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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
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
Input-Output Analysis: Motivation
• Input-Output Methodology– Build scenarios for the economy in a consistent way – To compute energy needs for different economic scenarios
Input-Output Analysis: Motivation
• Building 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 1936and applied to US national accounts inthe 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
R03Fish and other fishing products; aquaculture products; support services to fishing
0,0 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
R16Wood and of products of wood and cork, except furniture; articles of straw and plaiting materials
30,4 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 2
Output from sector i to final demand
Total 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 2
Output from sector i to final demand
Total 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 2
Output from sector i to final demand
Total 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 dimensionLower case bold letters for column vectorsUpper 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
ijij
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
ijij
j
za
x
Input-Output: Matrix A of technical coefficients
Rewritting the system of equations using aij:
ijij
j
za
x
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: Matrix A of technical coefficients
Rewritting the system of equations using aij:
• How can it be written in a matrix form?
ijij
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
ijij
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
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
ijij
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 iInputs 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
ijij
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, a21X1
units 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
ijij
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, sector or economy as a function of inputs z1, z2, …:
• Examples:
– Produces x units using z1 units of sector 1, z2 units of sector 2, … , zn units of sector n
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
1 2( , ,...)x z z
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 ca 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 X11 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
ijij
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 X11 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
ijij
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 X11 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
ijij
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
IntermediateInputs
(square matrix)
Fina
l Dem
and
Tota
l out
put
Outputs
Inpu
ts
Sectors
Sec
tors
Z
ij ij jz a x f x
• Intermediate inputs: intersector and intrasector inputs
• Final Demand: exports & consumption from households and government & investment
• Primary inputs: payments (wages, rents, interest) for primary factors of production (labour, land, capital) & taxes & imports
Input-Output Analysis: The model
• The input-ouput model
IntermediateInputs
(square matrix)
Primary Inputs
Fina
l Dem
and
Tota
l out
put
Outputs
Inpu
ts
Sectors
Sec
tors
Z
ij ij jz a x f x
• Intermediate inputs: intersector and intrasector inputs
• Final Demand: exports & consumption from households and government & investment
• Primary inputs: payments (wages, rents, interest) for primary factors of production (labour, land, capital) & taxes & imports
Input-Output Analysis: The model
• The input-ouput model
IntermediateInputs
(square matrix)
Primary Inputs
Total Inputs or Total Costs
Fina
l Dem
and
Tota
l out
put
Outputs
Inpu
ts
Sectors
Sec
tors
Z
ij ij jz a x f x
pi
Input-Output Analysis: The model
• The input-ouput model
Lines & columns are related by:
IntermediateInputs
(square matrix)
Primary Inputs
Total Inputs or Total Costs
Fina
l Dem
and
Tota
l out
put
Outputs
Inpu
ts
Sectors
Sec
tors
1 1
1 1
n n
ij i i ji ij j
n n
ij i i i i ji i ij 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:
IntermediateInputs
(square matrix)
Primary Inputs
Total Inputs or Total Costs
Fina
l Dem
and
Tota
l out
put
Outputs
Inpu
ts
Sectors
Sec
tors
1 1
1 1
n n
ij i i ji ij j
n n
ij i i i i ji i ij 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 Lf1 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 Lf1 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 Lf1 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
• For the transactions between sectors:
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
ijij
j
za
x11 12
1 2
...z z
x x
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 newc c
new new new
new newcn 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. orInputs from Agriculture to Industry
Exercise
• Matrix of technical coefficients:
ijij
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
ijij
j
za
x
Exercise
• Matrix of technical coefficients:
• Compute the Leontief inverse matrix:
ijij
j
Za
X
1
0,
j
j
I A A
Exercise
• Matrix of technical coefficients:
• Compute the Leontief inverse matrix:
1
0,
j
j
I A A
What is the meaning of this?x1=l11f1+l12f2+… L i
ijj
xl
f
Exercise
• Matrix of technical coefficients:
• Compute the Leontief inverse matrix:
1
0,
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:
1
0,
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:
1
0,
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:
1
0,
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 21.6z a x Initial z21=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 10080.8
20 40 30122 92.69
75 120 100101.6
c c cva va va
va
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
IntermediateInputs
(square matrix)
Primary Energy Inputs
Total Energy in Inputs
Embo
died
Ene
rgy
in
Fina
l Dem
and
Tota
l Ene
rgy
in o
utpu
ts
Outputs
Inpu
ts
Sectors
Sec
tors
=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 En
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
•
– Nº equations: 7– Nº unknonws: 7
•
1ˆ ´ S A x f x
Input-Output Analysis: Motivation
• Direct and indirect carbon emissions
