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# 数学代写|数值方法作业代写numerical methods代考|Dynamic Structural Models

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## 数学代写|数值方法作业代写numerical methods代考|Dynamic Behavior of Endogenous Variables

Let us suppose that, starting from initial values for output $Y_{0}, Y_{-1}$, government expenditures were fixed at a given value $G^{}, G_{t}=G^{} \forall t$. Even then, output would not be constant, in general. In fact, we would have
\begin{aligned} &Y_{1}=\left(\alpha_{2}+\beta_{2}\right) Y_{0}-\beta_{2} Y_{-1}+\left(\alpha_{1}+\beta_{1}\right)+G^{} \ &Y_{2}=\left(\alpha_{2}+\beta_{2}\right) Y_{1}-\beta_{2} Y_{0}+\left(\alpha_{1}+\beta_{1}\right)+G^{} \ &Y_{3}=\left(\alpha_{2}+\beta_{2}\right) Y_{2}-\beta_{2} Y_{1}+\left(\alpha_{1}+\beta_{1}\right)+G^{} \end{aligned} and whether output converges or explodes, i.e., whether it is stable or unstable, and whether it displays oscillations or not, depends just on the values of $\alpha_{2}$ and $\beta_{2}$. It is interesting to point out that there is an equilibrium value of output, defined precisely as that level of output such that if the economy started there, it would never move away from it. When it exists, that point is also called the steady-state of the system. This equilibrium level can in fact be easily obtained. To do so, we assume output to be constant over time in (1.9), to obtain $$Y^{}=\frac{\left(\alpha_{1}+\beta_{1}\right)+G^{}}{1-\alpha_{2}}$$ which can be seen to be directly related to the level chosen for government expenditures. Corresponding to these equilibrium values of government expenditures and output, there would be associated equilibrium values for private consumption and investment: $C^{}=\alpha_{1}+\alpha_{2} \frac{\left(\alpha_{1}+\beta_{1}\right)+G^{}}{1-\alpha_{2}}, I^{}=\beta_{1}$. An economy could stay at equilibrium values $G^{}, Y^{}, C^{}, I^{}$ forever.

## 数学代写|数值方法作业代写numerical methods代考|Dynamic Multipliers

In the response of an endogenous variable to a change in the value of an exogenous variable, we distinguish between the initial effect (the impact multiplier), the response over time (the dynamic multipliers), and the aggregate response over time (the total long-run multiplier). We must also distinguish between the response to a transitory change in an exogenous variable and the response to a permanent change. In the case of the multiplier-accelerator model, the second order difference output equation can be written as
$$Y_{t}=\left(\alpha_{2}+\beta_{2}\right) Y_{t-1}-\beta_{2} Y_{t-2}+\left(\alpha_{1}+\beta_{1}\right)+G_{t},$$
which in first differences becomes
$$\Delta Y_{t}=\left(\alpha_{2}+\beta_{2}\right) \Delta Y_{t-1}-\beta_{2} \Delta Y_{t-2}+\Delta G_{t},$$
as can be seen by subtracting the versions of Eq. (1.10) corresponding to time $t$ and $t-1$.

## 数学代写|数值方法作业代写NUMERICAL METHODS代考|DYNAMIC BEHAVIOR OF ENDOGENOUS VARIABLES

\begin{aligned} &Y_{1}=\left(\alpha_{2}+\beta_{2}\right) Y_{0}-\beta_{2} Y_{-1}+\left(\alpha_{1}+\beta_{1}\right)+G^{} \ &Y_{2}=\left(\alpha_{2}+\beta_{2}\right) Y_{1}-\beta_{2} Y_{0}+\left(\alpha_{1}+\beta_{1}\right)+G^{} \ &Y_{3}=\left(\alpha_{2}+\beta_{2}\right) Y_{2}-\beta_{2} Y_{1}+\left(\alpha_{1}+\beta_{1}\right)+G^{} \end{aligned} and whether output converges or explodes, i.e., whether it is stable or unstable, and whether it displays oscillations or not, depends just on the values of $\alpha_{2}$ and $\beta_{2}$. It is interesting to point out that there is an equilibrium value of output, defined precisely as that level of output such that if the economy started there, it would never move away from it. When it exists, that point is also called the steady-state of the system. This equilibrium level can in fact be easily obtained. To do so, we assume output to be constant over time in (1.9), to obtain $$Y^{}=\frac{\left(\alpha_{1}+\beta_{1}\right)+G^{}}{1-\alpha_{2}}$$ which can be seen to be directly related to the level chosen for government expenditures. Corresponding to these equilibrium values of government expenditures and output, there would be associated equilibrium values for private consumption and investment: $C^{}=\alpha_{1}+\alpha_{2} \frac{\left(\alpha_{1}+\beta_{1}\right)+G^{}}{1-\alpha_{2}}, I^{}=\beta_{1}$. An economy could stay at equilibrium values $G^{}, Y^{}, C^{}, I^{}$ f 永远。

## 数学代写|数值方法作业代写NUMERICAL METHODS代考|DYNAMIC MULTIPLIERS

Δ是吨=(一种2+b2)Δ是吨−1−b2Δ是吨−2+ΔG吨,

## 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。