Application to Leontief input-output model:
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 n interdependent
industries (or sectors) S1,…,Sn. That
means that each industry consumes some of the goods produced by the other
industries, including itself (for example, a power-generating plant uses some
of its own power for production). We say that such an economy
is closed if it satisfies its own needs; that is, no goods leave or
enter the system. Letmij be the number of units produced
by industry Si and necessary to produce one unit of
industry Sj. If pk is the
production level of industry Sk, then mij pj represents
the number of units produced by industry Si and
consumed by industry Sj . Then the total number of
units produced by industry Si is given by:
p1mi1+p2mi2+…+pnmin.
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. p1= production
level for the service industry
2. p2= production
level for the power-generating plant (electricity)
3. p3= 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 p3=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 di be the demand from the ith outside
industry, pi, and mij be
as in the closed model above, then
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 P are
nonnegative and therefore they are acceptable as solutions for this model. We
say in this case that the matrix A is productive.
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.
