Question

A hypothetical economy consisting of two industries is: Industry 1 Industry 2 Final demand Total production 1200 1500 Industry 1 Industry 2. 240 720 750 450 210 330 a) Find the input-output matrix if demand changes to 312 units in industry 1 and 299 units in industry 2. b) Find the new total production c) Determine the new input-output matrix

Answer 1

The input-output matrix 'A' is a square matrix with elements a_{ij ,} i = row number, j = column number, representing the amount of input 'i' which is required to produce per unit of output 'j'. The matrix column depicts the inputs needed for production of a certain output.

Each column entry is divided by the total production/output for that column in order to get the a_ij 's of the input-output matrix A.

This is because, the table states, that 240 units of industry 1 inputs is required to yield a production/output of 1200 units of industry 1 and 750 units of industry 2 inputs is required to yield a production/output of 1200 units in industry 1. Similarly, 720 units of industry 1 inputs is used to yield 1500 units of output in industry 2 and 450 units of industry 2 inputs is used to yield 1500 units of output in industry 2. So, we divide each of these input numbers by the total production/output in order to obtain the amount of input required to obtain per unit output.

a) So, if we have to obtain the input-output matrix A :

a₁₁ = 240/1200 = 0.2 , a₁₂ = 750/1200 = 0.625

a₂₁ = 720/1500 = 0.48 , a₂₂ = 450/1500 = 0.3

so, A =

10.2 0.625 10.48 03

b) Now, when demand changes, we know:

X = AX + F , X = production matrix, F = Demand matrix

So, if X₁ be production in industry 1 and X₂ be production in industry 2,

then we can say:

X1 X2
=  
10.2 0.625 10.48 03
*
X1 X2
+ $\begin{bmatrix} 312 \\ 299 \end{bmatrix}$

So, 0.2 X1 + 0.625 X2 + 312 = X1

0.48 X1+ 0.3 X2 + 299 = X2

So, -0.8 X1 + 0.625 X2 = -312 - (1)

0.48 X1 - 0.7 X2 = -299 - (2)

Then multuplying eqn (1) by 0.6 and adding it to (2):

-0.48 X1 + 0.375 X2 = -187.2

0.48 X1 - 0.7 X2 = -299

-0.325X2 = -486.2 ,

X2 = 486.2/0.325 = 1496 ,

0.48 X1 = -299 + 0.7 * 1496 = -299 + 1047

0.48 X1 = 748, X1 = 1558

So, the new production levels are 1558 in industry 1 and 1496 in industry 2.

c) So, the new input-output ratio:  

a₁₁ = 240/1558 = 0.15 , a₁₂ = 750/1558 = 0.48

a₂₁ = 720/1496 = 0.48 , a₂₂ = 450/1496 = 0.3

So, A = $\begin{bmatrix} 0.15 & 0.48 \\ 0.48 & 0.3 \end{bmatrix}$