Gestão de Sistemas Energéticos
2015/2016
Energy Analysis: Input-Output
Prof. Tânia Sousa
[email protected]
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
• Process Analysis 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
• Process Analysis 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
– To compute energy needs for different economic scenarios
because it allow us to build scenarios for the economy in a
consistent way
Input-Output Analysis: Motivation
• Building a scenario for the economy in a consistent
way is difficult because of the interdependence within
the economic system
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
Refinery
– Example:
Chemical Industry
Power Plant
Coal Mine
– To increase the output of chemical industry there is a direct &
indirect (electr. & refined oil products) increase in demand
for coal
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
R01
R02
(CPA*64)
Products of agriculture, hunting and related services
Products of forestry, logging and related services
Fish and other fishing products; aquaculture products; support services to
R03 fishing
RB Mining and quarrying
R10_12 Food products, beverages and tobacco products
R13_15 Textiles, wearing apparel and leather products
R16
R17
R18
R19
R20
R21
Wood and of products of wood and cork, except furniture; articles of straw
and plaiting materials
Paper and paper products
Printing and recording services
Coke and refined petroleum products
Chemicals and chemical products
Basic pharmaceutical products and pharmaceutical preparations
R01
954,9
0,0
R02
18,4
103,4
R03
0,0
0,0
RB
R10_12
0,0 4275,2
0,0
0,0
0,0
0,0
38,4
0,0
40,5
0,5
1284,7
21,1
0,0
0,1
0,0
0,0
3,9
4,0
152,7
1,1
5,3
10,6
3012,0
1,2
30,4
0,0
0,0
1,8
58,5
8,2
4,0
224,8
225,9
6,3
0,0
0,3
14,3
10,2
0,0
1,3
1,8
38,6
0,8
0,0
2,2
4,3
144,3
31,8
0,1
304,3
49,5
99,4
106,5
12,1
Input-Output: Basics
For the “Tire Factory”
Output from sector 1 to sector 2
Output from sector 1
to final demand
Total Production
from sector 1
Individual
Consumers
Tire Factory
Automobile
Factory
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
Individual
Consumers
Tire Factory
Automobile
Factory
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
Total production
from sector i
Output from sector i
to final demand
Individual
Consumers
Electricity Sector
Automobile
Factory
Input-Output: Basics
What is the meaning of this?
For the Electricity Sector:
xi= zi1+ zi2+… + zii+… + zin+ fi
Output from sector i
to sector 2
Total production
from sector i
Output from sector i
to final demand
Individual
Consumers
Electricity Sector
Automobile
Factory
Input-Output: Basics
Electricity consumed within the
electricity sector: hydraulic pumping &
electric consumption at the power
plants & losses in distribution
For the Electricity Sector:
xi= zi1+ zi2+… + zii+… + zin+ fi
Output from sector i
to sector 2
Total production
from sector i
Output from sector i
to final demand
Individual
Consumers
Electricity Sector
Automobile
Factory
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)
Input-Output: Basics
For all sectors:
x1  z11  z12  ...  f1
x2  z21  z22  ...  f 2
xn  zn1  zn 2  ...  f n
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?
Input-Output: Basics
For all sectors:
x1  z11  z12  ...  f1
x  Zi  f
x2  z21  z22  ...  f 2
xn  zn1  zn 2  ...  f n
x vector of sector output
f vector of final demand
Z matrix with intersectorial transactions
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
Input-Output: Matrix A
of technical coefficients
Let’s define:
zij
aij 
xj
• 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
Input-Output: Matrix A
of technical coefficients
Let’s define:
zij
aij 
xj
• 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
Input-Output: Matrix A
of technical coefficients
Rewritting the system of equations using aij:
x1  z11  z12  ...  f1
x2  z21  z22  ...  f 2 aij 
xn  zn1  zn 2  ...  f n
zij
xj
Input-Output: Matrix A
of technical coefficients
Rewritting the system of equations using aij:
x1  z11  z12  ...  f1
x2  z21  z22  ...  f 2 aij 
xn  zn1  zn 2  ...  f n
zij
xj
x1  a11 x1  a12 x2  ...  f1
x2  a21 x1  a22 x2  ...  f 2
xn  an1 x1  an 2 x2  ...  f n
• How can it be written in a matrix form?
x vector of sector output
f vector of final demand
A matrix of technical coefficients
Input-Output: Matrix A
of technical coefficients
Rewritting the system of equations using aij:
x1  z11  z12  ...  f1
x2  z21  z22  ...  f 2 aij 
xn  zn1  zn 2  ...  f n
zij
xj
x1  a11 x1  a12 x2  ...  f1
x2  a21 x1  a22 x2  ...  f 2
xn  an1 x1  an 2 x2  ...  f n
• In a matrix form:
x  Ax  f
x vector of sector output
x  Zi  f
f vector of final demand
A matrix of technical coefficients
Input-Output: Matrix A
of technical coefficients
• The meaning of matrix of technical coefficients A:
Input-Output: Matrix A
of technical coefficients
• The meaning of matrix of technical coefficients A:
x1  a11 x1  a12 x2  ...  f1
x2  a21 x1  a22 x2  ...  f 2
xn  an1 x1  an 2 x2  ...  f n
 x1   a11
 x  a
 2    21
 ...   ...
  
 xn   an1
a12
a22
...
an 2
– What is the meaning of this column?
... a1n 
... ... 

... ... 

... ann 
 x1   f1 
x   f 
 2   2
 ...   ... 
   
 xn   f n 
aij 
zij
xj
Input-Output: Matrix A of technical
coefficients
• The meaning of matrix of technical coefficients A:
x1  a11 x1  a12 x2  ...  f1
x2  a21 x1  a22 x2  ...  f 2
xn  an1 x1  an 2 x2  ...  f n
 x1   a11
 x  a
 2    21
 ...   ...
  
 xn   an1
a12
a22
...
an 2
... a1n 
... ... 

... ... 

... ann 
Inputs to sector 1
– Column i represents the inputs to sector i
 x1   f1 
x   f 
 2   2
 ...   ... 
   
 xn   f n 
aij 
zij
xj
Input-Output: Matrix A of technical
coefficients
• The meaning of matrix of technical coefficients A:
x1  a11 x1  a12 x2  ...  f1
x2  a21 x1  a22 x2  ...  f 2
xn  an1 x1  an 2 x2  ...  f n
 x1   a11
 x  a
 2    21
 ...   ...
  
 xn   an1
Inputs to sector 1
a12
a22
...
an 2
... a1n 
... ... 

... ... 

... ann 
 x1   f1 
x   f 
 2   2
 ...   ... 
   
 xn   f n 
aij 
zij
xj
– 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
Production Functions: a review
• Production functions specify the output x of a factory,
industry, sector or economy as a function of inputs z1,
z2, …:
x  f ( z1 , z2 ,...)
• Examples:
x  az1b z2 c ....
Cobb-Douglas Production Function
x  a  bz1  cz2  ....
Linear Production Function
– Produces x units using z1 units of sector 1, z2 units of sector 2,
… , zn units of sector n
x  ( z1 , z2 ,...)
Production Functions: a review
• Production functions specify the output x of a factory,
industry or economy as a function of inputs z1, z2, …:
x  f ( z1 , z2 ,...)
• Examples:
x  az1b z2 c ....
Cobb-Douglas Production Function
x  a  bz1  cz2  ....
Linear Production Function
• Which of these productions functions allow for substitution between
production factors?
Production Functions: a review
• Production functions specify the output x of a factory,
industry or economy as a function of inputs z1, z2, …:
x  f ( z1 , z2 ,...)
• Examples:
x  az1b z2 c ....
Cobb-Douglas Production Function
x  a  bz1  cz2  ....
Linear Production Function
• Which of these productions functions allow for substitution
between production factors?
•
Cobb-Douglas and Linear production functions
x  a  bz1  cz2  a  b  0.8 z1   c  dz2 
with d  1  0.2
bz1
cz2
Production Functions: a review
• Production functions specify the output x of a factory,
industry or economy as a function of inputs z1, z2, …:
x  f ( z1 , z2 ,...)
• Examples:
x  az1b z2 c ....
Cobb-Douglas Production Function
x  a  bz1  cz2  ....
Linear Production Function
• Which of these productions functions allow for scale economies?
Production Functions: a review
• Production functions specify the output x of a factory,
industry or economy as a function of inputs z1, z2, …:
x  f ( z1 , z2 ,...)
• Examples:
x  az1b z2 c ....
Cobb-Douglas Production Function
x  a  bz1  cz2  ....
Linear Production Function
• Which of these productions functions allow for scale economies?
•
Cobb-Douglas (if b+c >1)
a  2 z1   2 z2   a 2bc  z1   z2   2bc x  2 x if
b
c
b
c
b  c 1
Input-Output: Matrix A
of technical coefficients
• The meaning of matrix of technical coefficients A:
x1  a11 x1  a12 x2  ...  f1  x1   a11 a12 ... a1n   x1   f1 
x2  a21 x1  a22 x2  ...  f 2  x2   a21 a22 ... ...   x2   f 2 
 
   
 ...   ... ... ... ...   ...   ... 
xn  an1 x1  an 2 x2  ...  f n  xn   an1 an 2 ... ann   xn   f n 
Inputs to sector 1
aij 
– 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?
zij
xj
Input-Output: Matrix A
of technical coefficients
• The meaning of matrix of technical coefficients A:
x1  a11 x1  a12 x2  ...  f1  x1   a11 a12 ... a1n   x1   f1 
x2  a21 x1  a22 x2  ...  f 2  x2   a21 a22 ... ...   x2   f 2 
 
   
 ...   ... ... ... ...   ...   ... 
xn  an1 x1  an 2 x2  ...  f n  xn   an1 an 2 ... ann   xn   f n 
Inputs to sector 1
aij 
– 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
x1  min  z11 a11 , z21 a21 ,....
Leontief Production Function
zij
xj
Input-Output: Matrix A
of technical coefficients
• The meaning of matrix of technical coefficients A:
x1  a11 x1  a12 x2  ...  f1  x1   a11 a12 ... a1n   x1   f1 
x2  a21 x1  a22 x2  ...  f 2  x2   a21 a22 ... ...   x2   f 2 
 
   
 ...   ... ... ... ...   ...   ... 
xn  an1 x1  an 2 x2  ...  f n  xn   an1 an 2 ... ann   xn   f n 
Inputs to sector 1
aij 
– 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)
zij
xj
Input-Output Analysis: The model
• The input-ouput model
Sectors
(square matrix)
zij  aij x j
Total output
Intermediate
Inputs
•
Final Demand
Z
Inputs
Sectors
Outputs
f
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
Sectors
(square matrix)
zij  aij x j
Primary Inputs
Total output
Intermediate
Inputs
•
Final Demand
Z
Inputs
Sectors
Outputs
f
x
•
•
Intermediate inputs: intersector
and intrasector inputs
Final Demand: exports &
consumption from households
and government & investment
Primary inputs?
Input-Output Analysis: The model
• The input-ouput model
Sectors
(square matrix)
zij  aij x j
Primary Inputs
Total output
Intermediate
Inputs
•
Final Demand
Z
Inputs
Sectors
Outputs
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
Sectors
(square matrix)
zij  aij x j
Primary Inputs
Total Inputs or
Total Costs
pi
Total output
Intermediate
Inputs
•
Final Demand
Z
Inputs
Sectors
Outputs
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
Sectors
(square matrix)
Primary Inputs
pi
Total output
Intermediate
Inputs
Final Demand
Z
Inputs
Sectors
Outputs
f
x
n
z
j 1
Total Inputs or
Total Costs
n
ij
n
z
j 1
 f i  xi   zij  pi j
i 1
n
ij
 ci  gi  ei  invi   zij  av j  i j
i 1
Input-Output Analysis: The model
• The input-ouput model
Sectors
Zi  f  pi  i´Z
(square matrix)
Total output
Intermediate
Inputs
Final Demand
Z
Inputs
Sectors
Outputs
f
x
Ax  f  x
Ax  Zi
A  Zxˆ 1
Lines & columns are related by:
Primary Inputs
pi
n
z
j 1
Total Inputs or
Total Costs
n
ij
n
z
j 1
 f i  xi   zij  pi j
i 1
n
ij
 ci  gi  ei  invi   zij  av j  i j
i 1
Input-Output Analysis:
Leontief inverse matrix
• How to relate final demand to production?
Ax  f  x
x vector of sector output
f vector of final demand
A matrix of technical coefficients
Input-Output Analysis:
Leontief inverse matrix
• How to relate final demand to production?
Ax  f  x
x vector of sector output
f  x  Ax
f vector of final demand
f  I  A x
A matrix of technical coefficients
I  A f  x
1
I  A
1
Leontief inverse matrix
Lf  x
• x  Lf is useful for which types of questions?
Input-Output Analysis:
Leontief inverse matrix
• How to relate final demand to production?
Ax  f  x
x vector of sector output
f  x  Ax
f vector of final demand
f  I  A x
A matrix of technical coefficients
I  A f  x
1
I  A
1
Leontief inverse matrix
Lf  x
• x  Lf is useful for which types of questions?
– If final demand in sector i, fi, (e.g. automobile industry) is to
increase 10% next year how much output from each of the
sectors would be necessary to supply this final demand?
Input-Output Analysis:
Leontief inverse or total requirements matrix
2nd Indirect effects
1st Indirect effects
Direct effects
intersectorial needs to
produce the following
intersectorial needs
intersectorial needs to
produce these cars
cars for final demand
Input-Output Analysis:
Leontief inverse or total requirements matrix
• Leontief inverse matrix which can be obtained as:
I  A
1

 I  A  A 2  A3  ...   A j
j 0
• Total Output is:
x   I  A  f   I  A  A 2  A 3  ... f
1
– 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
Input-Output Analysis:
Leontief inverse or total requirements matrix
• Impacts in output from marginal increases in final
demand from f to fnew:
x new  Lf new
 x1  x1   l11
...
   ...

 
 xn  xn  ln1
 x1   l11 ...
...    ... ...

 
 xn  ln1 ...
x  Lf
... l1n   f1  f1 

... ...  ...


... lnn   f n  f n 
l1n   f1 
...  ... 
 
lnn   f n 
Input-Output: Multipliers
• Total output is:
x  Lf
 x1   l11 ... l1n   f1 
...    ... ... ...  ... 
  
 
 xn  ln1 ... lnn   f n 
lij 
xi
f j
If sector 1 is paints and sector 2 is
cars what is the meaning of l12?
x1  l11 f1  l12 f 2  ...
xn  ln1 f1  ln 2 f 2  ...
Input-Output: Multipliers
• Total output is:
Total production of paints
x  Lf
 x1   l11 ... l1n   f1 
...    ... ... ...  ... 
  
 
 xn  ln1 ... lnn   f n 
lij 
xi
f j
Production of paints required directly and
indirectly for 1 unit of final demand of cars
x1  l11 f1  l12 f 2  ...
xn  ln1 f1  ln 2 f 2  ...
Input-Output: Multipliers
• Total output is:
x  Lf
 x1   l11 ... l1n   f1 
...    ... ... ...  ... 
  
 
 xn  ln1 ... lnn   f n 
lij 
xi
f j
x1 needed for one unit of f2
x1  l11 f1  l12 f 2  ...
xn  ln1 f1  ln 2 f 2  ...
xn needed for one unit of f1
– 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?
Input-Output: Multipliers
• Total output is:
x1 needed for one unit of f2
x  Lf
x1  l11 f1  l12 f 2  ...
x
l
...
l
f
 1   11
1n   1 
...    ... ... ...  ... 
  
   x  l f  l f  ...
n
n1 1
n2 2
 xn  ln1 ... lnn   f n 
xi
xn needed for one unit of f1
lij 
f j
– 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
Input-Output: Multipliers
• Total output is:
x  Lf
 x1   l11 ... l1n   f1 
...    ... ... ...  ... 
  
 
 xn  ln1 ... lnn   f n 
lij 
xi
f j
x1 needed for one unit of f2
x1  l11 f1  l12 f 2  ...
xn  ln1 f1  ln 2 f 2  ...
xn needed for one unit of f1
– 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?
Input-Output: Multipliers
• Total output is:
x  Lf
 x1   l11 ... l1n   f1 
...    ... ... ...  ... 
  
 
 xn  ln1 ... lnn   f n 
lij 
xi
f j
x1 needed for one unit of f1
x1  l11 f1  l12 f 2  ...
xn  ln1 f1  ln 2 f 2  ...
xn needed for one unit of f1
– 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)
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
x  Lf
 x1   l11 ... l1n   f1 
...    ... ... ...  ... 
  
 
 xn  ln1 ... lnn   f n 
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
x  Lf
 x1   l11 ... l1n   f1 
...    ... ... ...  ... 
  
 
 xn  ln1 ... lnn   f n 
• Where do you expect the multiplier of the wind energy
sector to be higher: in a country that imports the wind
turbines or in a country that develops and produces
wind turbines?