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# The

Leontief
Production Model
Janet Aquino, Jonathan Guzman, Szuhara Jen
MATH 247Dr. Jen-Mei Chang
Leontiefs Question
In the 1930s, economist Wassily LeonJef divided the economy into 500
sectors, wriJng linear equaJons that described how each sector
distributed its output to other sectors. By creaJng this system of linear
equaJons, he created a 500 by 500 matrix and began developing the
Leon%ef Input-Output (or Produc%on) Model.
By creaJng this model, LeonJef aTempted to answer the following
quesJon: Is there a produc%on vector (also known as the producJon
level), x, that equals the total demand for producJon in an economy? In
other words, is there an x such that

x = i+d
where i = (intermediate demand), and d = (external demand)
There are two variaJons of this modela closed and an open model.
This presentaJon will deal with the open model with an iniJal d.

## Purpose: To nd an x that sa%ses both the internal demand and

external demands of the hypothe%cal economy of Nomansland.
Given the equaJon for the producJon vector

x = i + d,

we dene the intermediate demand vector as the augmented consumpJon
vectors Jmes x, or Cx; thus, we can solve for x:

x = Cx + d I x Cx = d (I C)x = d.
5 5

This expression works because I5-C can be inverted So long as C contains non-
nega3ve entries and its column sums are less than 1, (I5-C) is inver3ble. A
column sum less than 1 implies that not all producJon is being used within the
economy of Nomansland and allows for an external demand of producJon (i.e., d).

The Initial Conditions
The LeonJef ProducJon Model makes three assumpJons:
The prices of goods and services are held
input and output is measured in millions of dollars (hence
referred to as units of)
and the model assumes that for every sector of the economy
there is a unit consumpJon vector c, that lists the inputs needed
for the units of output in the sector.

## Let n equal the number of sectors in the economy. For

simplicity the following examples will hold n at 5the ve
sectors of manufacturing, agriculture, communicaJons,
services (labor) and nances.
The Model
Assume for the economy of Nomansland, the units each sector
consumes per unit of output are as follows (n = 5):

Inputs Purchased Manufacturing Agriculture Communica3ons Labor Finances
From... (c1) (c2) (c3) (c4) (c5)

## Finances: .10 .15 .15 .10 .05

If manufacturing were to produce 100 units, then it would consume 30 units from
manufacturing; 15 from agriculture; 25 from communicaJons; 15 from labor; and
10 from nances.
The columns of the table the consump%on vectors for each sectordenoted cj. If xj
is the units produced by each sector, then xjcj is the intermediate demand for the
sector. Given the previous example:
" .3 % " 30 %
\$ ' \$ '
\$ .15 ' \$ 15 '
x1 = 100 x1c1 = 100 \$ .25 '=\$ 25 '.
\$ .15 ' \$ 15 '
\$ ' \$ '
\$# .1 '& \$# 10 '&
In general, the total consumpJon of Nomansland is
5

x c j j = x1c1 + x2 c 2 + x3c3 + x4 c 4 + x5c 5 = Cx.
j=1

Thus, for the economy of Nomansland,
" 1.3 .15 .2 .15 .05 % " .7 .15 .2 .15 .05 %
\$ ' \$ '
\$ .15 1.35 .15 .35 .05 ' \$ .15 .65 .15 .35 .05 '
I 5 C = \$ .25 .05 1.3 .2 .4 ' = \$ .25 .05 .7 .2 .4 '.
\$ .15 .25 .15 1.1 .4 ' \$ .15 .25 .15 .9 .4 '
\$
.1 .15 .15 .1 1.05
' \$
.1 .15 .15 .1 .95
'
\$# '& \$# '&

To nd the inverse, augment (I5-C) with I5 and obtain the row-reduced
echelon form; the right-side matrix will be (I5-C)-1:
" 1 0 0 0 0 4.130 2.913 3.009 2.803 2.818 %
\$ '
rref \$
0 1 0 0 0 3.707 4.959 3.701 3.769 3.601 '
[ I5 C | I5 ] ~ \$ 0 0 1 0 0 3.776 3.482 4.834 3.490 3.887 '.
\$ 0 0 0 1 0 3.216 3.328 3.226 4.036 3.402 '
\$ '
\$# 0 0 0 0 1 1.955 1.990 2.004 1.866 2.890 '&

Then, assume that the nal demand vector dthe producJon demanded by
external sectorsis ! 250 \$
# &
# 350 &

d = # 200 &.
# 300 &
# &
#" 150 &%
Finally, perform matrix mulJplicaJon to obtain x:

# 4.130 2.913 3.009 2.803 2.818 &# 250 &
% (% (
% 3.707 4.959 3.701 3.769 3.601 (% 350 (

1
x = (I 5 C) d = % 3.776 3.482 4.834 3.490 3.887 (% 200 (
% (% (
%
3.216 3.328 3.226 4.036 3.402
(%
300
(
%\$ 1.955 1.990 2.004 1.866 2.890 ('%\$ 150 ('

" 3917.45 % Manufacturing

\$ '
\$ 5073.45 ' Agriculture

x \$ 4759.55 ' Communications
\$ '
4335.10 Labor
\$ '
\$# 2579.35 '& Finances

This is the producJon vectorthe level of output at which sectors of the
economy can saJsfy their demand for inputs and the outputs of external
enJJes. It is important to note that only at this producJon vector will the
aforemenJoned demands be saJsed.
In fact, (I
5
C)x
= d
will always yield a unique producJon vector.

Is the Economy of Nomansland
Productive?
By nding a unique soluJon for x, it is certain that Nomanslands economy is
producJvesaJsfying its internal and external demand for producJon. However,
producJvity can also be generalized by analyzing eigenvalues.
If Anxn has a largest eigenvalue , then the corresponding eigenvalue of A-1 is -1;
then eigenvalue of Anxn+aIn is (+a), where a is any real number; lastly, since the
eigenvectors of A and A-1 are the same, the eigenvalues of (I5-C) can be analyzed:
(I 5 C) = (C 1I 5 ) : a = 1
eigenvalue(I5 C ) : ( 1) = 1
1
eigenvalue(I C )1 :
1
5

## Consider (I5-C)-1x = (1-)-1x:

1. If < 1, then (I5-C)-1 is inverJble and thus the economy saJses internal
and external demand.
2. If = 1, then (I5-C)-1 is non-inverJble and the economy is closed.
3. And if > 1, then (I5-C)-1 is not non-negaJve and thus the economy
saJses no demand.
Conclusion
While 100-Jmes smaller, this sampling of the LeonJef ProducJon
Model gives insight into how the true, larger one funcJons: In the
midst of simultaneous internal and external demand, there always
exists a producJon level to saJsfy them.
The concepts in linear algebra and the study of economics are
closely intertwined; applying vectors and matrices to economics is a
concise way to organize economic schedules and make
The text passage describes the Leontief Production Model, which is a mathematical model used to describe the interdependencies between different sectors of an economy. It includes a detailed example of how to solve for the production vector \( x \) using matrix operations, and it provides a specific numerical example with a solution.
single_question_with_answer

1. the passage contains a mathematical question: "is there a production vector \( x \) that equals the total demand for production in an economy?"
2. the answer to this question is provided through the example, where the production vector \( x \) is calculated to satisfy both internal and external demands. the specific solution is given as:
   \[
   x = \begin{pmatrix}
   3917.45 \\
   5073.45 \\
   4759.55 \\
   4335.10 \\
   2579.35
   \end{pmatrix}
   \]
   this vector represents the production levels for manufacturing, agriculture, communications, labor, and finances, respectively.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Anda di halaman 1dari 11

# The

Leontief
Production Model
Janet Aquino, Jonathan Guzman, Szuhara Jen
MATH 247Dr. Jen-Mei Chang
Leontiefs Question
In the 1930s, economist Wassily LeonJef divided the economy into 500
sectors, wriJng linear equaJons that described how each sector
distributed its output to other sectors. By creaJng this system of linear
equaJons, he created a 500 by 500 matrix and began developing the
Leon%ef Input-Output (or Produc%on) Model.
By creaJng this model, LeonJef aTempted to answer the following
quesJon: Is there a produc%on vector (also known as the producJon
level), x, that equals the total demand for producJon in an economy? In
other words, is there an x such that

x = i+d
where i = (intermediate demand), and d = (external demand)
There are two variaJons of this modela closed and an open model.
This presentaJon will deal with the open model with an iniJal d.

## Purpose: To nd an x that sa%ses both the internal demand and

external demands of the hypothe%cal economy of Nomansland.
Given the equaJon for the producJon vector

x = i + d,

we dene the intermediate demand vector as the augmented consumpJon
vectors Jmes x, or Cx; thus, we can solve for x:

x = Cx + d I x Cx = d (I C)x = d.
5 5

This expression works because I5-C can be inverted So long as C contains non-
nega3ve entries and its column sums are less than 1, (I5-C) is inver3ble. A
column sum less than 1 implies that not all producJon is being used within the
economy of Nomansland and allows for an external demand of producJon (i.e., d).

The Initial Conditions
The LeonJef ProducJon Model makes three assumpJons:
The prices of goods and services are held
input and output is measured in millions of dollars (hence
referred to as units of)
and the model assumes that for every sector of the economy
there is a unit consumpJon vector c, that lists the inputs needed
for the units of output in the sector.

## Let n equal the number of sectors in the economy. For

simplicity the following examples will hold n at 5the ve
sectors of manufacturing, agriculture, communicaJons,
services (labor) and nances.
The Model
Assume for the economy of Nomansland, the units each sector
consumes per unit of output are as follows (n = 5):

Inputs Purchased Manufacturing Agriculture Communica3ons Labor Finances
From... (c1) (c2) (c3) (c4) (c5)

## Finances: .10 .15 .15 .10 .05

If manufacturing were to produce 100 units, then it would consume 30 units from
manufacturing; 15 from agriculture; 25 from communicaJons; 15 from labor; and
10 from nances.
The columns of the table the consump%on vectors for each sectordenoted cj. If xj
is the units produced by each sector, then xjcj is the intermediate demand for the
sector. Given the previous example:
" .3 % " 30 %
\$ ' \$ '
\$ .15 ' \$ 15 '
x1 = 100 x1c1 = 100 \$ .25 '=\$ 25 '.
\$ .15 ' \$ 15 '
\$ ' \$ '
\$# .1 '& \$# 10 '&
In general, the total consumpJon of Nomansland is
5

x c j j = x1c1 + x2 c 2 + x3c3 + x4 c 4 + x5c 5 = Cx.
j=1

Thus, for the economy of Nomansland,
" 1.3 .15 .2 .15 .05 % " .7 .15 .2 .15 .05 %
\$ ' \$ '
\$ .15 1.35 .15 .35 .05 ' \$ .15 .65 .15 .35 .05 '
I 5 C = \$ .25 .05 1.3 .2 .4 ' = \$ .25 .05 .7 .2 .4 '.
\$ .15 .25 .15 1.1 .4 ' \$ .15 .25 .15 .9 .4 '
\$
.1 .15 .15 .1 1.05
' \$
.1 .15 .15 .1 .95
'
\$# '& \$# '&

To nd the inverse, augment (I5-C) with I5 and obtain the row-reduced
echelon form; the right-side matrix will be (I5-C)-1:
" 1 0 0 0 0 4.130 2.913 3.009 2.803 2.818 %
\$ '
rref \$
0 1 0 0 0 3.707 4.959 3.701 3.769 3.601 '
[ I5 C | I5 ] ~ \$ 0 0 1 0 0 3.776 3.482 4.834 3.490 3.887 '.
\$ 0 0 0 1 0 3.216 3.328 3.226 4.036 3.402 '
\$ '
\$# 0 0 0 0 1 1.955 1.990 2.004 1.866 2.890 '&

Then, assume that the nal demand vector dthe producJon demanded by
external sectorsis ! 250 \$
# &
# 350 &

d = # 200 &.
# 300 &
# &
#" 150 &%
Finally, perform matrix mulJplicaJon to obtain x:

# 4.130 2.913 3.009 2.803 2.818 &# 250 &
% (% (
% 3.707 4.959 3.701 3.769 3.601 (% 350 (

1
x = (I 5 C) d = % 3.776 3.482 4.834 3.490 3.887 (% 200 (
% (% (
%
3.216 3.328 3.226 4.036 3.402
(%
300
(
%\$ 1.955 1.990 2.004 1.866 2.890 ('%\$ 150 ('

" 3917.45 % Manufacturing

\$ '
\$ 5073.45 ' Agriculture

x \$ 4759.55 ' Communications
\$ '
4335.10 Labor
\$ '
\$# 2579.35 '& Finances

This is the producJon vectorthe level of output at which sectors of the
economy can saJsfy their demand for inputs and the outputs of external
enJJes. It is important to note that only at this producJon vector will the
aforemenJoned demands be saJsed.
In fact, (I
5
C)x
= d
will always yield a unique producJon vector.

Is the Economy of Nomansland
Productive?
By nding a unique soluJon for x, it is certain that Nomanslands economy is
producJvesaJsfying its internal and external demand for producJon. However,
producJvity can also be generalized by analyzing eigenvalues.
If Anxn has a largest eigenvalue , then the corresponding eigenvalue of A-1 is -1;
then eigenvalue of Anxn+aIn is (+a), where a is any real number; lastly, since the
eigenvectors of A and A-1 are the same, the eigenvalues of (I5-C) can be analyzed:
(I 5 C) = (C 1I 5 ) : a = 1
eigenvalue(I5 C ) : ( 1) = 1
1
eigenvalue(I C )1 :
1
5

## Consider (I5-C)-1x = (1-)-1x:

1. If < 1, then (I5-C)-1 is inverJble and thus the economy saJses internal
and external demand.
2. If = 1, then (I5-C)-1 is non-inverJble and the economy is closed.
3. And if > 1, then (I5-C)-1 is not non-negaJve and thus the economy
saJses no demand.
Conclusion
While 100-Jmes smaller, this sampling of the LeonJef ProducJon
Model gives insight into how the true, larger one funcJons: In the
midst of simultaneous internal and external demand, there always
exists a producJon level to saJsfy them.
The concepts in linear algebra and the study of economics are
closely intertwined; applying vectors and matrices to economics is a
concise way to organize economic schedules and make
user
The text passage describes the Leontief Production Model, which is a mathematical model used to describe the interdependencies between different sectors of an economy. It includes a detailed example of how to solve for the production vector \( x \) using matrix operations, and it provides a specific numerical example with a solution.

TYPE: SINGLE_QUESTION_WITH_ANSWER

1. The passage contains a mathematical question: "Is there a production vector \( x \) that equals the total demand for production in an economy?"
2. The answer to this question is provided through the example, where the production vector \( x \) is calculated to satisfy both internal and external demands. The specific solution is given as:
   \[
   x = \begin{pmatrix}
   3917.45 \\
   5073.45 \\
   4759.55 \\
   4335.10 \\
   2579.35
   \end{pmatrix}
   \]
   This vector represents the production levels for manufacturing, agriculture, communications, labor, and finances, respectively.