Answer :
Answer:
Let’s solve this step by step using the input-output model.
Part (a): Total Output of Goods Needed
We can represent the economy with the following system of linear equations:
Let ( x_1 ) be the total output of services and ( x_2 ) be the total output of manufacturing products.
The equations based on the given consumption rates are: [ x_1 = 0.2x_1 + 0.3x_2 + 100 ] [ x_2 = 0.4x_1 + 0.3x_2 + 160 ]
Rearranging these equations to isolate ( x_1 ) and ( x_2 ): [ x_1 - 0.2x_1 - 0.3x_2 = 100 ] [ x_2 - 0.4x_1 - 0.3x_2 = 160 ]
Simplifying further: [ 0.8x_1 - 0.3x_2 = 100 ] [ -0.4x_1 + 0.7x_2 = 160 ]
We can solve this system of equations using matrix methods or substitution. Let’s use matrix methods:
The matrix form is: [ \begin{pmatrix} 0.8 & -0.3 \ -0.4 & 0.7 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix} = \begin{pmatrix} 100 \ 160 \end{pmatrix} ]
To find ( x_1 ) and ( x_2 ), we need to find the inverse of the coefficient matrix and multiply it by the demand vector.
The inverse of the coefficient matrix is: [ \begin{pmatrix} 0.8 & -0.3 \ -0.4 & 0.7 \end{pmatrix}^{-1} = \frac{1}{(0.8 \cdot 0.7 - (-0.3) \cdot (-0.4))} \begin{pmatrix} 0.7 & 0.3 \ 0.4 & 0.8 \end{pmatrix} ] [ = \frac{1}{(0.56 - 0.12)} \begin{pmatrix} 0.7 & 0.3 \ 0.4 & 0.8 \end{pmatrix} ] [ = \frac{1}{0.44} \begin{pmatrix} 0.7 & 0.3 \ 0.4 & 0.8 \end{pmatrix} ] [ = \begin{pmatrix} 1.59 & 0.68 \ 0.91 & 1.82 \end{pmatrix} ]
Multiplying this by the demand vector: [ \begin{pmatrix} x_1 \ x_2 \end{pmatrix} = \begin{pmatrix} 1.59 & 0.68 \ 0.91 & 1.82 \end{pmatrix} \begin{pmatrix} 100 \ 160 \end{pmatrix} ] [ = \begin{pmatrix} (1.59 \cdot 100 + 0.68 \cdot 160) \ (0.91 \cdot 100 + 1.82 \cdot 160) \end{pmatrix} ] [ = \begin{pmatrix} 159 + 108.8 \ 91 + 291.2 \end{pmatrix} ] [ = \begin{pmatrix} 267.8 \ 382.2 \end{pmatrix} ]
So, the total output needed is: [ x_1 = 267.80 \text{ million units of services} ] [ x_2 = 382.20 \text{ million units of manufacturing products} ]
Part (b): Value of Goods Consumed Internally
To find the value of goods consumed internally, we need to calculate the intermediate consumption based on the total output.
For services: [ \text{Internal consumption of services} = 0.2 \cdot 267.80 + 0.4 \cdot 382.20 ] [ = 53.56 + 152.88 ] [ = 206.44 \text{ million units of services} ]
For manufacturing products: [ \text{Internal consumption of manufacturing products} = 0.3 \cdot 267.80 + 0.3 \cdot 382.20 ] [ = 80.34 + 114.66 ] [ = 195.00 \text{ million units of manufacturing products} ]
the answer is table
(a) Services: $200.00 million Manufacturing: $280.00 million
(b) Services: $100.00 million Manufacturing: $120.00 million
