Application to Leontief input-output model

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** Introduction **In order to understand and be
able to manipulate the economy of a country or a region, one needs to come up
with a certain model based on the various sectors of this economy. The Leontief
model is an attempt in this direction. Based on the assumption that each
industry in the economy has two types of demands: external demand (from outside
the system) and internal demand (demand placed on one industry by another in
the same system), the Leontief model represents the economy as a system of
linear equations. The Leontief model was invented in the 30’s by Professor
Wassily Leontief (picture above) who developed an economic model of the United
States economy by dividing it into 500 economic sectors. On October 18, 1973,
Professor Leontief was awarded the Nobel Prize in economy for his effort.

**1) **** The
Leontief closed Model** Consider an economy consisting of

*p _{1}m_{i1}+p_{2}m_{i2}+…*+

* *

In order to have a balanced economy, the total production of each industry must be equal to its total consumption. This gives the linear system:

_{}

If

_{}

then the above system can be written as *AP=P*, where

_{}.

*A *is called the **input-output matrix.**

We are then looking for a vector *P* satisfying *AP=P *and* *with nonnegative components, at least one of which is positive.

** Example ** Suppose that the economy of a certain region depends on three
industries: service, electricity and oil production. Monitoring the operations
of these three industries over a period of one year, we were able to come up
with the following observations:

1. To produce 1 unit worth of service, the service industry must consume 0.3 units of its own production, 0.3 units of electricity and 0.3 units of oil to run its operations.

2. To produce 1 unit of electricity, the power-generating plant must buy 0.4 units of service, 0.1 units of its own production, and 0.5 units of oil.

3. Finally, the oil production company requires 0.3 units of service, 0.6 units of electricity and 0.2 units of its own production to produce 1 unit of oil.

Find the production level of each of these industries in
order to satisfy the external and the internal demands assuming that the above
model is **closed**, that is, no goods leave or enter the system.

** Solution** Consider the following variables:

1.
*p _{1}*= production level for the service
industry

2.
*p _{2}*= production level for the
power-generating plant (electricity)

3.
*p _{3}*= production level for the oil production
company

Since the model is closed, the total consumption of each industry must equal its total production. This gives the following linear system:

_{}_{}

The input-output matrix is

_{}

and the above system can be written as *(A-I)P=0.* Note that this
homogeneous system has infinitely many solutions (and consequently a nontrivial
solution) since each column in the coefficient matrix sums to 1. * *The
augmented matrix of this homogeneous system is

_{}

which can be
reduced to

_{}.

To solve the
system, we let *p _{3}=t* (a parameter), then the general solution is

_{}

and as we mentioned
above, the values of the variables in this system must be nonnegative in order
for the model to make sense; in other words, *t≥0.* Taking *t=100 *for example would
give the solution

_{}

**2) **** The
Leontief open Model** The first Leontief model treats the case where no
goods leave or enter the economy, but in reality this does not happen very
often. Usually, a certain economy has to satisfy an outside demand, for example, from bodies like the government
agencies. In this case, let

_{}

for each *i*. This gives the
following linear system (written in a matrix form):

_{}

where *P *and *A *are as above and

_{}

is the **demand vector.**

One way to solve this linear system is

_{}

Of course, we require here that
the matrix *I-A* be invertible, which might not be always the case. If, in
addition, *(I-A) ^{-1}* has nonnegative entries, then the
components of the vector

** Example **Consider an open economy with three
industries: coal-mining operation, electricity-generating plant and an
auto-manufacturing plant. To produce $1 of coal, the mining operation must
purchase $0.1 of its own production, $0.30 of electricity and $0.1 worth of
automobile for its transportation. To produce $1 of electricity, it takes $0.25
of coal, $0.4 of electricity and $0.15 of automobile. Finally, to produce $1
worth of automobile, the auto-manufacturing plant must purchase $0.2 of coal,
$0.5 of electricity and consume $0.1 of automobile. Assume also that during a
period of one week, the economy has an exterior demand of $50,000 worth of coal, $75,000 worth of
electricity, and $125,000 worth of autos. Find the production level of each of
the three industries in that period of one week in order to exactly satisfy
both the internal and the external demands.

** Solution **The input-output matrix of this economy is

_{}

and the demand vector
is

_{}

By equation (*)
above,

_{}

where

_{}

Using the
Gaussian-elimination technique (or the formula *B ^{-1}=(1/det(B))adj(B)*), we find that

_{}

which gives

_{}

So, the total
output of the coal-mining operation must be $229921.59, the total output for
the electricity-generating plant is $437795.27 and the total output for the
auto-manufacturing plant is $237401.57.

If you like to know
more about the subject and the life and achievements of W. Leontielf, check the
following links:

·
Wassily Leontief
(life and achievements of W. Leontielf)

·
http://online.redwoods.cc.ca.us/instruct/darnold/laproj/Fall2001/Iris/lapaper.pdf