leontief 1
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Transcript of leontief 1
Leontief Input-Output Model
Example: Suppose that we have an
economy with labor, transportation,
and food industries. Let $1 in labor
require 40 cents in transportation and
20 cents in food; while $1 in trans-
portation takes 50 cents in labor and
30 cents in transportation; and $1 in
food production uses 50 cents in la-
bor, 5 cents in transportation, and 35
cents in food.
Let the demand for the current pro-
duction period be $10,000 labor,$20,000
transportation, and $10,000 food.
Find the production schedule for the
economy.1
Solution: Let x1, x2, and x3 be the
dollar values of labor, transportation,
and food produced, respectively. Let
X =
x1x2x3
be the production vector,
and d =
10,00020,00010,000
be the demand vector.
Total production – Consumption= outside demand
.
That is, X − CX = d .
The consumption matrix is
C =
Labor Transpo. Food
0 0.5 0.50.4 0.3 0.050.2 0 0.35
LaborTranspo.Food
2
We have X − CX = d. It gives
(I−C)X = d. Thus, X = (I − C)−1d .
I − C =
1 0 00 1 00 0 1
−
0 0.5 0.50.4 0.3 0.050.2 0 0.35
=
1 −0.5 −0.5−0.4 0.7 −0.05−0.2 0 0.65
Now, we need to find (I − C)−1.
1 −0.5 −0.5 1 0 0−0.4 0.7 −0.05 0 1 0−0.2 0 0.65 0 0 1
∼1 −0.5 −0.5 1 0 00 0.5 −0.25 0.4 1 00 −0.1 0.55 0.2 0 1
3
∼1 −0.5 −0.5 1 0 00 1 −0.5 0.8 2 00 0 0.5 0.28 0.2 1
∼1 −0.5 −0.5 1 0 00 1 −0.5 0.8 2 00 0 1 0.56 0.4 2
∼1 −0.5 0 1.28 0.2 10 1 0 1.08 2.2 10 0 1 0.56 0.4 2
∼1 0 0 1.82 1.3 1.50 1 0 1.08 2.2 10 0 1 0.56 0.4 2
So,
(I − C)−1 =
1.82 1.3 1.51.08 2.2 10.56 0.4 2
.
4
Thus,
X = (I − C)−1d
=
1.82 1.3 1.51.08 2.2 10.56 0.4 2
10,00020,00010,000
=
18,200 + 26,000 + 15,00010,800 + 44,000 + 10,000
5600 + 8,000 + 20,000
=
592006480033600
So, the production schedule should
be $59,200 labor, $64,800 transporta-
tion, and $33,600 food.
5
Example: A company has two inter-
acting branches, B1 and B2. Branch
B1 consumes $0.5 of its own output
and $0.2 of B2-output for every $1 it
produces. Branch B2 consumes $0.6
of B1-output and $0.4 of its own out-
put per $1 of output.
The company wants to know how much
each branch should produce per month
in order to meet exactly a monthly
external demand of $50,000 for B1-
product and $40,000 for B2-product.
(a) Set up (without solving) a linear
system whose solution will represent
the required production schedule.
6
(b) Find a production schedule for
the above external demand.
(c) Determine whether or not every
nonnegative external demand could be
satisfied by a nonnegative production
schedule.
Solution: Let x1 and x2 be the dollar
values of outputs of branch B1 and
B2, respectively. Let X =
x1
x2
be
the production vector, and
d =
50,00040,000
be the demand vector.
(a) Consumption matrix is
B1 B2
C =
0.5 0.60.2 0.4
B1
B2
7
We know that (I − C)X = d. So,
1 00 1
−
0.5 0.60.2 0.4
x1
x2
=
50,00040,000
Then, 0.5 −0.6−0.2 0.6
x1
x2
=
50,00040,000
So,
0.5x1 − 0.6x2 = 50,000
−0.2x1 + 0.6x2 = 40,000
8
(b) We know that X = (I − C)−1d.
So, we need to find (I − C)−1.
I − C =
0.5 −0.6−0.2 0.6
.
Then,
(I − C)−1 =1
0.3− 0.12
0.6 0.60.2 0.5
=1
0.18
0.6 0.60.2 0.5
=
60/18 60/1820/18 50/18
=
10/3 10/310/9 25/9
So,
X =
x1
x2
= (I − C)−1d
9
=
10/3 10/310/9 25/9
50,00040,000
=
300,000500,000/3
So, the production schedule is $300,000
of outputs of branch B1, and $166,666.67
of outputs of branch B2.
(c) Since all entries of (I − C)−1 are
nonnegative, then a nonnegative pro-
duction vector can be found for any
given nonnegative demand.
In this case, the economy (and the
consumption matrix C) is said to be
productive.10
REMARK: In the consumption ma-
trix, if a column sums to less than 1,
then the corresponding industry con-
sumes less than $1 in order to pro-
duce $1 of output.
In this case, the industry (or the sec-
tor)is said to be profitable.
If all the industries are profitable then
the economy is productive.
If all row sums are less than 1, then
the economy can output $1 of each
industry while internally using less. So
the economy will be productive.
11
Example: 0.5 0.40.7 0.1
11
=
0.5 + 0.40.7 + 0.1
=
0.90.8
.
The above economy has two sectors.
While each having $1 value of prod-
uct, the first sector spends $0.9 and
the second sector spends $0.8.
Example: Let the consumption ma-
trix of an economy be
A M L
C =
0.3 0 0.20.2 0.6 0.30.4 0.2 0
AML
.
Since each column is less than 1, each
industry is profitable. So, the econ-
omy is productive.12
Example: Let the consumption ma-
trix of an economy be
A M L
C =
0.4 0 0.20.1 0.5 0.30.6 0.2 0
AML
.
The first column sum is 1.1. So, the
first industry is not profitable. The
second and the third industries are
profitable. Since each row sum is less
than 1, the economy can output $1
of each industry while internally using
less. So, the economy is productive.
13
Example: Let an economy contain
agriculture, manufacturing, and labor
industries. Let $1 of agriculture re-
quire 50 cents in agriculture, 20 cents
in manufacturing, and 100 cents in la-
bor. Let $1 of manufacturing use 80
cents in manufacturing and 40 cents
labor, while $1 labor takes 25 cents
agriculture and 10 cents manufactur-
ing.
Show that the economy is productive,
and find the production schedule if
demand is for $100 agriculture, $500
manufacturing, and $700 labor.
14
Solution: Let x1, x2 and x3 be the
dollar values of outputs of manufac-
turing, agriculture, and labor, respec-
tively. Let X =
x1x2x3
be the produc-
tion vector, and let d =
100500700
be the
demand vector.
The consumption matrix is
A M L0.5 0 0.250.2 0.8 0.11 0.4 0
AML
Two of the columns and two of the
rows have sums greater than 1.15
We need to find (I − C)−1:
0.5 0 −0.25 1 0 0−0.2 0.2 −0.1 0 1 0−1 −0.4 1 0 0 1
∼
1 0 −0.5 2 0 0−0.2 0.2 −0.1 0 1 0−1 −0.4 1 0 0 1
∼1 0 −0.5 2 0 00 0.2 −0.2 0.4 1 00 −0.4 0.5 2 0 1
∼1 0 −0.5 2 0 00 1 −1 2 5 00 −0.4 0.5 2 0 1
∼1 0 −0.5 2 0 00 1 −1 2 5 00 0 0.1 2.8 2 1
16
∼1 0 −0.5 2 0 00 1 −1 2 5 00 0 1 28 20 10
∼1 0 0 16 10 50 1 0 30 25 100 0 1 28 20 10
.
Since all the entries are nonnegative,
the economy is productive. Then
X =
x1x2x3
= (I − C)−1d
=
16 10 530 25 1028 20 10
100500700
=
10,10022,50019,800
.
17
Example: Let the consumption ma-
trix of an economy be
C =
0.5 0 0.250.2 0.8 0.81 0.4 0
=
1/2 0 1/41/5 4/5 4/51 2/5 0
.
Column sums are not less than 1. Row
sums are not less than 1 either. So,
we need to find (I − C)−1, which is
12/13 −10/13 −5/13−100/13 −25/13 −45/13−28/13 −20/13 −10/13
.
Since there are negative entries in
(I − C)−1, the economy is not pro-
ductive.
18
EXERCISES:
1) Consider an economy which has
stell plant, coal mine and transporta-
tion.
To produce $1 value of steel requires
50 cents from steel plant, 30 cents
from coal mine, and 10 cents from
transportation.
To produce $1 value of coal requires
10 cents from steel plant, 20 cents
from coal mine, and 30 cents trans-
portation.
$1 value of transportation uses 10 cents
from steel plant, 40 cents from coal
mine, and 5 cents from transporta-
tion.19
Assume that the outside demand for
the current production period is 2 mil-
lion dollars for steel, 1.5 million dol-
lars for coal, and $500,000 for trans-
portation. How much should each
industry produce to satisfy the de-
mands?
2) Let an economy be divided into
three sectors: manufacturing, agri-
culture, and services.
For each unit of output, manufac-
turing requires 0.10 unit from other
companies in the sector, 0.30 unit from
agriculture, and 0.30 unit from ser-
vices.20
For each unit of output, agriculture
uses 0.20 unit of its own output, 0.60
unit from manufacturing, and 0.10
unit of services.
For each unit of output, the services
sector consumes 0.10 unit of services,
0.60 unit from manufacturing, but no
agricultural products.
a) Construct the consumption ma-
trix for this economy, and determine
what intermediate demands are cre-
ated if agriculture plans to produce
100 units.
b) Determine the production levels
needed to satisfy a final demand of 18
units for manufacturing, 18 units for
agriculture, and 0 units of services.21
3) An economy has two interactive
sectors: S1 and S2.
To produce one dollar’s worth of out-
put, sector S1 consumes $.10 of its
own production and $.80 of S2-production.
To produce one dollar’s worth of out-
put, sector S2 consumes $.10 of its
own production and $.20 of S1-production.
Suppose that there is an outside de-
mand of $10,000 for S1-product, and
$20,000 for S2-product.
a) How much should each sector pro-
duce to satisfy this final demand?
b) Can every nonnegative external de-
mand be satisfied by a nonnegative
production schedule?22
4) Let an economy contain three in-
dustries: electricity, oil and pipeline.
For each $1 of electricity generated,
let there be charges of $.20 for elec-
tricity to run auxiliary equipment, $.40
for oil to power the generators, and
$.10 in pipeline usage. For each $1
of oil produced, suppose that $.10 is
spent on electricity and $.40 for oil
to produce steam that is pumped into
the well.
23
Finally, for each $1 in pipeline usage,
suppose that $.30 is spent on electric-
ity while $.20 is spent on oil to heat
the pipeline. Suppose that there is
an outside demand for $4200 worth
of electricity, $8400 worth of oil, and
$12,600 in pipeline usage.
a) How much should each industry
produce?
b) Is the consumption matrix produc-
tive?
24