投入产出矩阵分析的主要思想小结

Leontief的投入产出分析考虑了一个封闭（如果把最终消费和劳动力投入对应着的部门看成外生的，则这个系统是开放的）的多部门经济系统之间的投入关系，然后选择对某一个部门或者产品（通常是居民，或者说最终消费品，原则上可以任意选取）的投入做一个直接与间接影响分析：假设期望这个部门（例如居民）的消费部门中的某部门有一个增量，则所有部门应该如何变化；假设这个部门（例如居民）对某部门的投入有所增加，则所有部门会如何变化。

Define (x^{i}{j}) to be the quantity of product i in terms of a product unit from product i to product j, in short, (X^{From}{To}).
[\left[\begin{array}{cccc}x^{1}{1}& \cdots & x^{1}{N-1} & x^{1}{N}=y^{1} \ &\cdots&&\x^{N-1}{1}& \cdots & x^{N-1}{N-1} & x^{N-1}{N}=y^{N-1} \x^{N}{1}=y{1}& \cdots & x^{N}{N-1}=y{N-1} & x^{N}{N}=y^{N}=y{N} \end{array}\right] \hspace{2cm} (1)]
represents the full relation among all products in an economy.

Let us define also unit mass of every product, (M^{i}), and price of one product unit is (P^{i}).
[\hat{x}^{i}{j}=M^{i}x^{i}{j} \hspace{1cm} \mbox{ and } \hspace{1cm} \tilde{x}^{i}{j}=P^{i}x^{i}{j} \hspace{2cm} (2)]
One thing should be emphasized that whenever there is intellectual input/output, assuming it is the product (N), there is no well defined (M^{N}) for that and there is not even a good quantity (x^{N}_{j}) for that. We will keep this issue in our mind and just proceed from here anyway. If one is interested in price of a product per unit mass, then it can be calculated easily that (p^{i}=P^{i}/M^{i}) .

For convenience, we also define total output from product (i) and total input to product (i), [X^{i}=\sum_{j}x^{i}{j}\ \mbox{ , } X{i}=\frac{1}{M^{i}}\sum_{j}M^{j}x^{j}{i} \mbox{ and } X{i}=\frac{1}{P^{i}}\sum_{j}P^{j}x^{j}_{i}. \hspace{2cm} (3)]

For economics, since all products including populations and import/export, for every product the total input to that product equals to the total output from that product: (X^{i}=X_{i}), from which we have
[\sum_{j=1}^{N} M^{i}x^{i}{j} = \sum{j=1}^{N} \hat{x}^{i}{j} = \sum{j=1}^{N} \hat{x}^{j}{i} = \sum{j=1}^{N} M^{j}x^{j}{i}, \hspace{2cm} (4-1)]
[\sum{j=1}^{N} P^{i}x^{i}{j} = \sum{j=1}^{N} \tilde{x}^{i}{j} = \sum{j=1}^{N} \tilde{x}^{j}{i} = \sum{j=1}^{N} P^{j}x^{j}_{i}. \hspace{2cm} (4-2)]

However, there are other systems, where (X^{i}\neq X_{i}) in for example, flow of ideas and creativity and flow of happiness.

Now let us assume that we are focusing on the (N)th product (Because that (P^{N}) and (M^{N}) are not well defined, or because that we want to study impact of sector (N)), ie. we want to separate (x^{N}{j}) and (x^{j}{N}) from other (x^{i}{j}). Let us even give them a different name (y^{i}=x^{i}{N}) and (y_{i}=x_{i}^{N}). Those two are different.

Using those (y^{i}) and (X^{i}) we can write the output relation as,
[\sum_{j=1}^{N-1} x^{i}{j} + y^{i} = X^{i}, \forall i \neq N. \hspace{2cm} (5)]
Define (B^{i}{j}=\frac{x^{i}{j}}{X^{j}}) as the required product (i) in its own unit for producing one unit of product (j), then
[\sum{j=1}^{N-1} B^{i}{j}X^{j} + y^{i} = X^{i}, \forall i \neq N, \hspace{2cm} (6)] which can be written as
[BX+Y=X. \hspace{2cm} (7)]
If there is an expected (\Delta Y), after assuming all the coefficients are not changed (which implies that techniques and organization of production are the same), then
[\Delta X=(I-B)^{-1}\Delta Y = \sum{n=0}^{\infty} B^{n}\Delta Y, \hspace{2cm} (8)]
which has a very intuitive explanation as direct and indirect effect of (\Delta Y): (B \Delta Y) is the direct input and (BB\Delta Y) is an induced input and in principle induced input at all orders should be considered.

Let us now turn to consider input relation. One might guess that we should have
[\sum_{j=1}^{N-1}x^{j}{i} + y{i} = X_{i}, ] which is however not valid since (x^{j}{i}) and (x^{k}{i}) do not share a common unit. Therefore, we have to consider the input relation in terms of materials and money, thus
[\sum_{j=1}^{N-1} M^{j}x^{j}{i} + M^{N}y{i} = M^{i}X_{i}, \forall i \neq N, \hspace{2cm} (9-1)]
[\sum_{j=1}^{N-1} P^{j}x^{j}{i} + P^{N}y{i} = P^{i}X_{i}, \forall i \neq N. \hspace{2cm} (9-2)]
Define (F^{j}{i}=\frac{x^{j}{i}}{X^{j}}) as the input to product (i) from per unit of product (j), then Equ(9-2) becomes
[\sum_{j=1}^{N-1} F^{j}{i}P^{j}X^{j} + P^{N}y{i} = P^{i}X_{i}, \forall i \neq N, \hspace{2cm} (10)] which, under the condition of (X^{j}=X_{j}), leads to
[F^{T}\left(P.X\right)+\left(P^{N}Y\right)=\left(P.X\right), \hspace{2cm} (11) ] which in turn leads to [\Delta P.X=(I-F^{T})^{-1}\Delta P^{N}Y = \sum_{n=0}^{\infty} \left(F^{T}\right)^{n}\Delta P^{N}Y, \hspace{2cm} (12)] where (P.X) is the element-wise dot product between two column vectors (inner product is denoted as (P\cdot X) or (P^{T}X)).

Equ(8) and Equ(12) have different meanings: Equ(12) answers the question that given (\Delta y_{i}= \Delta x^{N}{i}), the input from product (N) to product (i), what will be the effect after such an initial momentum on other products, while Equ(8) is about that in order to reach an increase of (\Delta y^{i}= \Delta x{N}^{i}), additional input from product (i) to product (N), how much other products will end up with. Let us denote them as respectively (L^{B}=\left(1-B\right)^{-1}) and (L^{F}=\left(1-F\right)^{-1}). Here (B) and (F) refers to respectively Backwards and Forwards. (L) comes from the name of those matrices, the Leontief inverse.

Limitation:

1. Increasing on product \(i\) is assumed to lead to additional producing of product \(j\), while in reality, there is a problem of matching up: if a product \(k\) is required too in producing \(j\), this simply increasing on \(i\) should not have any effect on \(j\). However, Input-Output analysis ignore this matching problem.
2. Some systems do not have \(X_{i}=X^{i}\) and the current Input-Output analysis does not apply to those systems.