本文为《Linear algebra and its applications》的读书笔记

The economic model described in this section is the basis for more elaborate models used in many parts of the world.

The leontief input-output model

Suppose a nation’s economy is divided into $n$n sectors that produce goods or services, and let $\boldsymbol x$x be a production vector (产出向量) in $\mathbb R^n$Rn that lists the output of each sector for one year. Also, suppose another part of the economy (called the open sector) does not produce goods or services but only consumes them, and let $\boldsymbol d$d be a final demand vector (最终需求向量) (or bill of final demands (最终需求账单)) that lists the values of the goods and services demanded from the various sectors by the nonproductive part of the economy.

As the various sectors produce goods to meet consumer demand, the producers themselves create additional intermediate demand (中间需求) for goods they need as inputs for their own production. The interrelations between the sectors are very complex, and the connection between the final demand and the production is unclear. Leontief asked if there is a production level $\boldsymbol x$x such that the amounts produced (or “supplied”) will exactly balance the total demand for that production, so that

The basic assumption of Leontief’s input–output model is that for each sector, there is a unit consumption vector (单位消费向量) in $\mathbb R^n$Rn that lists the inputs needed $per$per $unit$unit $of$of $output$output of the sector. All input and output units are measured in millions of dollars, rather than in quantities such as tons or bushels. (Prices of goods and services are held constant.)

As a simple example, suppose the economy consists of three sectors—manufacturing, agriculture, and services—with unit consumption vectors $\boldsymbol c_1$c1​, $\boldsymbol c_2$c2​, and $\boldsymbol c_3$c3​, as shown in the table that follows.

If manufacturing decides to produce $x_1$x1​ units of output, then $x_1\boldsymbol c_1$x1​c1​ represents the $intermediate$intermediate $demands$demands of manufacturing. Likewise, if $x_2$x2​ and $x_3$x3​ denote the planned outputs of the agriculture and services sectors, $x_2\boldsymbol c_2$x2​c2​ and $x_3\boldsymbol c_3$x3​c3​ list their corresponding intermediate demands. The total intermediate demand from all three sectors is given by

where $C$C is the consumption matrix (消耗矩阵) $[\ \ \boldsymbol c_1\ \ \ \boldsymbol c_2\ \ \ \boldsymbol c_3\ \ ]$[  c1​   c2​   c3​  ], namely,

Equations (1) and (2) yield Leontief’s model.

Equation (4) may also be written as $I\boldsymbol x-C\boldsymbol x=\boldsymbol d$Ix−Cx=d, or
$(I-C)\boldsymbol x=\boldsymbol d\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$(I−C)x=d                (5)

If the matrix $I - C$I−C is invertible, then from the equation $(I-C)\boldsymbol x=\boldsymbol d$(I−C)x=d obtain $\boldsymbol x=(I-C)^{-1}\boldsymbol d$x=(I−C)−1d.

The theorem below shows that in most practical cases, $I - C$I−C is invertible and the production vector $\boldsymbol x$x is economically feasible, in the sense that the entries in $\boldsymbol x$x are nonnegative. In the theorem, the term column sum denotes the sum of the entries in a column of a matrix. Under ordinary circumstances, the column sums of a consumption matrix are less than 1 because a sector should require less than one unit’s worth of inputs to produce one unit of output.

The following discussion will suggest why the theorem is true and will lead to a new way to compute $(I-C)^{-1}$(I−C)−1.

A Formula for $(I-C)^{-1}$(I−C)−1

Imagine that the demand represented by $\boldsymbol d$d is presented(提供) to the various industries at the beginning of the year, and the industries respond by setting their production levels at $\boldsymbol x = \boldsymbol d$x=d, which will exactly meet the final demand. As the industries prepare to produce $\boldsymbol d$d, they send out orders for their raw materials and other inputs. This creates an intermediate demand of $C\boldsymbol d$Cd for inputs.

To meet the additional demand of $C\boldsymbol d$Cd, the industries will need as additional inputs the amounts in $CC\boldsymbol d=C^2\boldsymbol d$CCd=C2d. Of course, this creates a second round of intermediate demand, and when the industries decide to produce even more to meet this new demand, they create a third round of demand, namely, $C(C^2\boldsymbol d)=C^3\boldsymbol d$C(C2d)=C3d. And so it goes.

Theoretically, this process could continue indefinitely, although in real life it would not take place in such a rigid sequence of events. We can diagram this hypothetical situation as follows:

The production level $\boldsymbol x$x that will meet all of this demand is

To make sense of equation (6), consider the following algebraic identity:

It can be shown that if the column sums in $C$C are all strictly less than 1, then $I - C$I−C is invertible, $C^m$Cm approaches the zero matrix as $m$m gets arbitrarily large, and $I - C^{m+1}\rightarrow I$I−Cm+1→I . (This fact is analogous to the fact that if a positive number $t$t is less than 1, then $t^m \rightarrow 0$tm→0 as $m$m increases.) Using equation (7), write

The approximation in (8) means that the right side can be made as close to $(I - C)^{-1}$(I−C)−1 as desired by taking $m$m sufficiently large.

In actual input–output models, powers of the consumption matrix approach the zero matrix rather quickly. So (8) really provides a practical way to compute $(I - C)^{-1}$(I−C)−1. Likewise, for any $\boldsymbol d$d, the vectors $C^m\boldsymbol d$Cmd approach the zero vector quickly, and (6) is a practical way to solve $(I - C)\boldsymbol x =\boldsymbol d$(I−C)x=d. If the entries in $C$C and $\boldsymbol d$d are nonnegative, then (6) shows that the entries in $\boldsymbol x$x are nonnegative, too.

The Economic Importance of Entries in $(I - C)^{-1}$(I−C)−1

The entries in $(I - C)^{-1}$(I−C)−1 can be used to predict how the production $\boldsymbol x$x will have to change when the final demand $\boldsymbol d$d changes. In fact, the entries in column $j$j of $(I - C)^{-1}$(I−C)−1 are the increased amounts the various sectors will have to produce in order to satisfy an increase of 1 unit in the final demand for output from sector $j$j .