1. An economy has three sectors producing products Product 1, Product 2, and Prod- uct 3. • To produce 1 unit of Product 1 requires 0.20 units of Product 1, 0.15 units of Product 2, and 0.10 units of Product 3. • To product 1 unit of Product 2 it takes 0.14 units of Product 1, 0.05 units of Product 2, and 0.12 units of Product 3. • To produce 1 unit of Product 3 it takes 0.14 units of Product 1, and 0.08 units of Product 2. (a) What should the total production be set at in order to satisfy an external de- mand of 100 units of Product 1, 120 units of Product 2, and 150 units of Product 3? (b) Individually interpret each entry in the third column of (I − M)−1.

Part a: In order to meet the external demand of 100 units of Product 1, 120 units of Product 2 and 150 units of Product 3, the total production of 147.06 units of Product 1, 171.27 units of Product 2 and 184.29 units of Product 3 are to be produced.

Part b: The  individual entry of 3rd column (I-M)^-1 signifies the role of the demand of 3rd product for total estimation of product 1, product 2 and product 3 respectively.

Explanation:

The matrix form of the equation is given as

where

so the equation now becomes

From here it is given that

Or

Here

So now I-M is given as

Now the inverse is calculated as

So the adjoint of (I-M) is calculated as

Also the determinant is given as

So the inverse is given as

So the total demand of each product to meet the external demand is given as

So in order to meet the external demand of 100 units of Product 1, 120 units of Product 2 and 150 units of Product 3, the total production of 147.06 units of Product 1, 171.27 units of Product 2 and 184.29 units of Product 3 are to be produced.

Part b

The  individual entry of 3rd column (I-M)^-1 signifies the role of the demand of 3rd product for total estimation of product 1, product 2 and product 3 respectively.

3rd the reason why is because of management and leadership