Business, 14.02.2020 23:47 evansh78

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.

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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

$\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] =\left[\begin{array}{ccc}0.02&.0.15&0.10\\0.14&0.05&0.12\\0.14&0.08&0\end{array}\right] \left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] +\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] _{external}$

where

$\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] _{external}=\left[\begin{array}{c}100\\120\\150\end{array}\right]$

so the equation now becomes

From here it is given that

$P=MP+\Delta$

$P-MP=\Delta\\P(I-M)=\Delta\\P=(I-M)^{-1}\Delta$

Here

$P=\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right] \\M=\left[\begin{array}{ccc}0.02&.0.15&0.10\\0.14&0.05&0.12\\0.14&0.08&0\end{array}\right] \\\Delta=\left[\begin{array}{c}100\\120\\150\end{array}\right]$

So now I-M is given as

$\\I-M=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] -\left[\begin{array}{ccc}0.02&0.15&0.10\\0.14&0.05&0.12\\0.14&0.08&0\end{array}\right] \\ I-M=\left[\begin{array}{ccc}1-0.02&0-0.15&0-0.10\\0-0.14&1-0.05&0-0.12\\0-0.14&0-0.08&1-0\end{array}\right] \\I-M=\left[\begin{array}{ccc}0.98&-0.15&-0.10\\-0.14&0.95&-0.12\\-0.14&-0.08&1\end{array}\right]$

Now the inverse is calculated as

$(I-M)^{-1}=\frac{1}{det(I-M)}Adj(I-M)$

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

$adj(I-M)=adj(\left[\begin{array}{ccc}0.98&-0.15&-0.10\\-0.14&0.95&-0.12\\-0.14&-0.08&1\end{array}\right])\\adj(I-M)=\left[\begin{array}{ccc}0.9404&0.1580 & 0.1130\\0.1568&0.9660& 0.1316\\0.1442 &0.0994 & 0.9100\end{array}\right]\\$

Also the determinant is given as

$|I-M|=\left|\begin{array}{ccc}0.98&-0.15&-0.10\\-0.14&0.95&-0.12\\-0.14&-0.08&1\end{array}\right|\\|I-M|=0.8837$

So the inverse is given as

$(I-M)^{-1}=\frac{1}{det(I-M)}Adj(I-M)$

$(I-M)^{-1}=\frac{1}{0.8837}\left[\begin{array}{ccc}0.9404&0.1580 & 0.1130\\0.1568&0.9660& 0.1316\\0.1442 &0.0994 & 0.9100\end{array}\right]\\\\(I-M)^{-1}=\left[\begin{array}{ccc}1.0642 & 0.1788 & 0.1279\\ 0.1774 &1.0932 & 0.1489\\ 0.1632 & 0.1125 & 1.0298\end{array}\right]\\$

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

$P=(I-M)^{-1}\Delta$

$\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right]=\left[\begin{array}{ccc}1.0642 & 0.1788 & 0.1279\\ 0.1774 &1.0932 & 0.1489\\ 0.1632 & 0.1125 & 1.0298\end{array}\right]\left[\begin{array}{c}100\\120\\150\end{array}\right]\\\\\left[\begin{array}{c}P_1\\P_2\\P_3\end{array}\right]=\left[\begin{array}{c}147.06\\ 171.27 \\184.29\end{array}\right]\\$

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.

Answer from: Quest

being born in poverty would limit the people's chances, opportunities and benefits.

this happen because of several things, such as:

- poor people tend to gain less access to good education and nutrition. this mean that their chance to obtain high paying jobs would also be limited.

- poor people tend to socialize with other poor people while the rich tend to socialize with other rich people. this means that simply throguh their social circle, rich people would have more business opportunities compared to the poor people.

- rich people tend to have more capital in their disposal. this would benefit them in business since it is far easier for them to fund their business operation compared to poor people.

Answer from: Quest

I'm not 100% sure but i think it is service producing industries

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