数,则不等号将被反转,这仍然与 从 WAPM 得出的结果不同,因为这里的输出保持固定。)
一般来说,并非所有可能的投入都会被公司以正数使用;条件 $v / v_j=\mathrm{MP} i / \mathrm{MPj}$ 对于以正数使用的输入 必须成立;成本最小化的更一般的必要一阶条件可以从库恩-塔克定理(第 $4.8$ 节) 推导出来。在本案中, 与 $q^{\circ}$ 给定输出,要最大化的函数是 $-\mathbf{v} \mathbf{x}$ 并且约束是 $f(\mathbf{x})-q^{\circ} \geq 0$ ,和 $x_i \geq 0, \mathrm{i}=1, \ldots, \mathrm{n}$. 拉格朗日是 $-\mathbf{v} \mathbf{x}+i^{\circ}\left(f(\mathbf{x})-q^{\circ}\right)+\sum_{\mathrm{i}} \lambda_i x_i$ 表示 $\partial f f \partial x_i$ 作为 $f_i$ ,因此
一阶条件为:成本最小化的库恩-塔克一阶必要条件:
$$
-v_i+\lambda_i+\lambda_i=0, \quad i=1, \ldots, n,
$$
具有互补松弛条件 $\lambda^{\circ}\left(f(\mathbf{x})-q^{\circ}\right)=0$ 和 $\lambda_r x_i=0$ ,也就是说,如果 $\left(f(x)-q^{\circ}\right)>0$ ,然后 $\lambda^{\circ}=0$ , 而如果 $\lambda_i>0$ ,即如果 $v_i>\lambda f_i$ ,然后 $x_i=0$.

经济代写|微观经济学代写Microeconomics代考|Supply Curves: Short-Period Marshallian Analysis, Quasi-Rents

传统的马歇尔方法将企业的短期利润最大化与长期利润最大化以及短期与长期供应曲线区分开来。正如马歇尔所说,商品的长期价值或正常价值“是经济力量在长期内倾向于带来的价值。如果生活的一般条件在一段足够长的时间内保持静止,以使它们都能发挥其全部作用,那么经济力量将带来的平均值”(1972,V,iii,6;p.289) . 在单一企业层面,短期决策和长期决策之间的区别传统上描述如下:在短期内,企业不能改变某些投入的数量,在长期内,所有投入都可以改变.

这通常是根据公司使用的资本货物中更耐用的部分,即其固定工厂,是给定的还是可变的。但是,公司将其固定设备出售或出租给其他公司,或购买或租用其他公司的固定设备(公司是法人实体,不要与其工厂混淆)只需很少的时间:例如,如果特定生产或时期需要,公司可以租用建筑物或公寓。因此,短时期的特点是每个公司都有一个固定的工厂,这种观点很难辩护。如果重新阅读短期/长期区分的发明者阿尔弗雷德·马歇尔(Alfred Marshall),就会发现他所指的固定性不是指个别公司,而是指行业:短期是分析期间,在此期间没有足够的时间来显着改变整个行业可获得的行业所需的专门耐用投入的数量,并且需要大量时间来生产这些投入。因此,马歇尔对渔业的短期分析将渔船(和有经验的渔民)的数量视为给定,而不是企业的数量或渔船在不同企业之间的划分方式(1972,V,v,4;p . 307). 用工厂的语言来说,可以说在短期内,一个行业可用的固定工厂的总数是给定的,它们在公司之间的分配方式是次要的:

然而,一旦掌握了这一点,那么正是因为如果所有企业都是有效率的,工厂之间的分配方式是次要的,假设企业之间的工厂分布是给定的,对于确定行业的供应决策没有任何影响(这与确定正在研究的市场均衡有关。)。因此,我们毕竟可以遵循标准的教科书方法。

经济代写|微观经济学代写Microeconomics代考

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