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# 数学代写|数值方法作业代写numerical methods代考|The Deterministic, Discrete-Time Solow–Swan Model

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## 数学代写|数值方法作业代写numerical methods代考|The Exact Solution

Theoretical models are built not only to analyze a variety of positive and normative issues, but also to be confronted with actual data, in an attempt to validate their implications. The continuous-time version of the Solow-Swan model can be used to produce time series for physical capital, output, consumption and investment by sampling at discrete points in time, from the continuous time processes obtained from (2.30) and the implied expressions for the remaining variables. Discrete sampling amounts to giving discrete values: $t=1,2,3, \ldots$ to the time index in those expressions. This apparently innocuous procedure is subject, however, to potential pitfalls, that will be illustrated numerically in the next chapter.

An alternative method consists on analyzing directly the discrete version of the Solow-Swan model. To do so, we could think of directly translating the law of motion into discrete time by substituting a time difference $k_{t+1}-k_{t}$ for the time derivative $\dot{k}{t}$, like in: $$k{t+1}-k_{t}=s f\left(k_{t}\right)-(n+\delta+\gamma) k_{t}$$

## 数学代写|数值方法作业代写numerical methods代考|Approximate Solutions to the Discrete-Time Model

As the continuous-time model, the discrete-time version of the Solow-Swan economy can be solved exactly through the use of (2.36), as we will show in a section below. That is an exception, since nonlinearities in growth models will usually preclude the existence of an exact solution. To familiarize the reader with that practice, we proceed in this section to obtain the solution to the linear and the quadratic approximations to the model.

Considering the nonlinear difference equation in (2.36) as a function $k_{t+1}=$ $\Psi\left(k_{t} ; \theta\right)$ and using Taylor’s expansion and (2.38), the linear approximation to that equation around steady-state is,
\begin{aligned} k_{t+1}-k_{s s}=& \Psi\left(k_{s s}\right)+\left(\frac{\partial \Psi\left(k_{t} ; \theta\right)}{\partial k_{t}}\right){s s}\left(k{t}-k_{s s}\right) \ \Rightarrow k_{t+1} \simeq &\left(\frac{1}{(1+n)(1+\gamma)} f\left(k_{s s}\right)+\frac{1-\delta}{(1+n)(1+\gamma)} k_{s s}\right) \ &+\left(\frac{1}{(1+n)(1+\gamma)} s f^{\prime}\left(k_{s s}\right)+\frac{1-\delta}{(1+n)(1+\gamma)}\right)\left(k_{t}-k_{s s}\right) \ =& k_{s s}+\frac{s f^{\prime}\left(k_{s s}\right)+(1-\delta)}{(1+n)(1+\gamma)}\left(k_{t}-k_{s s}\right) \end{aligned}

## 数学代写|数值方法作业代写NUMERICAL METHODS代考|Numerical Exercise: Solving the Deterministic Solow–Swan Model

In the Discrete spreadsheet in the Solow_deterministic.xls file, time series are obtained for a deterministic, discrete-time version of the Solow-Swan economy from an initial capital stock of $k_{0}=20$. Aggregate technology is supposed to be of the Cobb-Douglas type, with a capital share of $\alpha=0.36$, and a technological constant $A=5.0$. Depreciation of physical capital is $\delta=7.5 \%$, savings are $36.0 \%$ of output each period, and we assume zero population growth, $n=0$. Since the savings rate is equal to the output elasticity of capital, the steady-state in this economy will be the Golden Rule. ${ }^{16}$ With these parameter values, steady state levels turn out to be: $k_{s s}=117.94, y_{s s}=27.85, c_{s s}=17.82, s_{s s}=i_{s s}=10.02$. Therefore, the economy starts to the left of the steady-state, with a stock of capital well below the steady-state level. The constant savings rate is relatively high, and capital accumulates quickly because the level of savings initially exceeds from total depreciation expenditures. ${ }^{17}$ After 16 periods, the economy has covered half the initial distance to steady-state, with a stock of capital above 70 units. The Discrete spreadsheet presents time series for 260 periods, and the discrete time model is solved using the exact solution (2.36), as well as using the solutions to the linear and quadratic approximations $(2.43),(2.44)$ to the discrete-time model. The resulting time series for the stock of capital under the different approaches are reported in the first panel. The time series for output, savings and consumption that are obtained under the exact solution are shown in Panel 2, while Panels 3 and 4 display the similar time series obtained under the linear and quadratic approximations to the model. Notice that, according to the model, output is obtained each period from the stock of capital accumulated at the end of the previous period. As in subsequent exercises, this is organized in the spreadsheet by making output to be a function of the stock of capital in the previous row. That is, in the row corresponding to time $t$ we have $k_{t+1}$ and variables like $y_{t}, c_{t}$. (The same exercise can be reproduced by MATLAB file: Solow_stochastic.m by setting the variance parameter sigmae to zero.) Consumers’ preferences do not play any role in this exercise. Nevertheless, to familiarize the reader with the type of welfare evaluation that will often be performed in the next chapters, consumers are supposed to have a constant relative risk aversion utility function, $U\left(c_{t}\right)=\frac{c_{t}^{1-\sigma}-1}{1-\sigma}$, with risk aversion coefficient of $\sigma=3.0$, and a time discount factor $\beta=0.95$, and we compute single-period as well as time-aggregate, discounted utility.

$$## 数学代写|数值方法作业代写NUMERICAL METHODS代考|APPROXIMATE SOLUTIONS TO THE DISCRETE-TIME MODEL 作为连续时间模型，Solow-Swan 经济的离散时间版本可以通过使用2.36，正如我们将在下面的部分中展示的那样。这是一个例外，因为增长模型中的非线性通常会排除精确解的存在。为了让读者熟悉这种做法，我们在本节中继续获得模型的线性和二次近似的解。 考虑非线性差分方程2.36作为一个函数ķ吨+1= Ψ(ķ吨;θ)并使用泰勒展开式和2.38，该方程在稳态附近的线性近似是，$$
\begin{aligned}
k_{t+1}-k_{s s}=& \Psi\left(k_{s s}\right)+\left(\frac{\partial \Psi\left(k_{t} ; \theta\right)}{\partial k_{t}}\right){s s}\left(k{t}-k_{s s}\right) \
\Rightarrow k_{t+1} \simeq &\left(\frac{1}{(1+n)(1+\gamma)} f\left(k_{s s}\right)+\frac{1-\delta}{(1+n)(1+\gamma)} k_{s s}\right) \
&+\left(\frac{1}{(1+n)(1+\gamma)} s f^{\prime}\left(k_{s s}\right)+\frac{1-\delta}{(1+n)(1+\gamma)}\right)\left(k_{t}-k_{s s}\right) \
=& k_{s s}+\frac{s f^{\prime}\left(k_{s s}\right)+(1-\delta)}{(1+n)(1+\gamma)}\left(k_{t}-k_{s s}\right)
\end{aligned}


## 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。