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THE LEONTIEF INPUT-OUTPUT MODEL

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Math213 Multidimensional
Mathematics

Ivy Tech Community College, Fall
2017

Have you ever wondered what matrix
algebra is used for? You might expect that a real-life problem involving linear
algebra would have only one solution, or perhaps no solutions. The purpose of
this paper is to show how linear algebra with many solutions arise naturally.

The application here comes from economics.

Wassily
Leontief, a professor of economics, wrote the input output economics. Leontief,
who was awarded the 1973 Nobel prize in Economic Science, open the door to a new
era in mathematical modeling in economics.

Matrix algebra played an essential
role in the Nobel prize winning work of Wassily Leontief, as mentioned in the
introduction. The economical modeling described in this paper is the basis for
more elaborate models used in many parts of the world.

A
May 2013 article in science news untitled “one of the most abstract fields in
math finds application in ‘real’ world” summarizes some of the most important
questions about finding application that perhaps, someday, will turn out to be
useful. Also supporting Leontief input-output model is those from a study which
took place at Syracuse University New York. The study conducted by Terry
McConnell, whom is professor and chair, of the mathematics department explains
that the model introduces an important application of matrix inversion in the modern
economy. Its goal is to predict the proper level of production for each of
several types of goods or service. The proper level of production is the one
which meets two requirements. There should be enough of each good to meet the
demand for it, and there should be no (leftovers) unused goods.

As
a simple example, suppose the economy consists of three sectors: manufacturing,
agriculture, and services with unit consumption vectors c1, c2, c3, as shown in
the table that follows

Inputs
consumed per unit of output

Purchased
from

Manufacturing

Agriculture

Services

Manufacturing

0.50

0.40

0.20

Agriculture

0.20

0.30

0.10

Services

0.10

0.10

0.30

c1

c2

c3

What
amount will be consumed by the manufacturing sector if it decides to produce
100 units? It will compute

100c1   =    100
=

to
produce 100 units, manufacturing will need 50 units from other parts of the
manufacturing sector, 20 units from agriculture, and 10 units from services.  The consumption matrix

C
=

Suppose
the final demand is 50 units for manufacturing, 30 units for agriculture, and
20 units for services. How to find the production level that will satisfy the
demand?

I-C
=  –  =

To
solve we will need to row reduce the augmented matrix

~  ~ …~

The
last column is rounded to the nearest whole unit. Manufacturing must produce
approximately 226 units, agriculture 119 units, and services only 78 units. The
matric I- C is invertible.

Overall, the basic assumption of Leontief’s
input-output model is that for each sector there is a unit consumption vector,
that lists the inputs needed per unit of output of the sector. All input and
output units are measured in millions of dollars, rather than in quantities
such as tons. Leontief input-output model discussed in this topic is useful for
understanding the concept and making consumption by hand or using computers. However,
it has many more parts to it and is suitable for large scale problems in real
life.

References

Leontief,
W. (1966). Input-output economics. N.Y.: Oxford University Press.

McConnell, T. (n.d.). The Leontief
Input-Output Model. Retrieved December 05, 2017, from
http://barnyard.syr.edu/mat183/l32/

Rehmeyer, J. (2013, May 23). One of
the most abstract fields in math finds application in the ‘real’ world.

Retrieved December 05, 2017, from
https://www.sciencenews.org/article/one-most-abstract-fields-math-finds-application-real-world

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