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