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## 金融代写|国际贸易理论代写Theory of International Trade代考|GDP maximization

Expenditure minimization: Looking at the behavior of a household from a different angle, let us consider the following problem in which the household minimizes the expenditure on the purchase of a consumption vector that guarantees a specified utility level: given $p$ and $u$,

GDP maximization: Given a price vector $p$ and a production vector $y$, we can calculate the value of the total production $p y \equiv \sum_{j=1}^n p_j y_j$, which is nothing but the Gross Domestic Product (GDP) of the country. Let us consider the GDP maximization problem under the resource constraint:
$$\max _y p y \text { subject to } y \in \mathscr{Y}(\bar{v}) \text {. }$$
We can show that the production point $\tilde{y}(p, \bar{v})$ of the production equilibrium for $p$ coincides with the solution to the GDP maximization problem. It suffices to show that the production equilibrium satisfies the conditions for the GDP maximization.

Taking account of the definition of $\mathscr{Y}(\bar{v})$, let us define the Lagrangian function for (1.15):
$$\mathscr{L} \equiv \sum_{j=1}^n p_j y_j+\sum_{j=1}^n \mu_j\left{F_j\left(v_{\bullet j}\right)-y_j\right}+\sum_{i=1}^m \xi_i\left{\bar{v}i-\sum{j=1}^n v_{i j}\right},$$
where $\mu_j(j=1,2, \ldots, n)$ and $\xi_i(i=1,2, \ldots, m)$ are the Lagrange multipliers. The first-order necessary conditions for the GDP maximization become as follows: for $i=1,2, \ldots, m$ and $j=1,2, \ldots, n$,
\begin{aligned} &\frac{\partial \mathscr{L}}{\partial y_j}=p_j-\mu_j=0 \ &\frac{\partial \mathscr{L}}{\partial \mu_j}=F_j\left(v_{\bullet j}\right)-y_j=0 \ &\frac{\partial \mathscr{L}}{\partial v_{i j}}=\mu_j \frac{\partial F_j\left(v_{\bullet j}\right)}{\partial v_{i j}}-\xi_i=0 \ &\frac{\partial \mathscr{L}}{\partial \xi_i}=\bar{v}i-\sum{k=1}^n v_{i k}=0 \end{aligned}

## 金融代写|国际贸易理论代写Theory of International Trade代考|GDP function

GDP function: The maximized GDP is written as a function of $p$ and $\bar{v}$ :
$$Y(p, \bar{v}) \equiv p \tilde{y}(p, \bar{v}) \equiv \max {p y \mid y \in \mathscr{Y}(\bar{v})} .$$
We call it the GDP function. ${ }^{18}$ The GDP function is a convenient analytical tool that describes intensively the behavior of the production sector of a country as a whole. We show some of the properties of the GDP function below.

The GDP function is linearly homogeneous, non-decreasing, and convex in $p$. The linear homogeneity of $Y$ in $p$ directly follows from the definition. To show $Y$ is non-decreasing in $p$, let us take two price vectors $p$ and $p^{\prime}$ such that $p_j0$ and $\tilde{y}_j(p, \bar{v}) \geq 0$, the second term is also non-negative, implying $Y$ is non-decreasing in $p$. Let us turn to the convexity in $p$. For arbitrary price vectors $p$ and $p^{\prime}$ and a real number $\lambda$ such that $0<\lambda<1$, define $p^\lambda \equiv \lambda p+[1-\lambda] p^{\prime}$. Then, by definition, we have $Y(p, \bar{v}) \geq p \tilde{y}\left(p^\lambda, \bar{v}\right)$ and $Y\left(p^{\prime}, \bar{v}\right) \geq p^{\prime} \tilde{y}\left(p^\lambda, \bar{v}\right)$. Multiplying the former by $\lambda$ and the latter by $[1-\lambda]$ and adding both sides, we obtain $\lambda Y(p, \bar{v})+[1-\lambda] Y\left(p^{\prime}, \bar{v}\right) \geq\left(\lambda p+[1-\lambda] p^{\prime}\right) \tilde{y}\left(p^\lambda, \bar{v}\right)=Y\left(p^\lambda, \bar{v}\right)$.

## 金融代写|国际贸易理论代写Theory of International Trade代考| GDP maximization

GDP最大化: 给定价格向量 $p$ 和生产向量 $y$ ，我们可以计算出总产量的价值 $p y \equiv \sum_{j=1}^n p_j y_j$ ，这只不过是该国的国内生产总值 (GDP)。让我们考虑一下资源约束下的GDP最大化问题:
$\max y p y$ subject to $y \in \mathscr{Y}(\bar{v})$. 我们可以证明生产点 $\bar{y}(p, \bar{v})$ 的生产均衡 $p$ 这与GDP最大化问题的解决方㝔相吻合。只要证明生产均衡满足GDP最大化的条件就足 够了。 考虑到 $\mathscr{Y}(\bar{v})$ ，让我们定义 (1.15) 的拉格朗日函数: 缺少或无法识别 \left 的分隔符 哪里 $\mu_j(j=1,2, \ldots, n)$ 和 $\xi_i(i=1,2, \ldots, m)$ 是拉格朗日乘数。GDP最大化的一阶必要条件如下: $i=1,2, \ldots, m$ 和 $j=1,2, \ldots, n$, $\frac{\partial \mathscr{L}}{\partial y_j}=p_j-\mu_j=0 \quad \frac{\partial \mathscr{L}}{\partial \mu_j}=F_j\left(v{\bullet j}\right)-y_j=0 \frac{\partial \mathscr{L}}{\partial v_{i j}}=\mu_j \frac{\partial F_j\left(v_{\bullet j}\right)}{\partial v_{i j}}-\xi_i=0 \quad \frac{\partial \mathscr{L}}{\partial \xi_i}=\bar{v} i-\sum k=1^n v_{i k}=0$

## 金融代写|国际贸易理论代写Theory of International Trade代考| GDP function

GDP函数：最大化的GDP写为 $p$ 和 $\bar{v}$ :
$$Y(p, \bar{v}) \equiv p \bar{y}(p, \bar{v}) \equiv \max p y \mid y \in \mathscr{Y}(\bar{v})$$

GDP 函数是线性㐎次的，不递减的，并且在 $p$ 线性均匀性 $Y$ 在 $p$ 直接䕊循定义。要显示 $Y$ 在 $p$ ，让伐们取两个价格向量 $p$ 和 $p^{\prime}$ 使得 $p_j 0$ 和 $\bar{y}_j(p, \bar{v}) \geq 0$ ，第二项也是非否定的，暗示 $Y$ 在 $p$. 让我们转向凸性 $p$.对于任意价格向量 $p$ 和 $p^{\prime}$ 和一个实数 $\lambda$ 使得 $0<\lambda<1$ 定义 $p^\lambda \equiv \lambda p+[1-\lambda] p^{\prime}$. 然后，根据定义，我们有 $Y(p, \bar{v}) \geq p \bar{y}\left(p^\lambda, \bar{v}\right)$ 和 $Y\left(p^{\prime}, \bar{v}\right) \geq p^{\prime} \bar{y}\left(p^\lambda, \bar{v}\right)$.侍前者乘以入后者由 $[1-\lambda]$ 并将两边相加，我们得到 $\lambda Y(p, \bar{v})+[1-\lambda] Y\left(p^{\prime}, \bar{v}\right) \geq\left(\lambda p+[1-\lambda] p^{\prime}\right) \bar{y}\left(p^\lambda, \bar{v}\right)=Y\left(p^\lambda, \bar{v}\right)$.

## 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。

Posted on Categories:Theory of International Trade, 国际贸易理论, 金融代写

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## 金融代写|国际贸易理论代写Theory of International Trade代考|Expenditure minimization

Expenditure minimization: Looking at the behavior of a household from a different angle, let us consider the following problem in which the household minimizes the expenditure on the purchase of a consumption vector that guarantees a specified utility level: given $p$ and $u$,
$$\min x p x \quad \text { subject to } U(x) \geq u \text {. }$$ The solution to the above problem depends on the pair of $(p, u)$ and is called the compensated demand function: $\bar{D}(p, u) \equiv\left(\bar{D}_1(p, u), \ldots, \widetilde{D}_n(p, u)\right) .^{13}$ By substituting $x=\widetilde{D}(p, u)$ into the objective function in (1.8), we obtain the expenditure function: $$E(p, u) \equiv p \widetilde{D}(p, u) .$$ The formal definitions of the compensated demand function and the expenditure function are parallel to those of the conditional factor demand function $\tilde{v}$ and the minimum cost function $C$. Therefore, similar to $\tilde{v}$ and $C$, we can show that the compensated demand function $\widetilde{D}(p, u)$ is homogeneous of degree zero in $p ;\left(p^{\prime}-p\right)\left{\widetilde{D}\left(p^{\prime}, u\right)-\widetilde{D}(p, u)\right} \leq 0$ for any $p$ and $p^{\prime}$; and $E(p, u)$ is monotonically increasing in $u$, concave and linearly homogeneous in $p$, and it satisfies Shephard’s lemma: $\partial E(p, u) / \partial p_i=\widetilde{D}_i(p, u)$ for $i=1,2, \ldots, n$. The $n \times n$ matrix $\left[E{i j}\right] \equiv\left[\partial^2 E / \partial p_i \partial p_j\right]{i, j=1, \ldots, n}$ consisting of the second-order derivatives of $E$ with respect to $p$ is negative semi-definite. Further, similar to Eq. (1.5), we can define the elasticity of substitution between good $i$ and good $j$ in consumption; with a slight abuse of notation, we shall denote it by $\sigma{i j}$. If $\sigma_{i j}>0\left(\sigma_{i j}<0\right)$, then good $i$ and good $j$ are mutually substitutes (complements, respectively) in consumption.

## 金融代写|国际贸易理论代写Theory of International Trade代考|Slutsky equation

Slutsky equation: As is obvious from the definitions of $D$ and $D$, we have $D_i(p, E(p, u)) \equiv \widetilde{D}_i(p, u)$ for any pair of $(p, u)$. By differentiating this identity with respect to $p_j$, we can show that the effect of a price change on the (ordinary) demand for good $i$ is decomposed into two effects. The decomposition is known as the Slutsky equation: for $i, j=1,2, \ldots, n$,
$$\frac{\partial D_i(p, I)}{\partial p_j}=\frac{\partial \widetilde{D}_i(p, u)}{\partial p_j}-x_j \frac{\partial D_i(p, I)}{\partial I},$$
where $x_j=D_j(p, I)=\widetilde{D}_j(p, u)$ is the quantity of demand for good $j$. The first term of the right-hand side of $(1.9), \partial \widetilde{D}_i(p, u) / \partial p_j$, is the substitution effect and the second term, $-x_j \partial D_i(p, I) / \partial I$, is the income effect. The substitution effect is positive (negative) if good $i$ and good $j$ are substitutes (complements, respectively). The substitution effect of its own price, $\partial \widetilde{D}_i(p, u) / \partial p_i$, is always negative. We say that good $i$ is a normal good (an inferior $g \circ o d)$ if $\partial D_i(p, I) / \partial I$ is positive (negative, respectively). Accordingly, the income effect in the Slutsky equation is negative (positive) if good $i$ is a normal good (inferior good, respectively). Therefore, the law of demand, which is equivalent to $\partial D_i(p, I) / \partial p_i<0$, holds true if good $i$ is a normal good.

## 金融代写国际贸易理论代写Theory of International Trade代考| Expenditure minimization

$\min x p x \quad$ subject to $U(x) \geq u$.

## 金融代写|国际贸易理论代写Theory of International Trade代考| Slutsky equation

$$\frac{\partial D_i(p, I)}{\partial p_j}=\frac{\partial \widetilde{D}_i(p, u)}{\partial p_j}-x_j \frac{\partial D_i(p, I)}{\partial I}$$

$-x_j \partial D_i(p, I) / \partial I$ ，是收入效应。如果良好，菖代效果为正 (负) $i$ 而且即好 $j$ 是菖代品 (分别是补品) 。自身价格的菖代效应， $\partial \widetilde{D}_i(p, u) / \partial p_i$ ，始终为负数。我们说好 $i$ 是正常商品 (忩质商品 $g \circ o d$ )如果 $\partial D_i(p, I) / \partial I$ 为阳性（分别为阴性) 。因此，斯卢 茨其方程中的收入效应是负的（正的），如果好的话 $i$ 是正常商品（分别为劣质商品）。因此，需求定律，其等价于 $\partial D_i(p, I) / \partial p_i<0$ ，如果良好，则成立 $i$ 是正常的好东西。

## 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。

Posted on Categories:Theory of International Trade, 国际贸易理论, 金融代写

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Production is a firm’s activity of converting a set of some objects (i.e., inputs) into different kinds of objects (i.e., outputs). In general, inputs are classified into two categories: intermediate goods and primary factors of production (production factors or, simply, factors). Intermediate goods are inputs to a production activity that themselves are the outputs of other production activities. On the other hand, primary factors are productive services provided by such things as labor, capital, and land that are not (considered to be) the outputs of other production activities (at least) within a certain period of time. We have to distinguish the productive services provided by a primary factor from the factor itself as a physical object that generates productive services. For example, when we refer to “capital” as a physical object, it means a stock variable such as machine tools, equipments, and factories, which are indeed the outputs of some production activities completed before the period of time we are concerned. On the other hand, when we refer to “capital” as a production factor, it means the workings or services generated by the physical capital and is considered as a flow variable defined for a certain period of time. ${ }^3$

## 金融代写|国际贸易理论代写Theory of International Trade代考|Description of Technology

Production function: Technology of producing a good is the entirety of firm’s knowledge of how to use and process inputs and how to organize and manage an appropriate system of production in order to obtain outputs efficiently. We can think of production technology as a certain relationship between possible combinations of inputs and the corresponding maximum outputs of the good. For simplicity, we often assume that inputs consist only of primary factors.

Suppose that there are $m$ primary factors indexed by $i=1,2, \ldots, m$. Let $v_i \in \mathbb{R}{+}$be the quantity of the ith primary factor (i.e., factor $i$ ). A combination of factor inputs, $v=\left(v_1, v_2, \ldots, v_m\right) \in \mathbb{R}{+}^m$, is called an input vector. Production technology of producing a certain good can be represented by a non-negative, real-valued function $F: \mathbb{R}{+}^m \rightarrow \mathbb{R}{+}$, which we call a production function:
$$F(v) \equiv F\left(v_1, v_2, \ldots, v_m\right),$$
which represents the maximum amount of the good obtained from putting an input vector $v$ into the production process under the current technology. We say that factor $i$ is indispensable in production if the good cannot be produced without factor $i$, that is, if $v_i=0$ implies $F\left(v_1, \ldots, v_i, \ldots, v_m\right)=0$

## 金融代写国际贸易理论代写Theory of International Trade代考| Description of Technology

$$F(v) \equiv F\left(v_1, v_2, \ldots, v_m\right),$$

## 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。