T_10_Input_Output 23 Nov.pdf

8/9/2019 T_10_Input_Output 23 Nov.pdf

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Energy Management :: 2012/13

Class # 10

Energy Analysis: Input-Output

Prof. Tânia Sousa

[email protected]


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Energy Management

Class # 10 :: Input-Output

Input-Output Analysis: Motivation

• Energy is needed in all production processes

• Different products have different embodiedenergies or specific energy consumptions– Can be assessed with Block Diagrams Methodology

– Change in time due to increases in energy efficiency

• Different Scenarios for the Economy havedifferent energy needs– Portuguese Scenarios for 2050:

http://www.cenariosportugal.com/


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Energy Management



• How can we compute (direct & indirect) changesin energy needs for different economic scenarios?• Example: if we want to increase the production of steel we

must at the same time increase production of coal, which inturn requires increased availability of steel, etc. in an infiniteseries - and similarly for the thousands of other products

which are directly or indirectly involved in the production ofsteel and coal.

difficulty is associated with the interdependencewithin the economic system


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Energy Management



• The interdependence of the Economic System:

Wassily Leontief, 1973 (Nobel Lecture)


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Direct and indirect carbon emissions


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Energy Management



• Input-Output Technique– Developed by Wassily Leontief in 1936

and applied to US national accounts inthe 40’s

– Input-output table is a matrix whose entries represent:

• the transactions occurring during 1 year between allsectors;

• the transactions between sectors and final demand;

• factor payments and imports.

– A tool of analysis for studying the complicatedinterdependence within the production system in amodern economy

– A tool to estimate (empirically) the change in demandfor inputs (e.g. energy) resulting from a change inproduction of the final good assuming that inputproportions are fixed.


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Energy Management


Input-Output: Basics

For the “Tire Factory”

X1= X11+ X12+… + X1n+ Y1

Output from sector 1 to sector 2

Output from sector 1 to final demandTotal Production from sector 1

Tire Factory

Automobile

Factory

Individual

Consumers


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Energy Management



For the Electricity Sector:

Xi= Xi1+ Xi2+… + Xii+… + Xin+ Yi

Output from sector i to sector 2

Output from sector i to final demandTotal Production from sector i

Electricity Sector

Automobile

Factory

Individual

Consumers

What is the meaning of this?


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Energy Management



For the Electricity Sector:

Xi= Xi1+ Xi2+… + Xii+… + Xin+ Yi

Output from sector i to sector 2

Output from sector i to final demandTotal Production from sector i

Electricity Sector

Automobile

Factory

Individual

Consumers

What is the meaning of this?Electricity consumed within the

electricity sector: hydraulic pumping

& electric consumption at the power

plants & losses in distribution


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Energy Management



For all sectors:

Xij is sales (ouput) from sector i to (input in) sector j (inmoney units)

Yi 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

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

X X X Y

X X X Y

X X X Y


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Energy Management


Input-Output: Matrix A of technical coefficients

Let’s define:

• What is the meaning of aij?

ij

ij

j

X a

X


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Energy Management



Let’s define:

• What is the meaning of aij?– a

ij input from sector i required to produce one (money)

unit worth of the product in sector j

– aij are the transaction or technical coefficients

ij

ij

j

X a

X

E M


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Energy Management



Rewritting the system of equations using aij :

• How can it be written in a matrix form?

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

X X X Y

X X X Y

X X X Y

ij

ij

j

X a

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 Y

X a X a X Y

X a X a X Y

=n×1 vector of sector output

=n×1 vector of final demand

=n×n matrix of technical

coefficients

X

Y

A

E M t


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Energy Management



Rewritting the system of equations using aij :

• How can it be written in a matrix form?

1 11 12 1

2 21 22 2

1 2

...

...

...n n n n

X X X Y

X X X Y

X X X Y

ij

ij

j

X a

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 Y

X a X a X Y

X a X a X Y



=n×n matrix of technical

coefficients

X

Y

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 Y

X a a X Y

X a a a X Y

X AX Y

E M t


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Energy Management


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 Y

X a X a X Y

X a X a X Y


• The meaning of matrix of technical coefficients A:

– The sector produces goods according to a fixedproduction function (recipe)

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

1, a21 units of sector 2, … ,an1 units of sector n

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 Y

X a a X Y

X a a a X Y

ij

ij

j

X a

X

Inputs to sector 1

E M t


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Energy Management


Production Functions: a review

• Production functions specify the output Q of a

factory, industry or economy as a function ofinputs X1, X2, …:

• Examples:

• Which of these productions functions allow for substitutionbetween production factors?

1 2( , ,...)Q f X X

1 2 ....b cQ aX X

1 2 ....Q a bX cX

1 2min , ,....Q aX bX

Cobb-Douglas Production Function

Linear Production Function

Leontief Production Function

Energy Management


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Energy Management





• Examples:

• Which of these productions functions allow for substitutionbetween production factors?

• Cobb-Douglas and Linear production fucntions

1 2( , ,...)Q f X X

1 2 ....b cQ aX X

1 2 ....Q a bX cX

1 2min , ,....Q aX bX




Energy Management


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Energy Management





• Examples:

• Which of these productions functions allow for scaleeconomies?

1 2( , ,...)Q f X X

1 2 ....b cQ aX X

1 2 ....Q a bX cX

1 2min , ,....Q aX bX




Energy Management


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Energy Management





• Examples:

• Which of these productions functions allow for scaleeconomies?

• Cobb-Douglas (if b+c >1)

1 2( , ,...)Q f X X

1 2 ....b cQ aX X

1 2 ....Q a bX cX

1 2min , ,....Q aX bX




Energy Management


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Energy Management


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 Y

X a X a X Y

X a X a X Y





1, a21 units of sector 2, … ,an1 units of sector n– Which type is this production function?

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 Y

X a a X Y

X a a a X Y

ij

ij

j

X a

X

Inputs to sector 1

Energy Management


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Energy Management


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 Y

X a X a X Y

X a X a X Y





1, a21 units of sector 2, … ,an1 units of sector n– Which type is this production function?

• Leontief which does not allow for 1) substitution betweenproduction factors or 2) scale economies

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

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 Y

X a a X Y

X a a a X Y

ij

ij

j

X a

X

Inputs to sector 1

Energy Management


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Energy Management


Input-Output: Leontief’s inverse matrix

• Leontief inverse matrix

• Where the Leontief inverse matrix can be obtained as:

1

1

AX Y X

Y X AX

Y I A X

I A Y X

X I A Y

X LY

1



=n×n matrix of technical coefficients

=n×n Leontief inverse matrix

X

Y

A

I A

1 2 3

0

... j

j

I A I A A A A

Energy Management


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Energy Management


Input-Output: Leontief’s inverse matrix

• Total output is:

– IY accounts for the final demand in total output (e.g. tiresconsumed by households) – direct effects

– AY accounts for the intersectorial needs to produce IY(e.g. rubber to produce the tires) – 1st indirect effects

– A[AY] accounts for the intersectorial needs to produce AY (e.g. electricity to produce the rubber) – 2nd indirecteffects

• It can be used to answer:– If final demand in sector i, Yi, (e.g. agriculture) is to

increase 10% what will be necessary changes in the final

outputs of all sectors, X1, X2, …?

1 X I A Y

2 3 ... X I A A A Y

Energy Management


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Energy Management


Input-Output: Multipliers

• Total output is:

• ij element of matrix matriz (I-A)-1 represents the

quantity of good i directly and indirectly needed (Xi) foreach unit of final demand of good j (Y j)

– Multiplier is the column sum which is the total outputneeded (X1+X2+…) to produce one unit of demand ofsector 1, Y1

2 3

1 11 1 1

1

...

...

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

...

n

n n nn n

X I A A A Y

X Y

X Y

X1 needed for one unit of Y1

Xn needed for one unit of Y1

Energy Management


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Energy Management


Input-Output: Multipliers

• The multiplier of sector j provides information on

the stimulus that 1 unit increase in Y j has on the(total output of) the economy (not on GDP)

• Multipliers change over time and over regionsbecause they depend on:– the economy structure, size, the way exports and sectors

are linked to each other and technology

2 3

1 11 1 1

1

...

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

...

n

n n nn n

X I A A A Y

X Y

X Y

X1 needed for one unit of Y1

Xn needed for one unit of Y1

Energy Management


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gy g


Input-Output Analysis: The model

• The input-ouput model• Primary inputs: added value(labor, rents, pay cap.) & imports

• Intermediate inputs: inputs forprocessing into outputs sold toother firms or to final purchasers.

• Consumption: exports & final

demand from households andgovernment

• Lines & columns are related by:

Intermediate

Inputs(square matrix)

Primary Inputs

Total Inputs or

Total Costs

C o n s u m p t i o

n

T o t a l o u t p u

t

Outputs

I n p u t s

Sectors

S e c t o r s

=n×1 vector of sector output X

=n×1 vector of final demandY

=n×n matrix of intersectorial transactions AX

AX Y X

1 1

n n

ij j i i ij j j j j

j i

A X Y X A X VA I X

Energy Management


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gy g


• For the primary inputs we define the coefficients:

– The meaning is the added value of sector j per unitproduction of sector j or imports of sector j per unit ofproduction

– These coefficients are assumed to be constant

• Relevance:

j

j

j

VAva

X

GDP= Added Values

Final consumption Exports ImportsGDP

j

j

j

I i

X

1T T VA va X va I A Y

Energy Management


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gy g


Exercise

• Considere the following Economy:


Energy Management


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gy g


Exercise

• Considere the following Economy:

• Compute the matrix A of the technicalcoeficients:

What is the meaning of this?What is the meaning of this?

Inputs of Agriculture into Industry

ij

ij

j

X a

X

Energy Management


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Exercise

• Matrix of technical coefficients:

ij

ij

j

X a

X


Energy Management


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Exercise


• What happens to the matrix of technicalcoefficients with time? Why?

ij

ij

j

X a

X


The amount of products (in money) from agriculture needed

to produce 1 unit worth of industry products

Energy Management


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• Compute the Leontief’s inverse matrix:

Exercise

ij

ij

j

X a

X

1

0,

j

j

I A A

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Exercise

1

0,

j

j

I A A


Energy Management


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Exercise

1

0,

j

j

I A A


the quantity of agriculture products

directly and indirectly needed for each

unit of final demand of industry goods

Energy Management


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Exercise

1

0,

j

j

I A A


Multiplier of the industry sector: the

total output needed for each unit of

final demand of industry goods

Energy Management


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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 X I A Y Exports Private Cons. Final Demand Final Demand20 30 50 55

30 40 70 70

10 30 40 40

Energy Management


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Exercise




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

1 X I A Y

Exports Private Cons. Final Demand Final Demand

20 30 50 55

30 40 70 70

10 30 40 40

1

2

3

55 80.8

70 122

40 101.6

X

X

X

Initial X

Energy Management


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Exercise




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

1 X I A Y

Exports Private Cons. Final Demand Final Demand

20 30 50 55

30 40 70 70

10 30 40 40

1

2

3

55 80.8

70 122

40 101.6

X

X

X

21 21 1 32.6 X a X Initial X21=20

Initial X

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• What is the new added value?

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• 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 100101.6

va va va

VA

Energy Management


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Input-Output Portugal

• O DPP (Departamento de Prospectiva e

Planeamento e Relações Internacionais) doMAMAOT desenvolveu um modelo de base input-output MODEM1 que tem sido utilizado para aavaliação do impacto macroeconómico, sectorial

e regional de políticas públicas e de grandesempreendimentos

• O DPP tem online a matriz input-output para2008 com 64 64 sectores

Energy Management


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Input-Output Portugal

• Input-Output matrix Portugal (2008)

PRODUCTS (CPA*64)

Products ofagriculture,hunting and

related services

Products offorestry, logging

and relatedservices

Fish and otherfishing

products;aquacultureproducts;

support servicesto fishing

Mining andquarrying

Food products,beverages and

tobaccoproducts

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 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 l eather 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 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 products224,8 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

Energy Management


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

Answer the following questions:

a) Write the matrix with the intersectorial flows.

b) Which is the sector with the highest added value?

c) Assuming that (I-A)-1=I+A, determine the sector that has to importmore to satisfy his own final demand.

Imports

A

C

B

20

5

30 3

5

2

6

2

5

95

65

150

120

500

Final Demand

Exercise

Energy Management


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a) Matrix:

b) Input- Output Model:

c) Matrix R=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

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c) (cont.) 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

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T_10_Input_Output 23 Nov.pdf