Question

If we consider more than three industries in an input-output model, it is cumbersome to represent all the demands in a diagram as in Exercise 39. Suppose we have the industries${\mathbf{I}}_{\mathbf{1}}{\mathbf{I}}_{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{I}}_{\mathbf{n}}$with outputs ${\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{.}\mathbf{.}\mathbf{.}{\mathbf{x}}_{\mathbf{n}}$. The output vector is,

$\overrightarrow{\mathbf{x}}\mathbf{=}\left[\begin{array}{c}{x}_1\\ x_2\\ .\\ .\\ .\\ x_n\end{array}\right]$

The consumer demand vector is,

$\overrightarrow{\mathbf{b}}\mathbf{=}\left[\begin{array}{c}{b}_1\\ b_2\\ .\\ .\\ .\\ b_n\end{array}\right]$

Where b_i is the consumer demand on industry I_i. The demand vector for the industry I_j is

$\overrightarrow{{\mathbf{v}}_{\mathbf{j}}}\mathbf{=}\left[\begin{array}{c}{a}_{1j}\\ a_{2j}\\ .\\ .\\ .\\ a_{\mathrm{nj}}\end{array}\right]$

Where a_ij is the demand industry I_j puts on industry I_i, for each \(1 of output industry I_j produces. For example, a₃₂= 0.5 means that industry I₂ needs 50 cents worth of product from industry I₃ for each \)1 worth of goods I₂ produces. The coefficient a_ii need not be 0: producing a product may require goods or services from the same industry.

1. Find the four demand vectors for the economy in exercise 39.
2. What is the meaning in economic terms of ${\mathbf{x}}_{\mathbf{j}}\overrightarrow{{\mathbf{v}}_{\mathbf{j}}}$?
3. What is the meaning economic terms of ${\mathit{x}}_{\mathbf{1}}\overrightarrow{{\mathbf{v}}_{\mathbf{1}}}\mathbf{+}{\mathbf{x}}_{\mathbf{2}}\overrightarrow{{\mathbf{v}}_{\mathbf{2}}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{x}}_{\mathbf{n}}\overrightarrow{{\mathbf{v}}_{\mathbf{n}}}\mathbf{+}\overrightarrow{\mathbf{b}}$?
4. What is the meaning in economic terms of ${\mathbf{x}}_{\mathbf{1}}\overrightarrow{{\mathbf{v}}_{\mathbf{1}}}\mathbf{+}{\mathbf{x}}_{\mathbf{2}}\overrightarrow{{\mathbf{v}}_{\mathbf{2}}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{x}}_{\mathbf{n}}\overrightarrow{{\mathbf{v}}_{\mathbf{n}}}\mathbf{+}\overrightarrow{\mathbf{b}}\mathbf{=}\overrightarrow{\mathbf{x}}$?

Short answer

1. The four demand vectors for an economy,

$\left[\begin{array}{c}1\\ 0.1\\ 0.2\end{array}\right]$, $\left[\begin{array}{c}0.2\\ 1\\ 0.5\end{array}\right]$, $\left[\begin{array}{c}0.3\\ 0.4\\ 1\end{array}\right]$ and $\left[\begin{array}{c}320\\ 90\\ 150\end{array}\right]$

2. The termdata-custom-editor="chemistry" ${\mathrm{x}}_{\mathrm{j}}\overrightarrow{{\mathrm{v}}_{\mathrm{j}}}$ represents the demand vector for the industry I_j in the economy.

3. The term data-custom-editor="chemistry" $x_1\overrightarrow{{\mathrm{v}}_1}+{\mathrm{x}}_2\overrightarrow{{\mathrm{v}}_2}+....+{\mathrm{x}}_{\mathrm{n}}\overrightarrow{{\mathrm{v}}_{\mathrm{n}}}+\overrightarrow{\mathrm{b}}$represents the total demand by the industries data-custom-editor="chemistry" ${\mathrm{I}}_1{\mathrm{I}}_2.....{\mathrm{I}}_{\mathrm{n}}$and the consumer.

4. The term data-custom-editor="chemistry" ${\mathrm{x}}_1\overrightarrow{{\mathrm{v}}_1}+{\mathrm{x}}_2\overrightarrow{{\mathrm{v}}_2}+.....+{\mathrm{x}}_{\mathrm{n}}\overrightarrow{{\mathrm{v}}_{\mathrm{n}}}+\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{x}}$represents how the total output from the given industries data-custom-editor="chemistry" ${\mathrm{I}}_1{\mathrm{I}}_2.....{\mathrm{I}}_{\mathrm{n}}$is distributed among the remaining industries and the consumer.

Step-by-step solution

Concept introduction

Two vectors are perpendicular to each other if their dot product is zero,

Let us suppose that $\overrightarrow{\mathbf{x}}$and $\overrightarrow{\mathbf{y}}$are two perpendicular then these two vectors are said to be perpendicular, if
$\overrightarrow{\mathbf{x}}\mathbf{.}\overrightarrow{\mathbf{y}}\mathbf{=}\mathbf{0}$

Find the four demand vectors for the economy

(a)

It is given that the x₁, x₂ and x₃ are the three outputs which is produced by the industries I₁, I₂ and I₃ respectively.

The system of linear equation from the Exercise 39,

x₁=0.2x₂+0.3x₃+320, x₂=0.1x₁+0.4x₃+90, x₃=0.2x₁+0.5x₂+150.

From the above equation we can write four demand vectors for an economy,

$\left[\begin{array}{c}1\\ 0.1\\ 0.2\end{array}\right]$, $\left[\begin{array}{c}0.2\\ 1\\ 0.5\end{array}\right]$, $\left[\begin{array}{c}0.3\\ 0.4\\ 1\end{array}\right]$and $\left[\begin{array}{c}320\\ 90\\ 150\end{array}\right]$

Solution for part (b)

(b)

The term ${\mathrm{x}}_{\mathrm{j}}\overrightarrow{{\mathrm{v}}_{\mathrm{j}}}$ represents the demand vector for the industry I_j in the economy.

Solution for part (c)

(c)

The term ${\mathrm{x}}_1\overrightarrow{{\mathrm{v}}_1}+{\mathrm{x}}_2\overrightarrow{{\mathrm{v}}_2}+.........+{\mathrm{x}}_{\mathrm{n}}\overrightarrow{{\mathrm{v}}_{\mathrm{n}}}+\overrightarrow{\mathrm{b}}$represents the total demand by the industries ${\mathrm{I}}_1,{\mathrm{I}}_2,....{\mathrm{I}}_{\mathrm{n}}$and the consumer.

Solution for part (d)

(d)

The term${\mathrm{x}}_1\overrightarrow{{\mathrm{v}}_1}+{\mathrm{x}}_2\overrightarrow{{\mathrm{v}}_2}+.........+{\mathrm{x}}_{\mathrm{n}}\overrightarrow{{\mathrm{v}}_{\mathrm{n}}}+\overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{x}}$represents how the total output from the given industries${\mathrm{I}}_1,{\mathrm{I}}_2,....{\mathrm{I}}_{\mathrm{n}}$is distributed among the remaining industries and the consumer.