The following problem refers to a closed Leontief modelSuppose the technology matrix for a closed model of a simple economy is given by matrix A. Find the gross productions for the industries. (Let H represent the number of household units producedand give your answers in terms of H.)GI HGovernment0.4 0.2 0.2IndustryA =0.2 0.5 0.5Households0.4 0.3 0.3unitsgovernmentindustryunitsunitshouseholds

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Asked Oct 20, 2019
The following problem refers to a closed Leontief model
Suppose the technology matrix for a closed model of a simple economy is given by matrix A. Find the gross productions for the industries. (Let H represent the number of household units produced
and give your answers in terms of H.)
GI H
Government
0.4 0.2 0.2
Industry
A =
0.2 0.5 0.5
Households
0.4 0.3 0.3
units
government
industry
units
units
households
help_outline

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The following problem refers to a closed Leontief model Suppose the technology matrix for a closed model of a simple economy is given by matrix A. Find the gross productions for the industries. (Let H represent the number of household units produced and give your answers in terms of H.) GI H Government 0.4 0.2 0.2 Industry A = 0.2 0.5 0.5 Households 0.4 0.3 0.3 units government industry units units households

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Expert Answer

Step 1

The technological equation for a closed Leontief model is given as follows.

(I-A)X 0
where A is the technology matrix
X is the gross production matrix
0 is the zero column matrix
I is the identity matrix
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(I-A)X 0 where A is the technology matrix X is the gross production matrix 0 is the zero column matrix I is the identity matrix

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Step 2

Construct an augmented matrix as follows.

Let x be the gross production of the govemment, x, be the gross production
of the industry andx, be the gross production of the households.
1 0 0
0.4 0.2 0.2
A0.2 0.5 0.5 X
I0 1
0
0 0 1
0.4 0.3 0.3
(-A)X0
0.4 0.2 0.2
10 0
0
0 1
0
0.2 0.5 0.5
0 0
0.4 0.3 0.3
1
x3
0.2
1-0.4
0.2
-0.2
1-0.5
0
-0.5
1-0.3
-0.4
0.3
-0.2 -0.2|x
0.6
0
0.2
-0.5 x2
0.5
-0.4 -0.3
0.7
0.2-0.2 0
0.6
The augmented matrix is-0.2
0.5
-0.5 0
0.7
-0.4 -0.3
0
O
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Let x be the gross production of the govemment, x, be the gross production of the industry andx, be the gross production of the households. 1 0 0 0.4 0.2 0.2 A0.2 0.5 0.5 X I0 1 0 0 0 1 0.4 0.3 0.3 (-A)X0 0.4 0.2 0.2 10 0 0 0 1 0 0.2 0.5 0.5 0 0 0.4 0.3 0.3 1 x3 0.2 1-0.4 0.2 -0.2 1-0.5 0 -0.5 1-0.3 -0.4 0.3 -0.2 -0.2|x 0.6 0 0.2 -0.5 x2 0.5 -0.4 -0.3 0.7 0.2-0.2 0 0.6 The augmented matrix is-0.2 0.5 -0.5 0 0.7 -0.4 -0.3 0 O

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Step 3

Reduce the augmented matrix using...

-2
0.6
-0.2-0.2 O
6
-2 0
0.5
Multiply each entry by 10
0.2
-0.5 0
5
-5 0
2
-0.4 -0.3
0.7
0
4-3
7
0
1
1
0
3
1
3
5
-5
0
-4
-3
7
0
1
0
3
1
3
RR,+2R
13
17
0
3
| R-R, +4R
17
0
13
0
3
3
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-2 0.6 -0.2-0.2 O 6 -2 0 0.5 Multiply each entry by 10 0.2 -0.5 0 5 -5 0 2 -0.4 -0.3 0.7 0 4-3 7 0 1 1 0 3 1 3 5 -5 0 -4 -3 7 0 1 0 3 1 3 RR,+2R 13 17 0 3 | R-R, +4R 17 0 13 0 3 3

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