Answer to Question #169012 in Algebra for Bernice maina

Question

​National exporters (NE) Ltd manufactures Agricultural, Industrial

and mineral products. One shilling worth of Agricultural output requires inputs worth Ksh 0.1, Ksh 0.15 and Ksh 0.20 from Agricultural, Industrial and mineral products respectively. One shilling worth of Industrial output requires inputs worth Ksh 0.25, Ksh 0.15 and Ksh 0.30 from Agricultural, Industrial and mineral products respectively. One shilling worth of Mineral output requires inputs worth Ksh 0.24, Ksh 0.16 and Ksh 0.16 from Agricultural, Industrial and mineral products respectively. In the next financial year, NE Ltd Plans to produce products worth Ksh 40 million, Ksh 20 million and ksh 50 million for the Agricultural, Industrial and mineral products respectively.

Required

i) Derive the technological matrix

ii) Write down the intermediate demand for each type of product

iii) Compute the final demand for each type of product.

iv) Compute the total worth of primary inputs

Answer 1

Let:

$x = \text{Agricultural Products}\\ y = \text{Industrial Products}\\ z = \text{Mineral Products}$

(i) Technology Matrix

$$\qquad \qquad Output\\\begin{matrix} \qquad \qquad \quad x\ &\quad y&\quad z \end{matrix}\\ input \quad \begin{matrix} x \\ y\\ z \end{matrix}\begin{bmatrix} 0.10 & 0.25 & 0.24 \\ 0.15 & 0.15 &0.16\\ 0.20 & 0.30 & 0.16 \end{bmatrix}\\$$

$$\begin{bmatrix} x \\ y\\ z \end{bmatrix} =\begin{bmatrix} 40 \\ 20\\ 50 \end{bmatrix}$$

(ii) Intermediate demand for each type of product

$$0.10x+0.25y+0.24z=x (output)\\ 0.15x + 0.15y+0.16z=y (output)\\ 0.20x+0.30y+0.16z=z (output)$$

Computing the intermediate demand, we do the following matrix multiplication:

$$\begin{bmatrix} 0.10 & 0.25 & 0.24 \\ 0.15 & 0.15 &0.16\\ 0.20 & 0.30 & 0.16 \end{bmatrix}\begin{bmatrix} 40 \\ 20\\ 50 \end{bmatrix} = \begin{bmatrix} 21 \\ 17\\ 22 \end{bmatrix}$$

The intermediate demand is 21, 17 and 22 for Agriculutural, Industrial and Mineral products respectively.

(iii) Final demand for each type of product

The final demand is computed thus:

$$\begin{bmatrix} 40 \\ 20\\ 50 \end{bmatrix} - \begin{bmatrix} 21 \\ 17\\ 22 \end{bmatrix} = \begin{bmatrix} 19 \\ 3\\ 28 \end{bmatrix}$$

(iv) Total worth of primary inputs

$$\begin{bmatrix} 0.10 \times 40 & 0.25 \times 20 & 0.24 \times 50 \\ 0.15 \times 40 & 0.15 \times 20 &0.16 \times 50\\ 0.20 \times 40 & 0.30 \times 20 & 0.16 \times 50 \end{bmatrix} = \begin{bmatrix} 4 & 5 & 12\\ 6 & 3 &8\\ 8 & 6 & 8 \end{bmatrix}$$

Summing the above vertically, we have the total worth of primary outputs to be:

18, 14 and 28 for Agriculuture, Industrial and Mineral products respectively.