**for**

**Introduction**

The mathematics model for the economy of a
country or a region is based on the various sectors of this
economy. The Leontief model is addresses this
problem. Assume that each industry in the economy has two
types of demands: an external demand (from outside the system) and an
internal demand (demand placed on one industry by another in the same
system), the Leontief model is a system of linear equations. The
Leontief model was invented in the 30's by Wassily Leontief who
developed an economic model of the United States economy by dividing
it into 500 economic sectors. Wassily Leontief received
the economics Nobel
Prize on October 18, 1973.

**The Closed Leontief Model**

Assume that an economy consists
of n interdependent
industries (or sectors) . Each
industry will consume 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). An economy is
called closed if it satisfies its own
needs; that is, no goods leave or enter the system. We
make the following conventions:

**(i)** is the
production level of industry ,

**(ii)** is
the number of units produced by industry
that is necessary to produce one unit by industry ,

**(iii)** is
the number of units produced by industry and
consumed by industry ,

**(iv)** is
the total number of units produced by industry .

Since the economy is closed, the total production for industry
equals it total consumption and we have the equations

for .

If the economy is balanced, the total production of each industry
must be equal to its total consumption. This results in the linear
system:

which can be written in matrix form

.

The matrix is
called the input-output matrix, and is
the production vector.

**Proof
** **Leontief
Model**** ****Leontief
Model**

**Example 1.** Consider
the closed three sector economy
consisting of say: Energy, Manufacturing, and Services
where the input-output matrix is given by

. Find
the production vector .

**Example 2.** Consider
the closed five sector economy
consisting of say: Agriculture, Construction, Energy, Manufacturing,
and Services where the input-output matrix is given by

. Find the production vector .

**The Open Leontief Model**

The closed Leontief model describes the case
when no goods leave or enter the economy. But often times,
an economy has to satisfy an outside demand. In this case,
let be
the demand from the
outside industry. Let and be
as in the closed model, then

for .

This results in the linear system:

which can be written in matrix form

,

where and
are
the same as in the closed model and is
the demand vector.

One way to solve this linear system is

.

Of course, the matrix must
be invertible, which might not be always the case. If, in
addition,
has nonnegative entries, then the components of the
vector are
nonnegative and therefore they are acceptable as solutions for this
model. We say in this case that the matrix
is
productive.

**Proof
** **Leontief
Model**** ****Leontief
Model**

**Example 3.** Consider
the open three sector economy
consisting of say: Energy, Manufacturing, and Services
where the input-output matrix is

and
the demand vector is . Find
the production vector .

**Example 4.** Consider
the open five sector economy
consisting of say: Agriculture, Construction, Energy, Manufacturing,
and Services where the input-output matrix is given by

and
the demand vector is . Find
the production vector .

**Research Experience for
Undergraduates**

**Leontief
Model** **Leontief
Model** Internet hyperlinks to web sites and
a bibliography of articles.

**Download this Mathematica
Notebook** Leontief
Economic Model

(c) John H. Mathews 2004