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# 数学代写|数值方法作业代写numerical methods代考|The Neoclassical Growth Model of Solow and Swan

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## 数学代写|数值方法作业代写numerical methods代考|Description of the Model

We assume that at the aggregate level, the available technology can be represented by a first-degree homogeneous production function $Y=F\left(K_{t}, N_{t}\right)$. As explained above, we identify employment with total population. Derivatives are: $F_{K_{t}}, F_{N_{t}}$, $F_{K_{t} N_{t}}>0, F_{N_{t} N_{t}}, F_{K_{t} K_{t}}<0$ and the Hessian is negative definite, so that $F$ is concave. We further assume: $F\left(K_{t}, 0\right)=F\left(0, N_{t}\right)=0$, so that we cannot produce anything without using positive amounts of the two inputs, and $\lim {K{t} \rightarrow 0} F_{K_{t}}=$ $\lim {N{t} \rightarrow 0} F_{N_{t}}=\infty, \lim {K{t} \rightarrow \infty} F_{K_{t}}=\lim {N{t} \rightarrow \infty} F_{N_{t}}=0$. These are usually known as Inada conditions.

The more restrictive aspect of this technology is the existence of decreasing returns to scale in each input, which, as we saw in Case 1 in the previous section, precludes the possibility of positive steady-state growth. The aggregate constant returns to scale assumption allows us to write,
$$Y_{t}=F\left(K_{t}, N_{t}\right)=N_{t} F\left(K_{t} / N_{t}, 1\right)=N_{t} f\left(k_{t}\right)$$
where $k_{t}=K_{t} / N_{t}$ denotes the per capita stock of productive capital or capital-labor ratio, and $f\left(k_{t}\right)=F\left(K_{t} / N_{t}, 1\right)$. The assumptions on $F$ imply: $f^{\prime}\left(k_{t}\right)>0, f^{\prime \prime}\left(k_{t}\right)<0, f(0)=0, \lim {k{t} \rightarrow 0} f^{\prime}=\infty, \lim {k{t} \rightarrow \infty} f^{\prime}=0 .$

## 数学代写|数值方法作业代写numerical methods代考|The Dynamics of the Economy

In this simple economy, output (or, equivalently, income) is used either as consumption or in the form of gross investment. The later is used in part to compensate for depreciated capital, and also as net additions to the stock of capital,

\begin{aligned} \text { Net investment } &=\dot{K}{t}=\frac{d K{t}}{d t}=\text { Gross investment }-\text { Depreciation } \ &=I_{t}-D_{t}=I_{t}-\delta K_{t}, \end{aligned}
where we have used the assumption on a constant rate $\delta$ of physical capital depreciation, independent of the stock of capital, $D_{t}=\delta K_{t}$.
So, we have the global constraint of resources:
$$Y_{t}=C_{t}+I_{t}=C_{t}+\dot{K}{t}+\delta K{t},$$
that is,
$$\dot{K}{t}=F\left(K{t}, N_{t}\right)-C_{t}-\delta K_{t}$$

n an exogenous growth economy, a steady-state is a vector of values for the rates of growth of the main variables physical capital, output and consumption in units of effective labor, that if it is ever reached, it can be maintained constant forever.

A steady-state is often referred to as a long-run equilibrium, because of the characteristic of having a constant rate of growth for appropriately defined variables.
Let us consider again the economy’s law of motion $(2.19)$, from which the growth rate of capital can be written,
$$\gamma_{k_{t}}=\frac{\dot{k}{t}}{k{t}}=s \frac{f\left(k_{t}\right)}{k_{t}}-(n+\delta+\gamma)$$

## 数学代写|数值方法作业代写NUMERICAL METHODS代考|THE DYNAMICS OF THE ECONOMY

\begin{aligned} \text { Net investment } &=\dot{K}{t}=\frac{d K{t}}{d t}=\text { Gross investment }-\text { Depreciation } \ &=I_{t}-D_{t}=I_{t}-\delta K_{t}, \end{aligned}

$$Y_{t}=C_{t}+I_{t}=C_{t}+\dot{K}{t}+\delta K{t},$$
that is,
$$\dot{K}{t}=F\left(K{t}, N_{t}\right)-C_{t}-\delta K_{t}$$

$$\gamma_{k_{t}}=\frac{\dot{k} {t}}{k {t}}=s \frac{f\leftk_{t}\对k_{t}\对}{k_{t}}-n+d+C$$

## 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。