Project 1
Linear Algebra 1B
Dr. Matt Insall
Determining output production with the Leontief Input-Output Model
By: Linzi Rector
Abstract:
Economists are able to determine a nation’s production vector of good and services in different
sectors of the economy with the use of the Leontief Input-Output model. By using a table of the
exchange of goods and services, economists are able to determine the consumption vector of the
different sectors. This paper discusses the process to determine the production vectors of the
following sector of the economy in 1992: agriculture, forestry, and fisheries, mining, petroleum
and natural gas, and construction.
Background:
In 1947, Nobel-prize winner Wassily Leontief began using linear algebra when analyzing
mathematical models in economic input and output. In his work, he divided the country into
production sectors that produce certain goods and services. From this, Leontief found a
production vector in the producing sectors and a final demand vector in the demanding sector
that lists the value of the goods and services. The Leontief Input-Output model is x=Cx+d,
where x is the amount produced, Cx is the intermediate demand, and d is the final demand, and C
is the consumption matrix. To find the consumption, use the table of Exchange of Goods and
Services in the U.S. for the given year. One of these tables can be found in Appendix A. We
can solve how much a sector will produce by matrices when we rewrite the model with Ix
equaling x, Ix being the identity matrix. This gives us (I-C)x=d.
To begin to solve for x, the amount produced, first find C and d. By using the table in Appendix
A, we are able to find the consumption matrix and the final demand vector. The final demand
vector is the values in the open sector:
d=
34940
-39241
787208
To find the consumption matrix, divide each column of the table by the total gross output for that
sector. See the table in Appendix A. Doing this gives the following matrix:
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Project 1
Linear Algebra 1B
Dr. Matt Insall
.9484 .0022 .4149
.0051 .8684 .5217
.0465 .1295 .0634
C=
Now, every variable in the equation besides x is known so it is possible to solve for x, the
production vector. To do this, start by taking the identity vector minus the consumption vector.
1
0
0
0
1
0
0
0
1
-
.9484 .0022 .4149
.0051 .8684 .5217
.0465 .1295 .0634
=
.0516 -.0022 -.4149
-.0051 .1316 -.5217
-.0465 -.1295 .9366
To solve for the production vector, x, solve the augmented matrix:
.0516
-.0051
-.0465
-.0022
.1316
-.1295
-.4149
-.5217
.9366
34940
-39241
787208
Using row reductions or a calculator or computer program, reduce the matrix to reduced row
echelon form. The reduced matrix is:
1
0
0
0
1
0
0
0
1
12374562.78
-594.8546
1454781.268
From this, the production vector can be determined.
x =
12374562.78
-594.8546
1454781.268
This shows that 12374562.78 units of agriculture, forestry, and fishery will be produced,
-594.8546 units of mining, petroleum, and natural gas will be produced, and 1454781.268 units
of construction will be produced in the given economy in 1992.
References:
1.
David C. Lay, Linear Algebra and it’s Applications, Third Edition, Addison Wesley,
2003, Pages 1-2, 57-58, 125-156.
2.
The Leontief Input-Output Model, Addison-Wesley, 2003. Online at
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Project 1
Linear Algebra 1B
Dr. Matt Insall
http://media.pearsoncmg.com/aw/aw_lay_linearalg_3/cs_apps/leontief.pdf
Appendix A:
This table is just a section of the entire table, taken off the Addison-Wesley website, of the
Exchange of Goods and Services in the U.S. for 1992 (in millions of 1992 dollars). This table is
just the information used in the example in this paper.
1
2
3
1
68681
78
5860
34940
2
367
31477
7368
-39241
3
3368
4693
895
787208
72416
36248
14124
TGO
OPEN
In the table 1 refers to the agriculture, forestry and fisheries sector, 2 refers to the mining,
petroleum and natural gas sector, and 3 refers to the construction sector. The TGO (Total Gross
Output) is the sum of the column and is used to determine the consumption matrix. To find the
consumption matrix divide each entry in the column by the TGO of that column. This gives you
the following table, which is used as the consumption matrix:
1
2
3
1
0.9484
0.0022
0.4149
2
0.0051
0.8684
0.5217
3
0.0465
0.1295
0.0634
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