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## 经济代写|Microeconomics代考微观经济学代写|Techniques Including Diferent Commodities

Now let us discuss the important case of choice between two alternative methods $\alpha$ and $\beta$ for the production of, say, commodity 1 , when method $\beta$ requires the use of commodities not appearing in technique $(\alpha)$.

To fix ideas, suppose that technique $(\alpha)$ includes $n$ commodities, and that technique $(\beta)$ has in common with technique $(\alpha)$ these $n$ commodities, as well as the methods to produce all of them except commodity 1 ; method $\beta$ for commodity 1 requires as inputs, in addition to some of the $n$ commodities common to both techniques, some commodities indexed as $n+1, n+2, \ldots, n+h$, which do not appear at all in technique $(\alpha)$. Technique ( $\beta$ ) must therefore also include industries producing these commodities, some of which might use themselves directly or indirectly in their own production. Therefore the determination of the supply price $p_{1}^{\beta(\alpha)}$ of commodity 1 at prices and distribution $(\alpha)$ requires the determination also of the supply prices of commodities $n+1, \ldots, n+h$. This means that we have $h+1$ equations to determine the $h+1$ supply prices of these $h$ commodities and of commodity 1. Formally, these $h+1$ equations are like price equations of non-basic commodities in technique $(\alpha)$, and therefore they can be solved. Thus $p_{1}^{\beta(\alpha)}$ is well determined and can be compared with $p_{1}^{(\alpha)}$. Now, the proof given above of Lemma $2.10$ that if $p_{1}^{\beta(\alpha)}w^{(\alpha)}$, did not need $\mathbf{A}$ to be indecomposable: the matrices $\mathbf{C}{a}$ and $\mathbf{C}{\beta a}$ were indecomposable anyway since all commodities need labour, directly or indirectly, to be produced. The key to that proof was that the subsistence vector was assumed to consist of numéraire. All that is needed to apply the same proof here is that commodity 1 , or one of the commodities which directly or indirectly use commodity 1 for their production, be included in the numéraire. (If commodity 1 is non-basic and not included in the numéraire not even indirectly, a method that decreases its cost has no effect on $w$.) Lemmas $2.11$ and $2.12$ too do not need an assumption that $\mathbf{A}$ be indecomposable. We conclude that those three lemmas are also valid for the case in which techniques differ in the commodities they include; only, the comparison of the profitability of alternative methods must be applied to the common commodities. Thus, the conclusion of Lemma $2.12$ that if two techniques are both on the outer envelope at the given $r$, then $\mathbf{p}^{(\alpha)}=\mathbf{p}^{(\beta)}$, must be intended to apply to the prices of the common commodities only.

## 经济代写|Microeconomics代考微观经济学代写|The Samuelson–Garegnani Model

As an important example of ‘enlarged’ techniques, let us study the model first proposed by Paul Samuelson in 1962 when presenting the notion of surrogate production function, and then much discussed in the Cambridge controversies in capital theory, in particular by Garegnani (1970), as discussed in Chap. 7. It is a two-industry model, where there is a single pure consumption good, which can be produced by alternative methods; each different method for the production of the consumption good utilizes labour and a different circulating capital good, which is produced by itself and labour via a single method specific to it. Thus each technique uses a different capital good, the sole commodity common to alternative techniques is the consumption good, and the comparison of alternative methods for the production of the consumption good necessarily involves considering also the capital good utilized and how it is produced.

Let the consumption good be commodity 2, and the capital good be commodity $1 \alpha$, or $1 \beta$, or $1 \gamma$, etc., depending on the method for the production of the consumption good, i.e. on the technique. Technique $(\alpha)$ is then a couple $\left(\mathbf{A}{\alpha}, \mathbf{a}{\mathbf{L} \alpha}\right)$ where the $2 \times 2$ matrix $\mathbf{A}{\alpha}$ has a positive first row $\mathbf{a}{1 \alpha}=\left(a_{1 \alpha, l \alpha}, a_{1 \alpha, 2}\right)$ and a second row consisting of zeros. Choose the consumption good as numéraire. The price equations are
$$p_{l \alpha}=(1+r) p_{l \alpha} a_{l \alpha, 1 \alpha}+w^{(\alpha)} a_{L 1 \alpha}$$
$$p_{2}^{(\alpha)}=1=(1+r) p_{l \alpha} a_{1 \alpha, 2}+w^{(\alpha)} a_{L 2, \alpha-}$$

# 微观经济学代考

## 经济代写|Microeconomics代考微观经济学代写|The Samuelson–Garegnani Model

$$p_{l \alpha}=(1+r) p_{l \alpha} a_{l \alpha, 1 \alpha}+w^{(\alpha)} a_{L 1 \alpha}$$
$$p_{2}^{(\alpha)}=1=(1+r) p_{l \alpha} a_{1 \alpha, 2}+w^{(\alpha)} a_{L 2, \alpha-}$$

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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