OR Consider a hypothetical economy of two industries A and B represented by the table given below : Producer User A User B Final demand Total output : : : : : A 500 350 150 1000 B 320 360 120 800…

Step 1: Identify the technical coefficients matrix

The technical coefficients $a_{ij}$ are calculated as: $$a_{ij}=\frac{\text{Input from industry }i\text{ to industry }j}{\text{Total output of industry }i}$$

For industry A: $$a_{AA}=\frac{500}{1000}=0.5,a_{BA}=\frac{320}{800}=0.4$$

For industry B: $$a_{AB}=\frac{350}{1000}=0.35,a_{BB}=\frac{360}{800}=0.45$$

Thus, the technical coefficients matrix $A$ is: $$A=\left[\begin{array}{cc}0.5& 0.35\\ 0.4& 0.45\end{array}\right]$$

Step 2: Determine the new final demand vector

The initial final demand vector $D$ is: $$D=\left[\begin{array}{c}150\\ 120\end{array}\right]$$

After changes:

• Final demand of A increases by 10%: $$150\times 1.10=165$$
• Final demand of B decreases by 10%: $$120\times 0.90=108$$

The new final demand vector $D^{\prime }$ is: $$D^{\prime }=\left[\begin{array}{c}165\\ 108\end{array}\right]$$

Step 3: Solve for the new output vector $X$

The input-output model equation is: $$X=AX+D^{\prime }$$ Rearranged as: $$X-AX=D^{\prime }⟹(I-A)X=D^{\prime }$$ Where $I$ is the identity matrix: $$I-A=\left[\begin{array}{cc}1-0.5& -0.35\\ -0.4& 1-0.45\end{array}\right]=\left[\begin{array}{cc}0.5& -0.35\\ -0.4& 0.55\end{array}\right]$$

Now, solve for $X$: $$\left[\begin{array}{cc}0.5& -0.35\\ -0.4& 0.55\end{array}\right]\left[\begin{array}{c}{X}_A\\ X_B\end{array}\right]=\left[\begin{array}{c}165\\ 108\end{array}\right]$$

Using Cramer’s rule or matrix inversion, we get: $$X=(I-A{)}^{-1}{D}^{\prime }$$

Calculating the inverse of $(I-A)$: $$(I-A{)}^{-1}=\frac{1}{(0.5)(0.55)-(-0.35)(-0.4)}\left[\begin{array}{cc}0.55& 0.35\\ 0.4& 0.5\end{array}\right]=\frac{1}{0.275-0.14}\left[\begin{array}{cc}0.55& 0.35\\ 0.4& 0.5\end{array}\right]=\frac{1}{0.135}\left[\begin{array}{cc}0.55& 0.35\\ 0.4& 0.5\end{array}\right]$$

Multiplying by $D^{\prime }$: $$X=\frac{1}{0.135}\left[\begin{array}{c}0.55\times 165+0.35\times 108\\ 0.4\times 165+0.5\times 108\end{array}\right]=\frac{1}{0.135}\left[\begin{array}{c}90.75+37.8\\ 66+54\end{array}\right]=\frac{1}{0.135}\left[\begin{array}{c}128.55\\ 120\end{array}\right]$$

Thus, the new output vector is: $$X=\left[\begin{array}{c}952.22\\ 888.89\end{array}\right]$$

Final Answer

The output vector of the economy after the change in final demand is: $$\boxed{\left[\begin{array}{c}{X}_A\\ X_B\end{array}\right]=\left[\begin{array}{c}952.22\\ 888.89\end{array}\right]}$$ (rounded to two decimal places).