Chapter 1.2, Problem 40E

If we consider more than three industries in an input-output model, it is cumbersome to represent all thedemands in a diagram as in Exercise 39. Supposewe have the industries $I_1,I_2,\mathrm{....}{I}_n$ , with outputs $x_1,x_2,\mathrm{...},x_n$ . The output vector is
$\stackrel{\to }{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right]$ .

The consumer demand vector is
$\stackrel{\to }{b}=\left[\begin{array}{c}{b}_1\\ b_2\\ ⋮\\ b_n\end{array}\right]$

where $b_i$ is the consumer demand on industry $I_i$ . Thedemand vector for industry $I_j$ is
${\stackrel{\to }{v}}_j=\left[\begin{array}{c}{a}_{1j}\\ a_{2j}\\ ⋮\\ a_{nj}\end{array}\right]$ ,
where $a_{ij}$ is the demand industry $I_j$ puts on industry $I_j$ , for each $1 of output industry $I_j$ produces. For exam ple, $a_{32}=0.5$ means that industry $I_2$ needs 50¢ worth of products from industry $I_3$ for each $1 worth of goods $I_2$ produces. The coefficient $a_{ii}$ need not be 0: Producing a product may require goods or services from the same industry.

a. Find the four demand vectors for the economy in Exercise 39.
b. What is the meaning in economic terms of $x_j{\stackrel{\to }{v}}_j$ ?
c. What is the meaning in economic terms of $x_1{\stackrel{\to }{v}}_1+x_2{\stackrel{\to }{v}}_2+\cdots +x_n{\stackrel{\to }{v}}_n+\stackrel{\to }{b}$ ?
d. What is the meaning in economic terms of the equation $x_1{\stackrel{\to }{v}}_1+x_2{\stackrel{\to }{v}}_2+\cdots +x_n{\stackrel{\to }{v}}_n+\stackrel{\to }{b}=\stackrel{\to }{x}$ ?