Leontief Input-Output Model

Becky Siegel

Carson Finkle

April 23 2008

What is the model about?

Used to describe the relationship of industries within a sectorInternationalNationalRegionalWithin a business

Assumptions of the Model

Fixed proportions of inputs Demand is met fully without a surplus

or shortage Internally for closed modelInternally + external demand for open

model

The Equation (closed) Let C be a consumption matrix such that and X be

the production vector such that

Economic Activities

Inputs to Energy

Inputs to Manufacturing

Inputs to Services

Total Production

Energy .2 .3 .2 ?

Manufacturing .5 .2 .3 ?

Services .3 .5 .5 ?

C X⏐ →⏐

= X⏐ →⏐

Closed Model

Energy Manufact-uring

Services

Ener. Manuf . Services

C =.2 .3 .2.5 .2 .3.3 .5 .5

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

EnergyManufacturingServices

= consumption matrix

Want to Find C x⏐ →⏐

= x⏐ →⏐so need to find eigenvector associated

with an eigenvalue of 1 (if it exists)

Characteristic polynomial: x3 −.9x2 −.12x+ .02λ =1,−.2,.1, want to find eigenvector for E1

C −λI =C−I =−.8 .3 .2 0.5 −.8 .3 0.3 .5 −.5 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

RREF =−.8 .3 .2 0.5 −.8 .3 0.3 .5 −.5 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥~

1 0−2549

0

0 1−3449

0

0 0 0 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

x1 =2549x3

x2 =3449x3

E1 =span

254934491

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

⎫

⎬

⎪⎪⎪

⎭

⎪⎪⎪

=span.51.691

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

The Equation (open) Let C be a consumption matrix such that and X be

the production vector such that

C X⏐ →⏐

+ D⏐ →⏐

= X⏐ →⏐

Economic

Activities

Inputs to

Labor

Inputs to

Transportation

Inputs to

Food

External

Demand

Total

Production

Labor 0 .5 .5 10,000 ?

Transportation .4 .3 .05 20,000 ?

Food .2 0 .35 10,000 ?

Open Model

Labor Transportation

FoodExternal

Demand

X⏐ →⏐

=C X⏐ →⏐

+ D⏐ →⏐

X⏐ →⏐

−C X⏐ →⏐

= D⏐ →⏐

(I −C) X⏐ →⏐

= D⏐ →⏐

X⏐ →⏐

=(I −C)−1 D⏐ →⏐

Solving for x vector

Labor Trans Food

C =0 .5 .5.4 .3 .05.2 0 .35

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

LaborTransportation

Food= consumption matrix

D=10,00020,00010,000

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ = external demand function

I −C =1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−

0 .5 .5.4 .3 .05.2 0 .35

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

1 −.5 −.5−.4 −.7 −.05−.2 0 .65

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

I −C−1 =1 −.5 −.5 1 0 0−.4 −.7 −.05 0 1 0−.2 0 .65 0 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥~

1 0 0 1.82 1.3 1.50 1 0 1.08 2.2 10 0 1 .56 .4 2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

X⏐ →⏐

=(1−C)−1D=1.82 1.3 1.51.08 2.2 1.56 .4 2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

10,00020,00010,000

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ =

59,20064,80033,600

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Just for fun…

Economic

Activities

Inputs to

Labor

Inputs to

Transportation

Inputs to

Food

External

Demand

Total

Production

Labor .5 .0 .25 10,000 ?

Transportation .2 .8 .8 20,000 ?

Food 1 .4 0 10,000 ?

Labor Trans Food

C =.5 0 .25.2 .8 .81 .4 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

LaborTransportation

Food= consumption matrix

D=10,00020,00010,000

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ = external demand function

I −C =1 0 00 1 00 0 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−

.5 0 .25

.2 .8 .81 .4 0

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥=

.5 0 .25

.2 .2 .81 .4 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

I −C−1 =.92 −.8 −.38−7.69 −1.92 −3.46−2.15 −1.54 .8

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

X⏐ →⏐

−C X⏐ →⏐

= D⏐ →⏐

⏐ →⏐ (I −C) X⏐ →⏐

= D⏐ →⏐

⏐ →⏐ X=(I −C)−1 D⏐ →⏐

(I −C)−1D=.92 −.8 −.38−7.69 −1.92 −3.46−2.15 −1.54 .8

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

10,00020,00010,000

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥ =

-75,300-152,200-44,300

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

Wrap Up

ContributionsLimitationsConclusions