# 投入产出分析

Wassily Wassilyovich Leontief (1906-1999)

## 投入产出表

1 z11 ... z1j ... z1N f1 x1
2 z21 ... z2j ... z2N f2 x2
... ... ... ... ... ... ... ...
N zN1 ... zNj ... zNN fN xN

$\displaystyle{ x_i=\sum_{k=1}^{N}z_{ik}+f_{i} }$

(1)

## 技术系数矩阵

$\displaystyle{ a_{ij}=\frac{z_{ij}}{x_j} }$

(2)

$\displaystyle{ z_{ij}=a_{ij}x_{j} }$

## Leontief矩阵

$\displaystyle{ \left\{\begin{array}{ll} x_1=&a_{1,1}x_1+a_{1,2}x_2+\cdot\cdot\cdot+a_{1,N}x_N\\ x_2=&a_{2,1}x_1+a_{2,2}x_2+\cdot\cdot\cdot+a_{2,N}x_N\\ \cdot\cdot\cdot&\cdot\cdot\cdot\\ x_N=&a_{N,1}x_1+a_{N,2}x_2+\cdot\cdot\cdot+a_{N,N}x_N\\ \end{array} \right. }$

$\displaystyle{ X=AX+F }$

$\displaystyle{ X=(I-A)^{-1}\cdot F }$

(3)

## 投入产出分析

$\displaystyle{ X+\Delta X=(I-A)^{-1}\cdot (F+\Delta F)=L F + L \Delta F }$

$\displaystyle{ \Delta X= L \Delta F }$

### 举例分析

Sector 1 Sector 2 Final Demand (fi) Total Output (xi)
Sector 1 150 500 350 1000
Sector 2 200 100 1700 2000

$\displaystyle{ A= \left( \begin{array}{cc} 0.15 & 0.25 \\ 0.20 & 0.05 \\ \end{array} \right) }$

$\displaystyle{ L= \left( \begin{array}{cc} 1.2541 & 0.3300 \\ 0.2640 & 1.1221 \\ \end{array} \right) }$

Znew Sector 1 Sector 2 Final Demand (fi) Total Output (xi)
Sector 1 187.13 460.40 600 1247.52
Sector 2 249.50 92.08 1500 1841.58

## 投入产出模型与马尔科夫链

### 谬误的比较

$\displaystyle{ m_{ij}=f_{ij}/\sum_{j=1}^{N+1}f_{ij} }$

$\displaystyle{ a_{ij}=z_{ij}/x_j }$

### 正确的比较

1 0 z1,1 z2,1 ... zN,1 x1k=1Nzk,1
... ... ... ... ... ... ...
j 0 z1,j z2,j ... zN,j xjk=1Nzk,j
... ... ... ... ... ... ...
N 0 z1,N z2,N ... zN,N xNk=1Nzk,N

#### 技术矩阵与马尔科夫矩阵

$\displaystyle{ m_{ij}=f_{ij}/\sum_{j=1}^{N+1}f_{ij} }$

$\displaystyle{ m_{ij}=z_{ji}/(\sum_{j=1}^{N}z_{ji}+x_i-\sum_{j=1}^{N}z_{ji})=z_{ji}/x_i }$

$\displaystyle{ M=A^T }$

## 参考文献

1. Miller, Ronald E.. Input–Output Analysis. Cambridge University Press.