Consumption, matrix ; Demand and production vectors

The idea of Leontief Input Output Model is based on a matrix $C$ which is called CONSUMPTION MATRIX . The $(i, j)$ entry of $C$,( that is $c_{i,j}$ ) is the ( monetary value of ) output from industry $i$ which is needed to produce one unit ( of monetary value ) of output of industry $j$.

In other words if industry $j$ wants to produce one unit of its own product, it needs to consume $c_{ij}$ units of the output of the industry $i$.

Suppose that we have $n$ industries. and each produce $x_i$ units. The production vector $p$ is defined as

$$p= \left[ \begin{array}{c} x_1\\ x_2\\ \vdots \\ x_n\\ \end{array} \right]$$

The consumption by industries will be $Cp$. And, $p-Cp = (I-C) p$ is the amount available for external use.

Lets define the demand, $d$, from outside of the system as

$$d = \left[ \begin{array}{c} d_1\\ d_2\\ \vdots \\ d_n\\ \end{array} \right]$$

where $d_i$ shows the demand for the output of the industry $i$.