Energy Management: 2013/2014

Energy Analysis: Input-OutputClass # 5

Prof. Tânia Sousa

taniasousa@ist.utl.pt

Input-Output Analysis: Motivation

• Energy is needed in all production processes

• Different products have different embodied energies

or specific energy consumptions

– How can we compute these?

Input-Output Analysis: Motivation

• Energy is needed in all production processes

• Block Diagrams Methodology

– To compute embodied energies or specific energy

consumptions of different products

– To compute the impact of energy efficiency measures in the

specific energy consumptions of a product

• Input-Output Methodology

– To compute the embodied energies for all products/sectors in

an economy simultaneously (no need to consider specific

consumption of inputs equal to zero)

– To compute the impact of energy efficiency measures across

the economy

– To compute energy needs for different economic scenarios

Input-Output Analysis: Motivation

• Build a scenario for the economy in a consistent way

is difficult because of the interdependence within the

economic system

– a change in demand of a product has direct and indirect

effects that are hard to quantify

Input-Output Analysis: Motivation

• Build a scenario for the economy in a consistent way

is difficult because of the interdependence within the

economic system

– a change in demand of a product has direct and indirect

effects that are hard to quantify

– Example:

– To increase the output of chemical industry there is a direct &

indirect (electr.) increase in demand for coal

Chemical Industry Power Plant

Coal Mine

Input-Output Analysis: Motivation

• Portuguese Scenarios for 2050:

http://www.cenariosportugal.com/

Input-Output Analysis: Basics

• Input-Output Technique

– A tool to estimate (empirically) the direct and indirect change

in demand for inputs (e.g. energy) resulting from a change in

demand of the final good

– Developed by Wassily Leontief in 1936

and applied to US national accounts in

the 40’s

– It is based on an Input-output table which is a matrix whose

entries represent:

• the transactions occurring during 1 year between all sectors;

• the transactions between sectors and final demand;

• factor payments and imports.

Input-Output Portugal

• Input-Output matrix Portugal (2008)

PRODUCTS (CPA*64) R01 R02 R03 RB R10_12

R01 Products of agriculture, hunting and related services 954,9 18,4 0,0 0,0 4275,2

R02 Products of forestry, logging and related services 0,0 103,4 0,0 0,0 0,0

R03

Fish and other fishing products; aquaculture products; support services to

fishing0,0 0,0 38,4 0,0 40,5

RB Mining and quarrying 0,5 0,0 0,0 152,7 10,6

R10_12 Food products, beverages and tobacco products 1284,7 0,1 3,9 1,1 3012,0

R13_15 Textiles, wearing apparel and leather products 21,1 0,0 4,0 5,3 1,2

R16

Wood and of products of wood and cork, except furniture; articles of straw

and plaiting materials30,4 0,0 0,0 1,8 58,5

R17 Paper and paper products 8,2 0,0 1,3 2,2 304,3

R18 Printing and recording services 4,0 0,3 1,8 4,3 49,5

R19 Coke and refined petroleum products 224,8 14,3 38,6 144,3 99,4

R20 Chemicals and chemical products 225,9 10,2 0,8 31,8 106,5

R21 Basic pharmaceutical products and pharmaceutical preparations 6,3 0,0 0,0 0,1 12,1

Input-Output Portugal

• DPP (Departamento de Prospectiva e Planeamento e

Relações Internacionais) that belongs to the MAOT

developed an input-output model MODEM1 which

has been used to evaluate the macroeconomic,

sectorial and regional impacts of public policies

• O DPP has online the input-output matrix for 2008

with 64 × 64 sectors

• World Input-Output Database for some countries from

1995 onwards:

http://www.wiod.org/database/nat_suts.htm

Input-Output: Basics

For the “Tire Factory”

x1= z11+ z12+… + z1n+ f1

Output from sector 1 to sector 2 Output from sector 1

to final demand

Total Production

from sector 1

Tire Factory

Automobile

Factory

Individual

Consumers

Input-Output: Basics

For the Electricity Sector:

xi= zi1+ zi2+… + zii+… + zin+ fi

Input-Output: Basics

For the Electricity Sector:

xi= zi1+ zi2+… + zii+… + zin+ fi

Output from sector i

to sector 2Output from sector i

to final demandTotal production

from sector i

Electricity Sector

Automobile

Factory

Individual

Consumers

Input-Output: Basics

For the Electricity Sector:

xi= zi1+ zi2+… + zii+… + zin+ fi

Output from sector i

to sector 2Output from sector i

to final demandTotal production

from sector i

Electricity Sector

Automobile

Factory

Individual

Consumers

What is the meaning of this?

Input-Output: Basics

For the Electricity Sector:

xi= zi1+ zi2+… + zii+… + zin+ fi

Output from sector i

to sector 2Output from sector i

to final demandTotal production

from sector i

Electricity Sector

Automobile

Factory

Individual

Consumers

Electricity consumed within the

electricity sector: hydraulic pumping &

electric consumption at the power

plants & losses in distribution

Input-Output: Basics

For all sectors:

zij is sales (ouput) from sector i to (input in) sector j (in ? units)

fi is final demand for sector i (in ? units)

xi is total output for sector i (in ? units)

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

x z z f

x z z f

x z z f

= + + +

= + + +

= + + +

⋮

Input-Output: Basics

For all sectors:

zij is sales (ouput) from sector i to (input in) sector j (in money

units)

fi is final demand for sector i (in money units)

xi is total output for sector i (in money units)

• The common unit in which all these inputs & outputs

can be measured is money

• Matrix form?

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

x z z f

x z z f

x z z f

= + + +

= + + +

= + + +

⋮

Input-Output: Basics

For all sectors:

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

x z z f

x z z f

x z z f

= + + +

= + + +

= + + +

⋮

= +x Zi f

i is a column vector of 1´s with the correct

dimension

Lower case bold letters for column vectors

Upper case bold letters for matrices

vector of sector output

vector of final demand

matrix with intersectorial transactions

x

f

Z

Input-Output: Matrix A

of technical coefficients

Let’s define:

• What is the meaning of aij?

zij is sales (ouput) from sector i to (input in) sector j

xj is total output for sector j

ij

ij

j

za

x=

Input-Output: Matrix A

of technical coefficients

Let’s define:

• The meaning of aij:

– aij input from sector i (in money) required to produce one unit

(in money) of the product in sector j

– aij are the transaction or technical coefficients

ij

ij

j

za

x=

Input-Output: Matrix A

of technical coefficients

Rewritting the system of equations using aij:

• How can it be written in a matrix form?

ij

ij

j

za

x=

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

x z z f

x z z f

x z z f

= + + +

= + + +

= + + +

⋮

vector of sector output

vector of final demand

matrix of technical coefficients

x

f

A

Input-Output: Matrix A

of technical coefficients

Rewritting the system of equations using aij:

• In a matrix form:

= +x Ax f

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

x z z f

x z z f

x z z f

= + + +

= + + +

= + + +

⋮

ij

ij

j

za

x=

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

vector of sector output

vector of final demand

matrix of technical coefficients

x

f

A= +x Zi f

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

Input-Output: Matrix A

of technical coefficients

• The meaning of matrix of technical coefficients A:

– What is the meaning of this row?

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

ij

ij

j

za

x=

Input-Output: Matrix A of technical

coefficients

• The meaning of matrix of technical coefficients A:

– Row i represents the outputs from sector i

• a12X2 output (sales) of sector 1 to sector 2

• a12 is output (sales) of sector 1 to sector 2 per unit of

sector 2

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

ij

ij

j

za

x=

Input-Output: Matrix A of technical

coefficients

• The meaning of matrix of technical coefficients A:

– What is the meaning of this column?

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

ij

ij

j

za

x=

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

Input-Output: Matrix A of technical

coefficients

• The meaning of matrix of technical coefficients A:

– Column i represents the inputs to sector i

– The sector i produces goods according to a fixed production

function (recipe)

• Sector 1 produces X1 units (money) using a11X1 units of sector 1, a21X1units of sector 2, … , an1X1 units of sector n

• Sector 1 produces 1 units (money) using a11 units of sector 1, a21 units

of sector 2, … , an1 units of sector n

Inputs to sector 1

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

ij

ij

j

za

x=

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

Production Functions: a review

• Production functions specify the output x of a factory,

industry or economy as a function of inputs z1, z2, …:

• Examples:1 2( , ,...)x f z z=

1 2....b cx az z=

1 2....x a bz cz= + + +

Cobb-Douglas Production Function

Linear Production Function

Production Functions: a review

• Production functions specify the output x of a factory,

industry or economy as a function of inputs z1, z2, …:

• Examples:

• Which of these productions functions allow for substitution between

production factors?

Cobb-Douglas Production Function

Linear Production Function

1 2( , ,...)x f z z=

1 2....b cx az z=

1 2 ....x a bz cz= + + +

Production Functions: a review

• Production functions specify the output x of a factory,

industry or economy as a function of inputs z1, z2, …:

• Examples:

• Which of these productions functions allow for substitution

between production factors?

• Cobb-Douglas and Linear production functions

Cobb-Douglas Production Function

Linear Production Function

1 2( , ,...)x f z z=

1 2....b cx az z=

1 2 ....x a bz cz= + + +

( )1 2 1 20.8 0.2 1b

x a bz cz a b z c zc

= + + = + + +

Production Functions: a review

• Production functions specify the output x of a factory,

industry or economy as a function of inputs z1, z2, …:

• Examples:

• Which of these productions functions allow for scale economies?

Cobb-Douglas Production Function

Linear Production Function

1 2( , ,...)x f z z=

1 2....b cx az z=

1 2....x a bz cz= + + +

Production Functions: a review

• Production functions specify the output x of a factory,

industry or economy as a function of inputs z1, z2, …:

• Examples:

• Which of these productions functions allow for scale economies?

• Cobb-Douglas (if b+c >1)

Cobb-Douglas Production Function

Linear Production Function

1 2( , ,...)x f z z=

1 2....b cx az z=

1 2....x a bz cz= + + +

( ) ( ) ( ) ( )1 2 1 22 2 2 2 2 if 1b c b cb c b c b c

a z z a z z x x b c+ + += = > + >

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

Input-Output: Matrix A

of technical coefficients

• The meaning of matrix of technical coefficients A:

– Production function assumed in the Input-Output Technique

• Sector 1 produces X1××××1 units (money) using X1×××× a11 units of sector 1, X1×××× a21 units of sector 2, … , X1×××× an1 units of sector n

• Is there substitution between production factors?

• Are scale economies possible?

Inputs to sector 1

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

ij

ij

j

za

x=

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

Input-Output: Matrix A

of technical coefficients

• The meaning of matrix of technical coefficients A:

– Production function assumed in the Input-Output Technique

• Sector 1 produces X1××××1 units (money) using X1×××× a11 units of sector 1, X1××××a21 units of sector 2, … , X1×××× an1 units of sector n

• Is there substitution between production factors? (input proportions are

fixed)

• Are scale economies possible?

Inputs to sector 1

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

ij

ij

j

za

x=

11 211

11 21

...z z

xa a

= = =

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

Input-Output: Matrix A

of technical coefficients

• The meaning of matrix of technical coefficients A:

– Production function assumed in the Input-Output Technique

• Sector 1 produces X1××××1 units (money) using X1×××× a11 units of sector 1, X1×××× a21 units of sector 2, … , X1×××× an1 units of sector n

• Leontief which does 1) not allow for substitution between production

factors and 2) not allow for scale economies

Inputs to sector 1

Leontief Production Function

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

ij

ij

j

za

x=

( )1 11 11 21 21min , ,....x z a z a=

Input-Output: Matrix A

of technical coefficients

• The meaning of matrix of technical coefficients A:

– Production function assumed in the Input-Output Technique

• Sector 1 produces X1××××1 units (money) using X1×××× a11 units of sector 1, X1×××× a21 units of sector 2, … , X1×××× an1 units of sector n

• Leontief which does not allow for 1) substitution between production

factors or 2) scale economies

• Matrix A is valid only for short periods (~5 years)

Inputs to sector 1

1 11 12 1 1 1

2 21 22 2 2

1 2

...

... ...

... ... ... ... ... ... ...

...

n

n n n nn n n

x a a a x f

x a a x f

x a a a x f

= +

1 11 1 12 2 1

2 21 1 22 2 2

1 1 2 2

...

...

...n n n n

x a x a x f

x a x a x f

x a x a x f

= + + +

= + + +

= + + +

⋮

ij

ij

j

za

x=

• Intermediate inputs: intersector

and intrasector inputs

• Final Demand: exports &

consumption from households

and government & investment

Input-Output Analysis: The model

• The input-ouput model

Intermediate

Inputs(square matrix)

Primary Inputs

Total Inputs or

Total Costs

Fin

al D

eman

d

Tota

l ou

tpu

t

Outputs

Inputs

Sectors

Sectors

Z

ij ij jz a x= f x

Input-Output Analysis: The model

• The input-ouput model

• Primary inputs: payments (wages,

rents, interest) for primary factors

of production (labour, land,

capital) & taxes & imports

Intermediate

Inputs(square matrix)

Primary Inputs

Total Inputs or

Total Costs

Fin

al D

eman

d

Tota

l ou

tpu

t

Outputs

Inputs

Sectors

Sectors

pi

xf

Z

Input-Output Analysis: The model

• The input-ouput model

Lines & columns are related by:

Intermediate

Inputs(square matrix)

Primary Inputs

Total Inputs or

Total Costs

Fin

al D

eman

d

Tota

l ou

tpu

t

Outputs

Inputs

Sectors

Sectors

1 1

1 1

n n

ij i i ji i

j j

n n

ij i i i i ji i i

j j

z f x z pi

z c g e inv z av i

= =

= =

+ = = +

+ + + + = + +

∑ ∑

∑ ∑

xf

Z

pi

Input-Output Analysis: The model

• The input-ouput model

Lines & columns are related by:

Intermediate

Inputs(square matrix)

Primary Inputs

Total Inputs or

Total Costs

Fin

al D

eman

d

Tota

l ou

tpu

t

Outputs

Inputs

Sectors

Sectors

1 1

1 1

n n

ij i i ji i

j j

n n

ij i i i i ji i i

j j

z f x z pi

z c g e inv z av i

= =

= =

+ = = +

+ + + + = + +

∑ ∑

∑ ∑

xf

Z

PI

´+ = +Zi f pi i Z

+ =Ax f x

1ˆ −

=

=

Ax Zi

A Zx

Input-Output Analysis:

Leontief inverse matrix

• How to relate final demand to production?

• Leontief inverse matrix which can be obtained as:

( )

( ) 1−

+ =

= −

= −

− =

=

Ax f x

f x Ax

f I A x

I A f x

Lf x

( ) 1

vector of sector output

vector of final demand

matrix of technical coefficients

Leontief inverse matrix−

−

x

f

A

I A

( ) 1 2 3

0

... j

j

∞−

=

− = + + + + =∑I A I A A A A

Input-Output Analysis: Leontief inverse or total requirements matrix

• can be used to answer:

– If final demand in sector i, fi, (e.g. agriculture) is to increase

10% next year how much output from each of the sectors

would be necessary to supply this final demand?

• Total Output is:

– If accounts for the final demand in total output (e.g. cars

consumed by households) – direct effects

– Af accounts for the intersectorial needs to produce If (e.g. steel to produce the cars) – 1st indirect effects

– A[Af] accounts for the intersectorial needs to produce Af(e.g. coal to produce the steel) – 2nd indirect effects

=x Lf

( ) ( )1 2 3 ...−

= − = + + + +x I A f I A A A f

Input-Output Analysis: Leontief inverse or total requirements matrix

• Impacts in output from marginal increases in final

demand from f to fnew:

1 1 11 1 1 1

1

1 11 1 1

1

...

... ... ... ... ...

...

...

... ... ... ... ...

...

new new

n

n n n nn n n

n

n n nn n

x x l l f f

x x l l f f

x l l f

x l l f

=

+ ∆ + ∆ =

+ ∆ + ∆ ∆ ∆ = ∆ ∆ ∆ = ∆

x Lf

x L f

Input-Output: Multipliers

• Total output is:

?

1 11 1 1

1

...

... ... ... ... ...

...

n

n n nn n

x l l f

x l l f

=

=

x Lf

1 11 1 12 2

1 1 2 2

...

...n n n

x l f l f

x l f l f

= + +

= + +

⋮

iij

j

xl

f

∂=∂

?

Input-Output: Multipliers

• Total output is:

– lij represents the production of good I, xi, that is directly and

indirectly needed for each unit of final demand of good j, fj

– What about lii?

x1 needed for one unit of f1

xn needed for one unit of f1

1 11 1 1

1

...

... ... ... ... ...

...

n

n n nn n

x l l f

x l l f

=

=

x Lf

1 11 1 12 2

1 1 2 2

...

...n n n

x l f l f

x l f l f

= + +

= + +

⋮

iij

j

xl

f

∂=∂

Input-Output: Multipliers

• Total output is:

– lij represents the production of good i, xi, that is directly and

indirectly needed for each unit of final demand of good j, fj

– lii > 1 represents the production of good i, xi, that is directly and

indirectly needed for each unit of final demand of good i, fi

x1 needed for one unit of f1

xn needed for one unit of f1

1 11 1 1

1

...

... ... ... ... ...

...

n

n n nn n

x l l f

x l l f

=

=

x Lf

1 11 1 12 2

1 1 2 2

...

...n n n

x l f l f

x l f l f

= + +

= + +

⋮

iij

j

xl

f

∂=∂

Input-Output: Multipliers

• Total output is:

– lij represents the production of good I, xi, that is directly and

indirectly needed for each unit of final demand of good j, fj

– What is the meaning of the i column sum?

x1 needed for one unit of f1

xn needed for one unit of f1

1 11 1 12 2

1 1 2 2

...

...n n n

x l f l f

x l f l f

= + +

= + +

⋮

iij

j

xl

f

∂=∂

1 11 1 1

1

...

... ... ... ... ...

...

n

n n nn n

x l l f

x l l f

=

=

x Lf

Input-Output: Multipliers

• Total output is:

– lij represents the production of good I, xi, that is directly and

indirectly needed for each unit of final demand of good j, fj

• Multiplier of sector i: the impact that an increase in

final demand fi has on total production (not on GDP)

x1 needed for one unit of f1

xn needed for one unit of f1

1 11 1 12 2

1 1 2 2

...

...n n n

x l f l f

x l f l f

= + +

= + +

⋮

iij

j

xl

f

∂=∂

1 11 1 1

1

...

... ... ... ... ...

...

n

n n nn n

x l l f

x l l f

=

=

x Lf

Input-Output: Multipliers

• Multipliers change over time and over regions because they depend on:

– the economy structure, size, the way exports and sectors are linked to each other and technology

1 11 1 1

1

...

... ... ... ... ...

...

n

n n nn n

x l l f

x l l f

=

=

x Lf

Input-Output: Primary Inputs

• For the primary inputs we define the coefficients:

– The added value of sector j per unit of production or imports

of sector j per unit of production are assumed to be constant

[ ] [ ][ ] [ ]

1 1 1

1 1 1

´ ... ...

´ ... ...

c n n c cn

c n n c cn

va x va x va va

m x m x m m

= =

= =

va

m

Input-Output: Primary Inputs

• For the primary inputs we define the coefficients:

– The added value of sector j per unit of production or imports

of sector j per unit of production are assumed to be constant

• To compute new values for added value or imports:

[ ] [ ][ ] [ ]

1 1 1

1 1 1

´ ... ...

´ ... ...

c n n c cn

c n n c cn

va x va x va va

m x m x m m

= =

= =

va

m

� �

�

1 1 1 10 0

... 0 ... 0 ...

0 0

new new

c c

new new new

new new

cn n cn n

new new

va x va x

va x va x

= = = =

=

c c

c

va va x va Lf

m m Lf

Input-Output: Primary Inputs

• Relevance:

GDP= Added Values

Final consumption Exports ImportsGDP = + −

∑∑ ∑ ∑

Exercise

• Considere the following Economy:

What is the meaning of this?

Exercise

• Considere the following Economy:

• Compute the matrix A of the technical coeficients:

Sales of Agric. to Indus. or

Inputs from Agriculture to

Industry

Exercise

• Matrix of technical coefficients:

ij

ij

j

za

x=

What is the meaning of this?

Exercise

• Matrix of technical coefficients:

• What happens to the matrix of technical coefficients

with time? Why?

The amount of agriculture products (in money)

needed to produce 1 unit worth of industry products

ij

ij

j

za

x=

Exercise

• Matrix of technical coefficients:

• Compute the Leontief inverse matrix:

ij

ij

j

Za

X=

( ) 10,

j

j

−

= ∞

− = ∑I A A

Exercise

• Matrix of technical coefficients:

• Compute the Leontief inverse matrix:

( ) 10,

j

j

−

= ∞

− = ∑I A A

What is the meaning of this?

x1=l11f1+l12f2+…[ ]L

Exercise

• Matrix of technical coefficients:

• Compute the Leontief inverse matrix:

( ) 10,

j

j

I A A−

= ∞

− = ∑

the quantity of agriculture products

directly and indirectly needed for each

unit of final demand of industry products

[ ]L

Exercise

• Matrix of technical coefficients:

• Compute the Leontief inverse matrix:

( ) 10,

j

j

I A A−

= ∞

− = ∑

What is the meaning of this?

x1=l11f1+l12f2+…

x2=l21f1+l22f2+…

x3=l31f1+l32f2+…

[ ]L

Exercise

• Matrix of technical coefficients:

• Compute the Leontief inverse matrix:

( ) 10,

j

j

I A A−

= ∞

− = ∑

Multiplier of the industry sector: the total

output needed for each unit of final

demand of industrial products[ ]L

Exercise

• Matrix of technical coefficients:

• Compute the Leontief inverse matrix:

( ) 10,

j

j

I A A−

= ∞

− = ∑

What is the sector whose increase in

final demand has the highest impact on

the production of the economy? [ ]L

Exercise

• If final demand in sector 1 (e.g. agriculture) is to

increase 10%

– What will be necessary changes in the final outputs of

agriculture, industry and services?=x Lf

Exports Private Cons. Final Demand Final Demand

20 30 50 55

30 40 70 70

10 30 40 40

= →

[ ]L =

Exercise

• If final demand in sector 1 (e.g. agriculture) is to

increase 10%

– What will be necessary changes in the final outputs of

agriculture, industry and services?

1

2

3

55 80.8

70 122

40 101.6

x

x

x

= = Initial x

Exports Private Cons. Final Demand Final Demand

20 30 50 55

30 40 70 70

10 30 40 40

= →

=x Lf

Exercise

• If final demand in sector 1 (e.g. agriculture) is to

increase 10%

– What will be necessary changes in the final outputs of

agriculture, industry and services?

– What will be the new sales of industry to agriculture?

1

2

3

5 5.8

0 2

0 1.6

x

x

x

∆ ∆ = = ∆

Exports Private Cons. Final Demand Final Demand

20 30 50 55

30 40 70 70

10 30 40 40

= →

=x Lf

Exercise

• If final demand in sector 1 (e.g. agriculture) is to

increase 10%

– What will be the new sales of industry to agriculture?

21 21 1 21.6x a x= = Initial x21=20

Exercise

• What is the new added value?

Exercise

• What is the new added value?

• GDP increased by 3%

1 2 3

20 40 30; ;

75 120 100

80.820 40 30

122 92.6975 120 100

101.6

c c cva va va

va

= = =

= =

∑

� �1 1

2

3 3

0 0

0 0 ...

0 0

new

c

new new new

c

new

c

va x

va

va x

= = =

c cva va x va Lf

Imports

A

C

B

20

5

30 3

5

2

6

2

5

95

65

150

120

500

Final Demand

Exercise

• Consider na economy based in 3 sectors, A, B e C.

• Write the matrix with the intersectorial flows and the

input-output model.

• Which is the sector with the highest added value?

Exercise

• Matrix:

• Input- Output Model:

A B C

A 5 30 6

B 2 3 2

C 5 20 5

A B C P. Final Total

A 5 30 6 120 161

B 2 3 2 150 157

C 5 20 5 500 530

Importação 65 0 95

Valor acrescentado 84 104 422

Total 161 157 530

Imports

A

C

B

20

5

30 3

5

2

6

2

5

95

65

150

120

500

Final Demand

Exercise

• Consider na economy based in 3 sectors, A, B e C.

• Write the matrix with the intersectorial flows.

• Which is the sector with the highest added value?

• Assuming that L=(I-A)-1=I+A, determine the sector that

has to import more to satisfy his own final demand.

Exercise

• Matrix:

• Input- Output Model:

• Matrix L=I+A

A B C

A 5 30 6

B 2 3 2

C 5 20 5

A B C P. Final Total

A 5 30 6 120 161

B 2 3 2 150 157

C 5 20 5 500 530

Importação 65 0 95

Valor acrescentado 84 104 422

Total 161 157 530

0.031 0.191 0.011 R= 1.031 0.191 0.011

0.012 0.019 0.004 0.012 1.019 0.004

0.031 0.127 0.009 0.031 0.127 1.009

1 im=IM i /X i = 0.404 0.000 0.179

1

1

[ ]L

Exercise

• For each vector of final demand we compute the

change in total output and the change in imports:

PF={1,0,0} PF={0,1,0} PF={0,0,1}

∆X ∆IM ∆X ∆IM ∆X ∆IM

1.031 0.416 0.191 0.000 0.011 0.005

0.012 0.000 1.019 0.000 0.004 0.000

0.031 0.006 0.127 0.000 1.009 0.181

∆ = ∆x L f

0.031 0.191 0.011 R= 1.031 0.191 0.011

0.012 0.019 0.004 0.012 1.019 0.004

0.031 0.127 0.009 0.031 0.127 1.009

1 im=IM i /X i = 0.404 0.000 0.179

1

1

[ ]L

Ti

[ ]´ 1 0 0∆ =f [ ]´ 0 1 0∆ =f [ ]´ 0 0 1∆ =f

� �∆ = ∆ = ∆c cm m x m L f

Input-Output

• Application to the energy sector?

Input-Output

• Energy needs for different economic scenarios

– Using the input-output analysis to build a consistent

economic scenarios and then combining that information with

the Energetic Balance

– Using the input-output analysis where one or more sectors

define the energy sector

Input-Output Analysis:

Embodied Energy

• The input-ouput model

Intermediate

Inputs(square matrix)

Primary Energy Inputs

Total Energy in Inputs

Em

bo

die

d E

ner

gy

in

Fin

al D

eman

d

Tota

l En

erg

y in

ou

tpu

ts

Outputs

Inputs

Sectors

Sectors

=n×1 vector of embodied energy in final demandEf

=n×n matrix of intersectorial transactions of embodied energyEZ

=1×n vector of direct energy inputs

(embodied energy in primary inputs, e.g,

direct primary energy consumption &

embodied energy in imports )

Epi

´E E E E+ = +Z i f pi i Z

Input-Output Analysis:

Embodied Energy

• Embodied energy intensity, CEi, in outputs from sector i

to final demand or to other sectors is constant, i.e.,

• The energy sector 1 receives (direct + indirect) energy

which is distributed to its intended output m1S1

[ ]

[ ]

1 11 1

21

1

1 1 11 1 1

2 21

1

... ...

... ... ... ...1 1 ... 1

... ... ... ... ...

... ...

... ...

... ... ... ...1 1 ... 1

... ... ... ... ...

... ...

E E

n

E

E En n nn

n

n n n n nn

pi z z

z

pi z z

pi CE m CE m

CE m

pi CE m CE m

+ ⇒

+

1 1 1

2 2 2

...

n n n

CE m S

CE m S

CE m S

=

´E E E E E+ = + =Z i f pi i Z x

Input-Output Analysis:

Embodied Energy

• Simplifying per unit of mass:

1 1 1 1 1 1 11 1 1

1

1 ... 0 0 1 ... 0 0 ... ...1

0 ... ... ... ... 0 ... ... ... ... ... ... ...1

0 0 ... ... ... 0 0 ... ... ... ... ... ......

0 ... ... 1 0 ... ... 1 ... .1

T

n

n n n n n n n

m S PI m S CE m CE m

m S PI m S CE m

+

1

2

...

.. n nn n

CE

CE

CE m CE

=

1,11 1 1 1 1

2,2 2

,1 1

... ...

... ... ... ...

...... ... ... ... ... ...

... ...

T

dirn n

dir

n dirn nn n n n

CEf S f S CE CE

CECE CE

CEf S f S CE CE

+ =

1 11 1 1 1 1 1 1

2

1

1 ... 0 0 ... ...

0 ... ... ... ... ... ... ... ... ...

0 0 ... ... ... ... ... ... ... ... ...

0 ... ... 1 ... ...

T

n

n n nn n n n n n

S f f CE PI m S CE

CE

S f f CE PI m S CE

+ =

Input-Output Analysis:

Embodied Energy

• Simplifying per unit of mass:

• We can compute the embodied energy intensities for

all sectors CEi because we have n equations with n

unknowns

– We must know mass flows, residue formation factors and

direct energies intensities

1,11 1 1 1 1

2,2 2

,1 1

... ...

... ... ... ...

...... ... ... ... ... ...

... ...

T

dirn n

dir

n dirn nn n n n

CEf S f S CE CE

CECE CE

CEf S f S CE CE

+ =

1, 1 1 1

2,

,

...

... ...

dir

dir

n dir n n n

CE pi m S

CE

CE pi m S

=

Input-Output Analysis:

Embodied Energy

• Simplifying per unit of mass:

• We can compute the change in embodied energy

intensities for all sectors with the change in direct

energy intensities

1,11 1 1 1 1

2,2 2

,1 1

... ...

... ... ... ...

...... ... ... ... ... ...

... ...

T

dirn n

dir

n dirn nn n n n

CEf S f S CE CE

CECE CE

CEf S f S CE CE

+ =

( ) 1´

´

dir

dir dir

dir

−

+ =

= − =

∆ = ∆

A* ce ce ce

ce I A* ce L*ce

ce L* ce

= +

=

∆ = ∆

x Ax f

x Lf

x L f1ˆ´ ´−=A* ce S A ce

Input-Output Analysis

• To compute embodied “something”, e.g., energy or

CO2, that is distributed with productive mass flows

use:

– x is the vector with specific embodied “CO2” for all outputs

assuming that outputs from the same operation have the same

specific embodied value

– f is the vector with specific direct emissions of “CO2” for

each operation

– S is the diagonal matrix with the residue formation factors for

each operation

– A is the matrix with the mass fractions

• There are things that should flow with monetary

values instead of mass flows

– Economic causality instead of physical causality

1ˆ ´− + =S A x f x

Input-Output Analysis: Motivation

• Direct and indirect carbon emissions