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# 数学代写|数学建模代写Mathematical Modeling代考|Samuelson’s Modified Investment Model

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## 数学代写|数学建模代写Mathematical Modeling代考|Samuelson’s Modified Investment Model

In this case, the rate of investment is slowed not only by excess capital as before, but it is also slowed by a high investment level so that Eqn. (71) becomes
$$\frac{d k}{d t}=I(t), \frac{d I}{d t}=-m k(t)-n I(t)$$
so that
$$\begin{gathered} I \frac{d I}{d k}+m k(t)+n I(t)=0 \ \frac{d^2 k}{d t^2}+n \frac{d k}{d t}+m k=0 \end{gathered}$$
which are the equations for damped harmonic motion corresponding to the case when a particle performing SHM is acted on by a resistance force proportional to the velocity.

Let $p_r(t), s(t)$, and $d_r(t)$ be the price, supply, and demand of a commodity in the $r$ th market, so that Evan’s price adjustment model mechanism suggests
$$\frac{d p_r}{d t}=-\mu_r\left(s_r-d_r\right), r=1,2, \ldots, n$$
Now we assume that the supply and demand of the commodity in the $r$ th market depends upon its price in all the markets, so that
$$s_r-d_r=c_r+\sum_{s=1}^n d_{r s} p_s$$
Let $p_r(t), s(t)$, and $d_r(t)$ be the price, supply, and demand of a commodity in the $r$ th market, so that Evan’s price adjustment model mechanism suggests
$$\frac{d p_r}{d t}=-\mu_r\left(s_r-d_r\right), r=1,2, \ldots, n$$
Now we assume that the supply and demand of the commodity in the $r$ th market depends upon its price in all the markets, so that
$$s_r-d_r=c_r+\sum_{s=1}^n d_{r s} p_s$$
where $c_r$ ‘s and $d_{r s}$ ‘s are constants. From Eqns. (80) and (81), we get
$$\frac{d p_r}{d t}=-\mu_r\left(c_r+\sum_{s=1}^n d_{r s} p_s\right), r=1,2, \ldots, n$$
If $p_{1 e^2}, p_{2 e}, \ldots, p_{n e}$ are the equilibrium prices in the $n$ markets and
$$p_r=p_r-p_{r e}$$
we get
$$\frac{d p_r}{d t}=-\mu_r \sum_{s=1}^n d_{r s} p_s=\sum_{s=1}^n e_{r s} p_s, r=1,2, \ldots, n$$
where
$$e_{r s}=-\mu_r d_{r s}$$
Substituting $P_r=A_r e^{\lambda t}$ and eliminating $A_1, A_2, \ldots, A_n$ we get
$$|\lambda I-E|=0, E=\left[e_{r s}\right]$$
Thus the equilibrium will be stable if all the eigenvalues of the matrix $E$ have negative real parts.

If $d_{r s}=0$ when $r \neq s$, the markets are independent so that nonzero value of some or all of these $d_{r s}^n$ ‘s introduce dependence among markets.

## 数学代写|数学建模代写Mathematical Modeling代考|Leontief’s Open and Closed Dynamical Systems for Inter-Industry Relations

We consider $n$ industries. Let
$x_{r s}=$ contribution from the $r$ th industry to the $s$ th industry per unit time
$x_r=$ contribution from the $r$ th industry to consumers per unit time
$X_r=$ total output of the $r$ th industry per unit time
$\xi_r=$ input of labor in the $r$ th industry
$p_r=$ price per unit of the product of the $r$ th industry
$w=$ wages per unit of labor per unit time
$Y=$ total labor input into the system
$S_{r s}=$ stock of the product of the $r$ th industry held by the $s$ th industry
$S_r=$ stock of the $r$ th industry
Thus we get the following equations:
(i) From the principle of continuity, the rate of change of stock of the $r$ th industry $=$ excess of the total output of the $r$ th industry per unit time over the contribution of the $r$ th industry to consumers and other industries per unit time, so that
$$\frac{d}{d t} S_r=X_r-x_r-\sum_{s=1}^n x_{r s}$$
and since
$$\begin{gathered} S_r=\sum_{s=1}^n S_{r s} \ \frac{d}{d t} \sum_{s=1}^n S_{r s}=X_r-x_r-\sum_{s=1}^n x_{r s},(r=1,2, \ldots, n) \end{gathered}$$

(ii) Since the total labor input into the system $=$ the sum of labor inputs into all industries, we get
$$Y=\sum_{r=1}^n \xi_r$$
(iii) Assuming the condition of perfect competition and no profit in each industry, we should have for each industry the value of input equal to the value of output so that
$$p_r X_r=\sum_{s=1}^n p_s x_{s r}+w \xi_r(r=1,2, \ldots, n)$$
(iv) We further assume that the input coefficients
$$a_{r s}=\frac{x_{r s}}{X_s}, b_{r s}=\frac{S_{r s}}{X_s}, b_r=\frac{\xi_r}{X_r}(r, s=1,2, \ldots, n)$$
are constants.
We then get the equations
$$\begin{gathered} \frac{d}{d t} \sum_{s=1}^n b_{r s} X_s=X_r-x_r-\sum_{s=1}^n a_{r s} X_s,(r=1,2, \ldots, n) \ Y=\sum_{s=1}^n b_s X_s \ p_r=\sum_{s=1}^n p_s a_{s r}+w b_r,(r=1,2, \ldots, n) \end{gathered}$$

## 数学代写|数学建模代写Mathematical Modeling代考|Samuelson’s Modified Investment Model

$$\frac{d k}{d t}=I(t), \frac{d I}{d t}=-m k(t)-n I(t)$$

$$\begin{gathered} I \frac{d I}{d k}+m k(t)+n I(t)=0 \ \frac{d^2 k}{d t^2}+n \frac{d k}{d t}+m k=0 \end{gathered}$$

$$\frac{d p_r}{d t}=-\mu_r\left(s_r-d_r\right), r=1,2, \ldots, n$$

$$s_r-d_r=c_r+\sum_{s=1}^n d_{r s} p_s$$

$$\frac{d p_r}{d t}=-\mu_r\left(s_r-d_r\right), r=1,2, \ldots, n$$

$$s_r-d_r=c_r+\sum_{s=1}^n d_{r s} p_s$$

$$\frac{d p_r}{d t}=-\mu_r\left(c_r+\sum_{s=1}^n d_{r s} p_s\right), r=1,2, \ldots, n$$

$$p_r=p_r-p_{r e}$$

$$\frac{d p_r}{d t}=-\mu_r \sum_{s=1}^n d_{r s} p_s=\sum_{s=1}^n e_{r s} p_s, r=1,2, \ldots, n$$

$$e_{r s}=-\mu_r d_{r s}$$

$$|\lambda I-E|=0, E=\left[e_{r s}\right]$$

## 数学代写|数学建模代写Mathematical Modeling代考|Leontief’s Open and Closed Dynamical Systems for Inter-Industry Relations

$x_{r s}=$ 来自 $r$ 这个行业 $s$ 单位时间内的工业
$x_r=$ 来自 $r$ 该行业的消费者每单位时间
$X_r=$ 总产出 $r$ 单位时间内的工业
$\xi_r=$ 劳动力的投入 $r$ 行业
$p_r=$ 每单位产品的价格 $r$ 行业
$w=$ 单位劳动每单位时间的工资
$Y=$ 投入系统的总劳动
$S_{r s}=$ 产品的库存 $r$ 该行业持有 $s$ 行业
$S_r=$ 股票 $r$ 行业

(i)从连续性原则来看，……的存货变化率 $r$ 行业 $=$ 总产出的剩余部分 $r$ 该行业单位时间内的贡献 $r$ 该行业与其他行业的消费者每单位时间，使
$$\frac{d}{d t} S_r=X_r-x_r-\sum_{s=1}^n x_{r s}$$

$$\begin{gathered} S_r=\sum_{s=1}^n S_{r s} \ \frac{d}{d t} \sum_{s=1}^n S_{r s}=X_r-x_r-\sum_{s=1}^n x_{r s},(r=1,2, \ldots, n) \end{gathered}$$

(ii)自投入系统的总劳动以来 $=$ 我们得到所有产业的劳动投入之和
$$Y=\sum_{r=1}^n \xi_r$$
(3)假设在完全竞争的条件下，每个行业都没有利润，则每个行业的投入价值应该等于产出价值，从而使
$$p_r X_r=\sum_{s=1}^n p_s x_{s r}+w \xi_r(r=1,2, \ldots, n)$$
(iv)我们进一步假设输入系数
$$a_{r s}=\frac{x_{r s}}{X_s}, b_{r s}=\frac{S_{r s}}{X_s}, b_r=\frac{\xi_r}{X_r}(r, s=1,2, \ldots, n)$$

$$\begin{gathered} \frac{d}{d t} \sum_{s=1}^n b_{r s} X_s=X_r-x_r-\sum_{s=1}^n a_{r s} X_s,(r=1,2, \ldots, n) \ Y=\sum_{s=1}^n b_s X_s \ p_r=\sum_{s=1}^n p_s a_{s r}+w b_r,(r=1,2, \ldots, n) \end{gathered}$$

## MATLAB代写

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