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数学代写|数学建模代写Mathematical Modeling代考|Samuelson’s Interaction Models

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数学代写|数学建模代写Mathematical Modeling代考|Samuelson’s Interaction Models

The basic equations for the first interaction model are:
$$Y(t)=C(t)+I(t), C(t)=\alpha Y(t-1), I(t)=\beta[C(t)-C(t-1)]$$

Here the positive constant $\alpha$ is the marginal propensity to consume with respect to income of the previous year and the positive constant $\beta$ is the relation given by the acceleration principle, i.e., $\beta$ is the increase in investment per unit of excess of this year’s consumption over the last year’s.
From Eqn. (75), we get the second order difference equation
$$Y(t)-\alpha(1+\beta) Y(t-1)+\alpha \beta Y(t-2)=0$$
In the second interaction model, there is an additional investment by the government and this investment is assumed to be a constant $\gamma$. In this case $(76)$ is modified to
$$Y(t)-\alpha(1+\beta) Y(t-1)+\alpha \beta Y(t-2)-\gamma=0$$
The solution of Eqns. (76) and (77) can show either an increasing trend in $Y(t)$, a decreasing trend in $Y(t)$, or an oscillating trend in it.

数学代写|数学建模代写Mathematical Modeling代考|Application to Actuarial Science

One important aspect of actuarial science is what is called mathematics of finance or mathematics of investment.

If a sum $S_0$ is invested at compound interest of $i$ per unit amount per unit time and $S_{\mathrm{t}}$ is the amount at the end of time $t$, then we get the difference equation
$$S_{t+1}=S_t+i S_t=(1+i) S_t$$
which has the solution
$$S_t=S_0(1+i)^t$$
which is the well-known formula for compound interest.
Suppose a person borrows a sum $S_0$ at compound interest $i$ and wants to amortize his dept, i.e., he wants to pay the amount and interest back by payment of $n$ equal installments, say $R$, the first payment to be made at the end of the first year.
Let $S_t$ be the amount due at the end of $t$ years, then we have the difference equation
$$S_{t+1}=S_t+i S_t-R=(1+i) S_t-R$$
Its solution is
\begin{aligned} S_t & =\left(S_0-\frac{R}{i}\right)(1+i)^t+\frac{R}{i} \ & =S_0(1+t)^t-R \frac{(1+i)^t-1}{i} \end{aligned}
If the amount is paid back in $n$ years, $S_n=0$, so that
$$R=S_0 \frac{i}{1-(1+i)^{-n}}=S_0 \frac{1}{a n T_i}$$
where, $a n i$ called the amortization factor is the present value of an annuity of 1 per unit time for $n$ periods at an interest rate $i$.

数学代写|数学建模代写Mathematical Modeling代考|Samuelson’s Interaction Models

$$Y(t)=C(t)+I(t), C(t)=\alpha Y(t-1), I(t)=\beta[C(t)-C(t-1)]$$

$$Y(t)-\alpha(1+\beta) Y(t-1)+\alpha \beta Y(t-2)=0$$

$$Y(t)-\alpha(1+\beta) Y(t-1)+\alpha \beta Y(t-2)-\gamma=0$$

数学代写|数学建模代写Mathematical Modeling代考|Application to Actuarial Science

$$S_{t+1}=S_t+i S_t=(1+i) S_t$$

$$S_t=S_0(1+i)^t$$

$$S_{t+1}=S_t+i S_t-R=(1+i) S_t-R$$

\begin{aligned} S_t & =\left(S_0-\frac{R}{i}\right)(1+i)^t+\frac{R}{i} \ & =S_0(1+t)^t-R \frac{(1+i)^t-1}{i} \end{aligned}

$$R=S_0 \frac{i}{1-(1+i)^{-n}}=S_0 \frac{1}{a n T_i}$$

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。