Policy Papers

Evolution of EROIs of Electricity Until 2050 Estimation and Implications on Prices

Adrien FABRE

PP 2018-09

Suggested citation A Fabre (2018) Evolution of EROIs of Electricity Until 2050 Estimation and Implications on Prices FAERE Policy Paper 2018-09

wwwfaerefr

Evolution of EROIs of Electricity Until 2050Estimation and Implications on Prices

Adrien Fabre1

Abstract

The EROI ndashfor Energy Returned On Investedndash of an energy technology measures its ability to provide energy efficiently Previousstudies draw a link between the affluence of a society and the EROI of its energy system and show that EROIs of renewablesare lower than those of fossil fuels Logically concerns have been expressed that system-wide EROI may decrease during arenewable energy transition First I explain theoretically that the EROIs of renewables themselves could then decrease asenergy-efficient fossil fuels would be replaced by less energy-efficient renewables in the supply-chain Then using the multire-gional input-output model THEMIS I estimate the evolution of EROIs and prices of electric technologies from 2010 to 2050 fordifferent scenarios Global EROI of electricity is predicted to go from 12 in 2010 to 11 in 2050 in a business-as-usual scenario butdown to 6 in a 100 renewable one Finally I study the economic implication of a declining EROI An inverse relation betweenEROI and price is suggested empirically even though theory shows that both quantities may move in the same direction

Keywords EROI input-output THEMIS MRIO sustainability energy transition

Acknowledgments I am grateful to three anonymous review-ers for their insightful comments and especially to one ofthem who spotted an important typo I hail the work of Kon-stantin Stadler on pymrio an invaluable open-source pythonlibrary when dealing with Multi-Regional Input-Output Ta-bles (MRIO) I am thankful to Thomas Pregger and his teamat DLR for graciously providing me the tables of future energyconsumption in the Greenpeace [R]evolution scenarios I amindebted to Thomas Gibon Anders Arvesen and the Norwe-gian University of Science and Technology (NTNU) for pro-viding me the data of THEMIS and helping me using it I amthankful to Arjan de Koning Carey King and Adam Brandt foranswering my questions I am grateful to Cyril Franccedilois forhis thoughtful comments on this paper I am thankful to theIsterre for providing me an office at Grenoble to conduct thisresearch I am grateful to Olivier Vidal and Mouez Fodha fortheir support

Code All the code is on-line and can be accessed from anotebook at bitlyfuture_eroi_code A substantial share ofthis work has been to contribute to the python library pym-

rio githubcombixioupymrio Using my fork of pymrio onecan now easily undertake EROIs and related computations onExiobase and THEMIS

1Paris School of Economics Universiteacute Paris 1 Pantheacuteon-Sorbonneadrienfabrepsemaileu 48 bd Jourdan 75014 Paris

1 Introduction

As the harmful impacts of climate change call for a promptenergy transition away from fossil fuels mdashnot to mention theirdepletion that shall ultimately make this transition unavoid-able concerns have been expressed that in a decarbonized 5

energy system the lower efficiency of renewable energy mightnot allow to sustain advanced standards of living (Lambert et al2014 Tverberg 2017)2 We measure the energy efficiency ofa technology or energy system using the Energy Returned OnInvested (EROI) which is the ratio between the energy it de- 10

livers throughout its lifetime and the energy required to buildoperate and dismantle it A minimal requirement for a tech-nology or energy system to be energetically sustainable is tohave an EROI above 1 meaning that it provides more energythan it requires 15

One issue to assess future energy systems is that the futureEROI of a given technology cannot be readily deduced fromcurrent estimates Indeed as King (2014) remarked the EROIof a technology is not intrinsic but depends on the wholetechnological structure of the economy Indeed suppose that 20

solar panels have a lower EROI than thermal power plants sothey require more energy to supply the same amount of en-ergy Then a plant producing solar panels will require moreenergy if the electricity it uses is produced by solar panelsrather than by thermal plants Ultimately solar panels built 25

using electricity from solar panels rather fossils will requiremore energy and have a lower EROI Some have called to

2The energy expert Jean-Marc Jancovici also expressed concerns overthis subject during a presentation at the Eacutecole Normale Supeacuterieure in 2018ldquoWhat happens to the EROI when you have only wind and solar panels tobuild wind and solar panels I think it crashesrdquo

Evolution of EROIs of Electricity Until 2050Estimation and Implications on Prices

Adrien Fabre1

Abstract

The EROI ndashfor Energy Returned On Investedndash of an energy technology measures its ability to provide energy efficiently Previousstudies draw a link between the affluence of a society and the EROI of its energy system and show that EROIs of renewablesare lower than those of fossil fuels Logically concerns have been expressed that system-wide EROI may decrease during arenewable energy transition First I explain theoretically that the EROIs of renewables themselves could then decrease asenergy-efficient fossil fuels would be replaced by less energy-efficient renewables in the supply-chain Then using the multire-gional input-output model THEMIS I estimate the evolution of EROIs and prices of electric technologies from 2010 to 2050 fordifferent scenarios Global EROI of electricity is predicted to go from 12 in 2010 to 11 in 2050 in a business-as-usual scenario butdown to 6 in a 100 renewable one Finally I study the economic implication of a declining EROI An inverse relation betweenEROI and price is suggested empirically even though theory shows that both quantities may move in the same direction

Keywords EROI input-output THEMIS MRIO sustainability energy transition

Acknowledgments I am grateful to three anonymous review-ers for their insightful comments and especially to one ofthem who spotted an important typo I hail the work of Kon-stantin Stadler on pymrio an invaluable open-source pythonlibrary when dealing with Multi-Regional Input-Output Ta-bles (MRIO) I am thankful to Thomas Pregger and his teamat DLR for graciously providing me the tables of future energyconsumption in the Greenpeace [R]evolution scenarios I amindebted to Thomas Gibon Anders Arvesen and the Norwe-gian University of Science and Technology (NTNU) for pro-viding me the data of THEMIS and helping me using it I amthankful to Arjan de Koning Carey King and Adam Brandt foranswering my questions I am grateful to Cyril Franccedilois forhis thoughtful comments on this paper I am thankful to theIsterre for providing me an office at Grenoble to conduct thisresearch I am grateful to Olivier Vidal and Mouez Fodha fortheir support

Code All the code is on-line and can be accessed from anotebook at bitlyfuture_eroi_code A substantial share ofthis work has been to contribute to the python library pym-

rio githubcombixioupymrio Using my fork of pymrio onecan now easily undertake EROIs and related computations onExiobase and THEMIS

1Paris School of Economics Universiteacute Paris 1 Pantheacuteon-Sorbonneadrienfabrepsemaileu 48 bd Jourdan 75014 Paris

1 Introduction

As the harmful impacts of climate change call for a promptenergy transition away from fossil fuels mdashnot to mention theirdepletion that shall ultimately make this transition unavoid-able concerns have been expressed that in a decarbonized 5

energy system the lower efficiency of renewable energy mightnot allow to sustain advanced standards of living (Lambert et al2014 Tverberg 2017)2 We measure the energy efficiency ofa technology or energy system using the Energy Returned OnInvested (EROI) which is the ratio between the energy it de- 10

livers throughout its lifetime and the energy required to buildoperate and dismantle it A minimal requirement for a tech-nology or energy system to be energetically sustainable is tohave an EROI above 1 meaning that it provides more energythan it requires 15

One issue to assess future energy systems is that the futureEROI of a given technology cannot be readily deduced fromcurrent estimates Indeed as King (2014) remarked the EROIof a technology is not intrinsic but depends on the wholetechnological structure of the economy Indeed suppose that 20

solar panels have a lower EROI than thermal power plants sothey require more energy to supply the same amount of en-ergy Then a plant producing solar panels will require moreenergy if the electricity it uses is produced by solar panelsrather than by thermal plants Ultimately solar panels built 25

using electricity from solar panels rather fossils will requiremore energy and have a lower EROI Some have called to

2The energy expert Jean-Marc Jancovici also expressed concerns overthis subject during a presentation at the Eacutecole Normale Supeacuterieure in 2018ldquoWhat happens to the EROI when you have only wind and solar panels tobuild wind and solar panels I think it crashesrdquo

compute the evolution of EROIs during a renewable energytransition (Brandt 2017) and this study aims to do so whileaccounting for their system dependency Indeed provided30

that EROIs of renewables are lower than EROIs of fossils andthat decreasing EROIs jeopardize prosperity the evolution ofEROIs during the energy transition is of critical importancelet us review these two hypotheses in turn

Many estimations of EROIs have been made and among35

the various different figures derived from diverse data setsand methodologies none stands out as singularly authorita-tive Dale (2010) reviews all EROI estimates until 2010 whileHall et al (2014) aggregate the estimates of the literature in ameta-analysis I choose to present the results of Weiszligbach et al40

(2013) (see Figure 1) because they compute the EROIs of dif-ferent technologies in a comparable manner In addition thebuffered EROIs of Weiszligbach et al (2013) take into account thesupplementary capacity grid and storage required for the de-ployment of renewable technologies which yields lower but45

presumably more accurate estimates for their EROIs As an-ticipated the EROIs of renewable electricity sectors they findare significantly lower than those of electricity from fossil fu-els except for hydro

Figure 1 Estimates of EROIs of different electricity technologies fromWeiszligbach et al (2013) where supplementary capacity and storage requiredfor the deployment of these technologies is accounted for

Some authors argue that the value of EROI is of primary50

concern as they draw a link between the system-wide EROIand affluence of a society (Hall et al 2009 Hall 2011 Lambert amp Lambert2011 Lambert et al 2014 Fizaine amp Court 2016) Here is howHall (2011) summarizes the argument

Think of a society dependent upon one resource55

its domestic oil If the EROI for this oil was 111then one could pump the oil out of the groundand look at it () Hall et al (2009) examined theEROI required to actually run a truck and foundthat if the energy included was enough to build60

and maintain the truck and the roads and bridgesrequired to use it (ie depreciation) one wouldneed at least a 31 EROI at the wellhead Now ifyou wanted to put something in the truck saysome grain and deliver it that would require an65

EROI of say 51 to grow the grain () 7 or 81to support the families If the children were to beeducated you would need perhaps 9 or 101 havehealth care 121 have arts in their life maybe 141and so on 70

The reasoning of Hall relies on the observation that all sectorsof the economy require energy and that the more efficient isthe energy production (ie the higher is the EROI) the moreenergy is available to the rest of the economy In strict logicHallrsquos argument relies on two questionable assumptions that 75

factors of production (and especially the labor force) are usedat their full capacity and that technical and organizationalprogress will not be sufficient to sustain current level of pros-perity with significantly less labor (or other factors of pro-duction in limited supply) If one rejects these assumptions 80

one can imagine a sustained level of prosperity with a lowersystem-wide EROI provided that a higher share of factors ofproduction be devoted to the energy sector for example un-employed people could be mobilized to sustain the energysurplus available to the rest of society In parallel to a shift 85

in the labor force Raugei (2019) explains that an increasedefficiency of energy use may also counteract the decrease inenergy services implied by a declining EROI That being saidgiven that current system-wide EROI is already declining dueto the decline in fossil fuels quality (Dale et al 2011 Poisson et al90

2013 Court amp Fizaine 2017) and that technical progress is in-cremental the aforementioned analyses should not be ne-glected Under the current system of production which willpersist in the short term EROI should not decrease too muchfor prosperous standards of living to be sustained 95

In view of the potential implications of a declining EROIthis paper provides an assessment of the EROI of differentelectricity technologies in various prospective scenarios whichincludes a 100 renewable electricity system To this endI employ input-output analysis and I rely on a prospective 100

series of multi-regional Input-Output Tables (IOT) THEMIS(Gibon et al 2015) which models two scenarios from the In-ternational Energy Agency (IEA 2010) Baseline and Blue MapIn addition I modify THEMISrsquo IOTs to embed two decarbonizedscenarios of power generation Greenpeacersquos Energy [R]evolution105

(ER) and Advanced Energy [R]evolution (ADV) (Teske et al2015) Although Pehl et al (2017) and Arvesen et al (2018) al-ready computed energy requirements of electricity technolo-gies for prospective scenarios they focused on life-cycle as-sessment coefficients such as future CO2 emissions and did 110

not provide results in terms of EROI let alone system-wideEROI Furthermore they did not study a scenario with 100renewable electricity I intend to fill this gap

Then I analyze the economic implications of a decliningEROI through its relation with price Previous studies suggest 115

an inverse relation between EROIs and energy prices and suchan average relation is retrieved empirically using prices ob-served and predicted from THEMIS However theoretical anal-ysis tempers this finding Indeed while explaining to whatextent EROI and price are related I show that they do not 120

necessarily move in opposite directions This calls for taking

2

prices predictions from input-output analysis with more cau-tion than EROI estimates because IOT is better suited to han-dle physical notions than economic ones Finally the eco-nomic analysis weakens the view that a decrease in EROI would125

necessarily lead to a surge in energy expenditures and henceto a contraction of GDP

Section 2 explains theoretically why the EROI of a tech-nology is not an intrinsic property section 3 presents the method-ology and the results section 4 studies the implications of de-130

clining EROIs on prices and GDP section 5 concludes

2 The EROI of a Technology Is Not Intrinsic

21 A Simple Model With A Unique Energy Technology

The element ai j of the technology matrix A representsthe quantity of input i required to produce one unit of output135

j Below is an illustrative technology matrix with three inputs(and the same three outputs) an energy technology mate-rials and energy me denotes the quantity of materials (m)required to produce one unit of energy technology (e) andthis notation extends naturally to all elements of A The nu-140

merical values of the coefficients have a purely pedagogicalpurpose and have been arbitrarily chosen

A =

0 0 1me mm 0Ee Em 0

=

0 0 1me 02 001 05 0

energy technomaterials

energy

The system-wide EROI or Energy Returned On Investedis the ratio between the energy delivered by the system andthe energy required to build operate maintain and disman-145

tle it In other words it is the inverse of the amount of energyrequired to produce one unit of energy when the series of allembodied inputs are taken into account

The embodied inputs x required for a final demand y canbe calculated using the Leontief inverse matrix (Leontief 1986Eurostat 2008 Miller amp Blair 2009)

x(

y)

= (I minus A)minus1middot y (1)

We denote by 1S the vector with 1 at the positions of thesectors s isin S and zeros everywhere else As energy E is the150

last input of our list 1E =

001

and the gross embodied energy

required for a final demand y is the last element of x1

TEmiddot (In minus A)minus1

middot y Thus the EROI is

EROI =delivered energy

net embodied energy

=1

1TEmiddot(

(I minus A)minus1 middot1E minus1E

) (2)

After some calculations (available on-line) we find

EROI =(1minusEe ) (mm minus1)+Emme

Ee (mm minus1)minusEmme

=072minus05me

008+05me(3)

Unsurprisingly one can see in Figure 2 that the EROI de-creases with the material intensity of the energy technology 155

because extracting and processing material requires energy

Figure 2 EROI in the simple model in function of the material intensity me

of the energy technology

For an intensity above 06 the EROI is below 1 An EROIbelow 1 means that the energy technology is not worth devel-oping because (in net) it consumes energy rather than pro-viding it Such a system is not sustainable (and not realistic) 160

for it to happen the society should have accumulated energyin the past from an energy source no more accessible andwould waste this energy in that absurd technology

For even higher intensities the EROI falls below 0 whichmeans that the energy (recursively) required to produce one 165

unit of energy is infinite Here free energy coming from thepast would not suffice to build the energy technology onewould also need to have free materials (ie materials requir-ing no energy to access them) Such a world is physically im-possible 170

22 A Simple Model With A Mix of Two Energy Technologies

Now let us consider two energy technologies with thesame energy intensity but different materials intensities

Even if this example is purely illustrative let us call themPV (for solar photovoltaic) and gas (for gas power-plant elec- 175

tricity) to grasp the motivation for this paper The numbersare completely made up but they respect the fact that PV ismore material intensive than gas (Hertwich et al 2015) Hereis our new technology matrix where p represents the share ofPV in the energy (or electricity) mix 180

3

prices predictions from input-output analysis with more cau-tion than EROI estimates because IOT is better suited to han-dle physical notions than economic ones Finally the eco-nomic analysis weakens the view that a decrease in EROI would125

necessarily lead to a surge in energy expenditures and henceto a contraction of GDP

Section 2 explains theoretically why the EROI of a tech-nology is not an intrinsic property section 3 presents the method-ology and the results section 4 studies the implications of de-130

clining EROIs on prices and GDP section 5 concludes

2 The EROI of a Technology Is Not Intrinsic

21 A Simple Model With A Unique Energy Technology

The element ai j of the technology matrix A representsthe quantity of input i required to produce one unit of output135

j Below is an illustrative technology matrix with three inputs(and the same three outputs) an energy technology mate-rials and energy me denotes the quantity of materials (m)required to produce one unit of energy technology (e) andthis notation extends naturally to all elements of A The nu-140

merical values of the coefficients have a purely pedagogicalpurpose and have been arbitrarily chosen

A =

0 0 1me mm 0Ee Em 0

=

0 0 1me 02 001 05 0

energy technomaterials

energy

The system-wide EROI or Energy Returned On Investedis the ratio between the energy delivered by the system andthe energy required to build operate maintain and disman-145

tle it In other words it is the inverse of the amount of energyrequired to produce one unit of energy when the series of allembodied inputs are taken into account

The embodied inputs x required for a final demand y canbe calculated using the Leontief inverse matrix (Leontief 1986Eurostat 2008 Miller amp Blair 2009)

x(

y)

= (I minus A)minus1middot y (1)

We denote by 1S the vector with 1 at the positions of thesectors s isin S and zeros everywhere else As energy E is the150

last input of our list 1E =

001

and the gross embodied energy

required for a final demand y is the last element of x1

TEmiddot (In minus A)minus1

middot y Thus the EROI is

EROI =delivered energy

net embodied energy

=1

1TEmiddot(

(I minus A)minus1 middot1E minus1E

) (2)

After some calculations (available on-line) we find

EROI =(1minusEe ) (mm minus1)+Emme

Ee (mm minus1)minusEmme

=072minus05me

008+05me(3)

Unsurprisingly one can see in Figure 2 that the EROI de-creases with the material intensity of the energy technology 155

because extracting and processing material requires energy

Figure 2 EROI in the simple model in function of the material intensity me

of the energy technology

For an intensity above 06 the EROI is below 1 An EROIbelow 1 means that the energy technology is not worth devel-oping because (in net) it consumes energy rather than pro-viding it Such a system is not sustainable (and not realistic) 160

for it to happen the society should have accumulated energyin the past from an energy source no more accessible andwould waste this energy in that absurd technology

For even higher intensities the EROI falls below 0 whichmeans that the energy (recursively) required to produce one 165

unit of energy is infinite Here free energy coming from thepast would not suffice to build the energy technology onewould also need to have free materials (ie materials requir-ing no energy to access them) Such a world is physically im-possible 170

22 A Simple Model With A Mix of Two Energy Technologies

Now let us consider two energy technologies with thesame energy intensity but different materials intensities

Even if this example is purely illustrative let us call themPV (for solar photovoltaic) and gas (for gas power-plant elec- 175

tricity) to grasp the motivation for this paper The numbersare completely made up but they respect the fact that PV ismore material intensive than gas (Hertwich et al 2015) Hereis our new technology matrix where p represents the share ofPV in the energy (or electricity) mix 180

3

A =

0 0 0 p

0 0 0 1minusp

mPV mg mm 0EPV Eg Em 0

=

0 0 0 p

0 0 0 1minusp

07 01 02 001 01 05 0

PVgas

materialsenergy

With some calculus (see on-line) we obtain

EROI =067minus03p

013+03p(4)

This corresponds to the system-wide EROI But now thatwe have two technologies we can compute the EROI of eachof them3

EROIPV = 1558minus0698p

EROIg as = 5154minus2308p (5)

Logically the EROI of PV is lower as compared to gas be-cause of its higher material intensity But it is worth noticingthat both EROIs depend on the energy mix p the EROI of atechnology is not an intrinsic property Indeed it dependson the whole economic system or more precisely of all tech-185

nologies used in their chain of production4 Here the higherthe share of PV in the mix the more the lower EROI of PV con-taminates each technology and the lower the EROI of bothtechnologies

One can see on Figure 3 that for highest penetration of PV190

the EROI falls below unity In other words a renewable energymix with 100 PV is not sustainable in this example Evenmore worryingly if one computes the EROI of PV in an en-ergy mix relying mostly on gas one would find a high-enoughEROI for PV (meaning above 1) Hence one cannot conclude195

that a technology is sufficiently efficient (or sustainable) justby computing its EROI in the current energy mix Yet EROIscomputations have always been done from actual data of oureconomy and could falsely represent the efficiencies of en-ergy technologies in another energy mix say a 100 renew-200

able one This uncertainty concerning the sustainability of adecarbonized energy system motivates the core of this paperthe estimation of EROIs after a global energy transition

3 Estimation of Current and Future EROIs Using THEMIS

31 Definitions and Setting205

Different notions of EROIs have been used in the litera-ture and some papers clarify them all (eg Brandt amp Dale

3Similarly to the system-wide EROI the EROI of a technology is the ratiobetween the energy delivered by one unit of this technology (over its life-time) and the energy required to build operate maintain and dismantle it

Furthermore one can show that 1EROI =

pEROIPV

+1minusp

EROIg as and this formula

generalizes to any number of technologies4Chain of production recursive or embodied inputs are synonyms their

analysis is known as structural path analysis in the literature

Figure 3 EROIs in the two-technology model in function of the share p of PVin the energy mix

2011 Murphy et al 2011) The most relevant notion for thisresearch is defined by Brandt amp Dale (2011) as the Gross En-ergy Ratio (GER) The GER measures the ratio of energy deliv- 210

ered over energy embodied in inputs net of the energy of thefuels transformed in the process Thus for example the de-nominator of the GER does not take into account the energyprovided by gas in a gas powered plant The term ldquogrossrdquo isused because all energy output is taken into account on the 215

contrary Net Energy Ratios subtract from the numerator allldquoself-userdquo output that is used in the pathway of production ofthe technology5 A related indicator that is sometimes used tocompute EROI (as it is already included in many input-outputdatabases) is the Cumulated Energy Demand (CED) I do not 220

use it because Arvesen amp Hertwich (2015) have shown that itis erroneous to use the CED directly for EROI computationswithout making adjustments

In most cases EROIs (or energy ratios) are defined usingquantities of primary energy However I adopt a different 225

approach in this paper and use only secondary energies inmy computations Indeed as Arvesen amp Hertwich (2015) putit ldquoEROI does not need to measure primary energy per sethe crucial point is to measure energy diverted from societyin a unit of equivalencerdquo Also the choice of secondary en- 230

ergy carriers is consistent with an energy system relying onrenewable electricity while for such systems the definition ofprimary energy is not harmonized and this can lead to incon-sistencies Frischknecht et al (2015) spot for example a factor6 between the cumulative (primary) energy demand for solar 235

5It is worth noting that the Gross Energy Ratio is called by King (2014)the net external energy ratio As the terminologies of these two papers arenot compatible I follow Brandt amp Dale (2011) who aim at harmonising theterminology For King ldquogrossrdquo energy is the total energy diverted from Na-ture while ldquonetrdquo is the output of energy from the technology what Brandtand Dale call ldquogrossrdquo Furthermore King would qualify ldquoexternalrdquo any no-tion that subtract the fuel transformed in production from the denominatorwhile Brandt and Dale always take this as a base case and employ ldquoexter-nalrdquo when self-use output is also subtracted it mirrors their notion of ldquonetrdquofor the denominator As we study EROIs of electricity technologies self-useoutput consists in electricity inputs in the pathway of production

4

photovoltaic computed according to different methods Al-though the sectors bringing energy are not the same in thetwo approaches (the primary approach uses crude oil whenthe secondary approaches uses gasoline for example) bothapproaches are equally valid240

Furthermore practitioners often use a factor of conver-sion (around 3) to account for the higher quality of electricityas compared to fossil fuels I follow the recommendation ofMurphy et al (2011) by undertaking my computations with-out and with a quality-adjustment factor of 26 However I245

prefer not to bring to the fore the quality-adjusted computa-tions provided in Appendix C and I focus instead on non-quality adjusted EROIs The reason for this is that the factorof conversion is not well established it represents the inverseof the yield of a thermal power station (about 38) but this250

yield depends on the technology and on the fuel used More-over for certain usage like heating the yield of fossil fuels isclose to that of electricity and fossil fuels are disproportion-ately used for these applications for which they have a higheryield therefore the difference in quality between fossils and255

electricity may be smaller than usually assumed Finally Ta-ble 1 summarizes the choices that have been made to addresscommon problems in Net Energy Analysis These choices areconsistent with the method of Brand-Correa et al (2017) tocompute national EROIs260

To avoid the possible ambiguity of sentences I reproducebelow the formulas used to compute the EROI for a technol-ogy (or an energy system) t which I denote GER2nd

t Let us re-call that y is the vector of final demand given by the scenarioand A is the technology matrix (or input-output table) E S is265

the row vector of unitary energy supply per sector meaningthat E S

t is the energy supplied by one unit of sector t henceE S

middot yt gives the energy supplied by the technology t6

supplyt = E Smiddot yt (7)

⊙ (resp ⊘) denotes the Hadamard (or entrywise) product(resp division) so that E S

⊙12nd is the vector of unitary sec-270

ondary energy supply The main term at the denominator ofthe GER is the secondary energy embodied in inputs net ofthe energy supplied by the technology

net embodiedt = E S⊙12nd middot

(

(I minus A)minus1middot yt minus yt

)

(8)

To this term we also need to subtract the energy suppliedby secondary fuels which are direct inputs to thermal electric-275

ity somewhere in the supply-chain including at the last stageIndeed such energy is not used to build or maintain the en-ergy system rather it is an energy transformed and deliveredby the electricity technology so including it would amount todouble-counting This term is especially important when t is280

some kind of thermal electricity

6In practice y is obtained from the scenario of energy demand from theIEA

yt =

(

demand⊘ES)

⊙1t (6)

fuels inputs to elect = E S⊙12nd fuelmiddotAmiddot1thermal elec⊙(I minus A)minus1

middotyt

(9)where 1thermal elec ⊙ (I minus A)minus1

middot yt is the embodied thermalelectricity

Finally we have

GER2ndt =

supplyt

net embodiedt minus fuel input to elect

(10)

32 Data Sources and Method 285

I apply these formulas to the IOTs (ie technology ma-trices A) and the vectors of unitary energy supply E S fromTHEMIS (Gibon et al 2015) THEMIS contains hybrid input-output tables precise data on electricity units (the foreground)is completed with data on other sectors that originates from 290

life cycle inventories and national accounts (the background)Gibon et al (2015) have compiled various life cycle invento-ries into the 609 sectors of the foreground including origi-nal and up-to-date life cycle inventories for electricity sec-tors Hertwich et al 2015 and its Supplementary Information 295

(SI) detail sources and values retained for the evolution ofcrucial parameters of electricity technologies such as energyefficiency and market shares of different photovoltaic mod-ules The background contains data in physical units for 4087sectors from the life cycle inventory ecoinvent and data in 300

monetary units for 203 sectors from the input-output databaseExiobase (Wood et al 2014) The 44 Exiobase regions are ag-gregated into 9 macro-regions that coincide with those of theInternational Energy Agency (IEA) so that the number of rowsand columns in each IOT is 9 times the number of sectors 305

44046 Starting from data of the 2010 IOT the 2030 and 2050IOTs of THEMIS embed expected technological efficiency im-provements of key background sectors produced by the NewEnergy Externalities Development for Sustainability project(NEEDS 2009) NEEDSrsquo realistic-optimistic scenario was iden- 310

tified as the closest match to the Blue Map and Greenpeacersquosscenarios assumptions namely the deployment of best avail-able techniques and reasonable efficiency trends while therealistic-pessimistic scenario matched the Baseline assump-tions Besides improvements in foreground processes are 315

modeled using (1) industry road maps (2) technology learn-ing curves and (3) expert opinion (see SI of Hertwich et al(2015) for more details) Furthermore it is worth noting thatTHEMIS IOTs are constructed as if the whole economy wereat a steady-state contrarily to national accounts which give 320

the flows between sectors for a given year This matches per-fectly our purpose because there is no need to adjust the EROIcomputations for the growth of some sector or for the life-times of some technologies Finally as THEMIS is multire-gional EROIs are given in total rather than internal terms 325

meaning that embodied energy contains energy embodied inimportsThe two scenarios native in THEMIS are the base-line (BL) and the Blue Map (BM) scenarios of the IEA (IEA2010) While the former posits an almost constant electricity

5

Table 1 How this paper deals with classical problems of Net Energy Analysis

Problem Reference Solution adoptedSystem boundary Suh (2004) Input-Output (exhaustive) approach

Dynamic vs steady state Muumlller et al (2014) Steady state with vintage capitalPredicting future coefficients Gibon et al (2015) Use of THEMIS modelingMeshing distinct energy types Raugei (2019) Compare only electricity technologiesPrimary vs secondary energy Arvesen amp Hertwich (2015) Secondary energy

Quality adjustment Murphy et al (2011) Emphasis on non-quality adjusted both doneDefinition of EROI Brandt amp Dale (2011) Gross Energy Ratio

Figure 4 Evolution of global EROIs and mixes of electricity for different scenarios

mix the latter is compatible with a 50 probability to con-330

tain the global mean temperature anomaly to +2degC in 2100As Blue Map still relies at 30 on fossil fuels based electric-ity in 2050 mdashincluding 17 with Carbon Capture and Stor-age (CCS) it does not allow to assess more decarbonized sce-narios Hence I combine with THEMIS the scenarios from335

Greenpeacersquos Energy [R]evolution report (Teske et al 2015)Greenpeace proposes a business as usual scenario (REF) closeto baseline as well as two scenarios compatible with the 2degCtarget Both exclude CCS and phase out from nuclear be-tween 2012 and 20507 The first Greenpeace scenario Energy340

[R]evolution (ER) comprises 93 of electricity from renew-able sources in 2050 while the second one Advanced En-ergy [R]evolution (ADV) attains 100 renewable As the dif-ference is small between these two scenarios I focus on the100 renewable one I describe my methodology for embed-345

ding the regional electricity mixes of Greenpeacersquos scenariosinto THEMIS in Appendix A

In the literature most EROIs estimations follow a bottom-up approach that use data from life cycle inventories Bottom-

7The study funded by Greenpeace was in fact conducted by researchersat the Institute of Engineering Thermodynamics of the German AerospaceCenter (DLR) who applied their model REMix Using the same modelBerrill et al (2016) minimize the cost of European electricity generation un-der different carbon prices Interestingly an outcome of the model was tophase nuclear out but to select coal with CCS This indicates that the choicesof Greenpeace were not solely motivated by a minimization of costs but alsoby expert judgment and ethical considerations

up studies describe in details the power facilities and the most 350

direct inputs to the energy technologies but they do not coverthe entire economy indirect inputs such as clerical work orRampD are often beyond their system boundaries (Suh 2004)On the contrary the input-output method allows to encom-pass all embodied inputs exhaustively As a consequence of 355

this more comprehensive account of embodied energy thanusual we expect estimates of EROIs lower than the averageof the literature That being said it is not a concern if ourestimates are not directly comparable to those of the litera-ture as we are mainly interested in comparing them inter- 360

nally among the different years and scenarios and to scruti-nize whether they vary substantially or not

Because renewable sources are intermittent and dispersedthe capacity grid extension and storage they require do notincrease linearly with the electricity delivered Hence as Green- 365

peace scenarios are not native in THEMIS they need furtheradjustments to account for these non-linearities I explain inAppendix A how the need for overcapacity is addressed Con-cerning transmission and storage however the requirementsare not given by the Greenpeace report (Teske et al 2015) so 370

they have not been taken into account Even if the report doesnot precise any plan relative to storage hydrogen producedfrom renewables seems to play a substantial role in Green-peace scenarios as its share in the electricity mix is 5 inADV 2050 However as the sector lsquoElectricity from hydrogenrsquo 375

is absent from THEMIS hydrogen has been excluded fromthis analysis These limitations should be addressed in future

6

Table 2 EROIs and share in electricity mix of electric technologies in the model THEMIS for different scenarios and yearsThe bottom line in columns mix gives the total secondary energy demand in PWha

Scenario Baseline (BL) Blue Map (BM +2deg) ADV (100 renewable)

Year 2010 2030 2050 2030 2050 2030 2050

Variable EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix

biomass w CCS ndash 000 ndash 000 ndash 000 46 000 40 001 ndash 000 ndash 000biomassampWaste 114 001 63 002 59 003 55 006 52 005 52 005 46 005

ocean 55 000 24 000 29 000 37 000 58 000 48 001 49 003geothermal 54 000 52 001 51 001 52 001 54 002 38 003 39 007

solar CSP 216 000 89 000 91 001 82 002 79 006 93 007 78 022solar PV 93 000 74 001 72 001 64 002 60 006 54 014 47 021

wind offshore 94 000 110 001 105 001 77 003 63 004 65 004 64 010wind onshore 95 001 93 004 81 004 71 008 73 008 72 017 58 024

hydro 132 016 119 014 119 012 128 018 131 014 110 013 109 008nuclear 105 014 73 011 70 010 73 019 74 024 83 002 ndash 000

gas w CCS ndash 000 ndash 000 75 000 79 001 91 005 ndash 000 ndash 000coal w CCS ndash 000 ndash 000 62 000 71 005 71 012 ndash 000 ndash 000

oil 84 006 98 002 99 001 95 003 73 001 100 001 ndash 000gas 139 021 150 021 149 023 173 014 197 011 165 018 ndash 000coal 129 042 115 045 115 045 116 018 124 001 104 016 115 000Total 122 1976 109 3429 107 4597 91 2801 80 4022 81 3674 58 6404

work together with the study of an energy transition in thetransportation sector (which also partly relies on hydrogen)Such extension will not be easy as the transportation sectors380

are still not sufficiently disaggregated in THEMIS to study achange in their technology Meanwhile other references canprovide information on orders of magnitude of storage andtransmission (Berrill et al 2016 Koskinen amp Breyer 2016 Scholz et al2017) Applying REMix the same optimization model that is385

used in the Greenpeace report Scholz et al (2017) show thatthe cost of storage and transmission combined is 46 of to-tal cost in a business-as-usual scenario and 106 in a 100renewable one The adjustment needed for the cost around6 gives a rough estimate of the upward bias of unadjusted390

EROI estimates (see section 42 on the relation between priceand EROI)

Finally data for Concentrated Solar Panels (CSP) had tobe adjusted because the original data mistakenly containedan energy supplied by unit of solar CSP of 0 in some regions395

(leading to abnormally low EROIs around 2) Backed by ThomasGibon core developer of THEMIS I corrected this error bysetting the unitary energy supplied for solar CSP in all regionsto its value in OECD North America (still letting the value de-pend on the scenario and the year)400

33 Main Results

Main results are shown in Figure 4 and in Table 2 Com-plementary results for quality-adjusted EROIs and all scenar-ios can be found in Appendix C Complete results are pro-vided in the Supplementary Information spreadsheet they405

include eg regional estimates and a decomposition of EROIsrsquodenominators between direct and indirect energy Some EROIsare missing because not all technologies already existed on

an industrial scale in 2010 and some technologies are dis-carded in the future by some scenarios Conversely some 410

EROIs are given for apparent shares of production of 0 thisis the case when the share is rounded to 0 but not 0

One can notice that as expected PV and wind panels havea lower EROI than electricity from fossil fuels The EROIs ofrenewables decrease as anticipated in the previous section 415

However they remain largely above 1 suggesting that renew-ables are energetically sustainable Recall that this was notevident as in theory nothing guarantees that EROIs stay above1 when the energy mix changes (see section 22) Values forcurrent EROIs range from 8 to 22 This range is in-line with 420

that from Hall et al (2014) but not with Weiszligbach et al (2013)who find more contrasts between renewables and fossils Suchdiscrepancy is common in the EROI literature may be due todifferences in the methodology (Weiszligbach uses bottom-updata from specific locations) and does not affect this paperrsquos 425

results on the evolution of EROIsThe system-wide EROI for the entire electricity sector is

given at the bottom line of Table 2 It is estimated at 122 in2010 it decreases slightly until 108plusmn01 in 2030 and 2050 inthe Baseline scenario An examination of regional estimates 430

(see SI) reveals that this decrease is driven by a compositioneffect in the global mix Indeed the largest energy producerin 2010 North America has higher EROIs and is replaced byChina in 2030 which has lower ones the EROIs in each worldregion remaining quite stable8 The decrease is more pro- 435

nounced when the penetration of renewable is higher downto 80 in 2050 in the Blue Map scenario and even 58 in the

8In Baseline EROIs of OECD Europe and India are very close to globalEROI while those of Africa amp Middle East and Rest of developing Asia arewithin plusmn3 to global ones

7

100 renewable one The magnitude of the decline is sub-stantial an expected halving of global EROI may prove to bea challenge for the success of an energy transition to renew-440

ablesOne may wonder whether our results are driven by con-

servative forecasts concerning the progress in renewable tech-nologies or any other hypothesis concerning the evolution ofthe technology matrix Of course the quality of input-output445

data is never perfect and making predictions is notoriouslydifficult as was recently proven by the unexpected fall in theprice of photovoltaic (PV) modules However there are sev-eral reasons to be more confident into future EROIs estimatesfrom THEMIS than into past predictions on prices from other450

sources First technical coefficients are more stable than pricesSecond THEMIS accounts for materials and energy efficiencygains for electricity technologies and uses ldquofairly favorableassumptions regarding wind conditions insolation and re-sulting load factorsrdquo which if anything would bias EROIs of455

renewables upward (see SI of Hertwich et al 2015) ThirdTHEMIS already includes recent industry road maps in its prospec-tive matrices (see section 31) eg concerning the shift of PVmarket shares from cristalline silicon modules towards moreefficient cadmium telluride (CdTe) or CIGS modules Overall460

the data from THEMIS seems most accurate concerning ma-terials metallurgy and energy sectors and further improve-ments should probably focus on other sectors like transportor services

4 Implications of a Decreasing EROI on Prices and GDP465

The forecast of declining EROIs made in the previous sec-tion calls for an assessment of its economic implications Themain channel through which a decrease in EROI could affectthe economy is arguably a rise in energy price (and correla-tively in energy expenditures) In this section I review the lit-470

erature on the relation between EROI and the price of energyestimate it empirically and extend a result from Herendeen(2015) to characterize this relation As in previous work aninverse relation is documented empirically Yet theoreticalanalysis shows that EROI and price might decrease together475

This theoretical result tempers the view that a decreasing EROInecessarily leads to a contraction of GDP

41 Inverse Relation Proposed in First Studies

King amp Hall (2011) point both theoretically and empiricallythat the price of a unit of energy pt and the EROI of a technol-480

ogy t are inversely related Defining the monetary return on

investment MROI (ie the financial yield $out$investment

) they de-rive the formula

pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(11)

Heun amp de Wit (2012) find an equivalent formula Theydesignate MROI as the mark-up mt consider production costs485

per gross output ct =$investmentEout+Ein

and use their own notion ofEROI

Table 3 Predicted average global price of electricity (in euroMWh)

year 2010 2030 2050scenario all BL BM ADV BL BM ADV

price 27 28 30 30 28 30 32

EROIHt =

Eout+EinEin

= EROIt +1 so that equation (11) rewrites

pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt

(12)

The problem with these formulas is that all variables movetogether when EROI varies so does the cost of production 490

so that we cannot predict the future price taking this cost asfixed Heun amp de Wit (2012) acknowledge this and thus studythe empirical link between EROI and price

42 Empirical Relation Between EROI and Price

Using US data on oil and EROI from (Cleveland 2005) 495

Heun amp de Wit (2012) regress pt on the EROI9 They obtain agood fit even in their simplest regression (R2

= 08) and findpoil =β0 middotEROIminus14

oil This result is interesting and documents a negative rela-

tionship between price and EROI which is close to an inverse 500

one As the authors do not regress price on the inverse ofEROI one cannot compare whether an inverse specificationwould provide as good a fit as a log-log one To undertake thiscomparison I run these two regressions using all estimates ofEROI computed using THEMIS one for each combination of 505

scenario year region and sector To obtain the price corre-sponding to each EROI which I take before taxes and subsi-dies on production I assemble from the columns compensa-

tion of employees and operating surplus of the characteriza-tion matrix of THEMIS a row vector v of value-added per unit 510

of each sector Indeed the vector of prices excluding tax p

can be seen as emerging from value-added according to

p = v middot (I minus A)minus1⊘E S (13)

because the price of energy in sector s ps is the sum ofthe value-added of inputs embodied in s v middot (I minus A)minus1

middot1s di-vided by the energy supplied by one unit of s E S

s To the ex- 515

tent that the physical constituents and processes of a giventechnology will not change in an unexpected way and as THEMISmodels technical progress but not behaviors nor general equi-librium effects prices forecast using the above formula seemless reliable than EROI estimates For this reason I report 520

only the global average electricity prices of the main scenar-ios (see Table 3) but I do not detail the substantial variationsbetween regions or sectors10

9Although they claim that their explained and explanatory variables arerespectively the cost of production and their notion of EROI EROIH the for-mer is indeed the producer price and the latter our notion of EROI accordingto the source of their data Cleveland (2005)

10The results are on-line and available on demand

8

Table 2 EROIs and share in electricity mix of electric technologies in the model THEMIS for different scenarios and yearsThe bottom line in columns mix gives the total secondary energy demand in PWha

Scenario Baseline (BL) Blue Map (BM +2deg) ADV (100 renewable)

Year 2010 2030 2050 2030 2050 2030 2050

Variable EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix

biomass w CCS ndash 000 ndash 000 ndash 000 46 000 40 001 ndash 000 ndash 000biomassampWaste 114 001 63 002 59 003 55 006 52 005 52 005 46 005

ocean 55 000 24 000 29 000 37 000 58 000 48 001 49 003geothermal 54 000 52 001 51 001 52 001 54 002 38 003 39 007

solar CSP 216 000 89 000 91 001 82 002 79 006 93 007 78 022solar PV 93 000 74 001 72 001 64 002 60 006 54 014 47 021

wind offshore 94 000 110 001 105 001 77 003 63 004 65 004 64 010wind onshore 95 001 93 004 81 004 71 008 73 008 72 017 58 024

hydro 132 016 119 014 119 012 128 018 131 014 110 013 109 008nuclear 105 014 73 011 70 010 73 019 74 024 83 002 ndash 000

gas w CCS ndash 000 ndash 000 75 000 79 001 91 005 ndash 000 ndash 000coal w CCS ndash 000 ndash 000 62 000 71 005 71 012 ndash 000 ndash 000

oil 84 006 98 002 99 001 95 003 73 001 100 001 ndash 000gas 139 021 150 021 149 023 173 014 197 011 165 018 ndash 000coal 129 042 115 045 115 045 116 018 124 001 104 016 115 000Total 122 1976 109 3429 107 4597 91 2801 80 4022 81 3674 58 6404

work together with the study of an energy transition in thetransportation sector (which also partly relies on hydrogen)Such extension will not be easy as the transportation sectors380

are still not sufficiently disaggregated in THEMIS to study achange in their technology Meanwhile other references canprovide information on orders of magnitude of storage andtransmission (Berrill et al 2016 Koskinen amp Breyer 2016 Scholz et al2017) Applying REMix the same optimization model that is385

used in the Greenpeace report Scholz et al (2017) show thatthe cost of storage and transmission combined is 46 of to-tal cost in a business-as-usual scenario and 106 in a 100renewable one The adjustment needed for the cost around6 gives a rough estimate of the upward bias of unadjusted390

EROI estimates (see section 42 on the relation between priceand EROI)

Finally data for Concentrated Solar Panels (CSP) had tobe adjusted because the original data mistakenly containedan energy supplied by unit of solar CSP of 0 in some regions395

(leading to abnormally low EROIs around 2) Backed by ThomasGibon core developer of THEMIS I corrected this error bysetting the unitary energy supplied for solar CSP in all regionsto its value in OECD North America (still letting the value de-pend on the scenario and the year)400

33 Main Results

Main results are shown in Figure 4 and in Table 2 Com-plementary results for quality-adjusted EROIs and all scenar-ios can be found in Appendix C Complete results are pro-vided in the Supplementary Information spreadsheet they405

include eg regional estimates and a decomposition of EROIsrsquodenominators between direct and indirect energy Some EROIsare missing because not all technologies already existed on

an industrial scale in 2010 and some technologies are dis-carded in the future by some scenarios Conversely some 410

EROIs are given for apparent shares of production of 0 thisis the case when the share is rounded to 0 but not 0

One can notice that as expected PV and wind panels havea lower EROI than electricity from fossil fuels The EROIs ofrenewables decrease as anticipated in the previous section 415

However they remain largely above 1 suggesting that renew-ables are energetically sustainable Recall that this was notevident as in theory nothing guarantees that EROIs stay above1 when the energy mix changes (see section 22) Values forcurrent EROIs range from 8 to 22 This range is in-line with 420

that from Hall et al (2014) but not with Weiszligbach et al (2013)who find more contrasts between renewables and fossils Suchdiscrepancy is common in the EROI literature may be due todifferences in the methodology (Weiszligbach uses bottom-updata from specific locations) and does not affect this paperrsquos 425

results on the evolution of EROIsThe system-wide EROI for the entire electricity sector is

given at the bottom line of Table 2 It is estimated at 122 in2010 it decreases slightly until 108plusmn01 in 2030 and 2050 inthe Baseline scenario An examination of regional estimates 430

(see SI) reveals that this decrease is driven by a compositioneffect in the global mix Indeed the largest energy producerin 2010 North America has higher EROIs and is replaced byChina in 2030 which has lower ones the EROIs in each worldregion remaining quite stable8 The decrease is more pro- 435

nounced when the penetration of renewable is higher downto 80 in 2050 in the Blue Map scenario and even 58 in the

8In Baseline EROIs of OECD Europe and India are very close to globalEROI while those of Africa amp Middle East and Rest of developing Asia arewithin plusmn3 to global ones

7

100 renewable one The magnitude of the decline is sub-stantial an expected halving of global EROI may prove to bea challenge for the success of an energy transition to renew-440

ablesOne may wonder whether our results are driven by con-

servative forecasts concerning the progress in renewable tech-nologies or any other hypothesis concerning the evolution ofthe technology matrix Of course the quality of input-output445

data is never perfect and making predictions is notoriouslydifficult as was recently proven by the unexpected fall in theprice of photovoltaic (PV) modules However there are sev-eral reasons to be more confident into future EROIs estimatesfrom THEMIS than into past predictions on prices from other450

sources First technical coefficients are more stable than pricesSecond THEMIS accounts for materials and energy efficiencygains for electricity technologies and uses ldquofairly favorableassumptions regarding wind conditions insolation and re-sulting load factorsrdquo which if anything would bias EROIs of455

renewables upward (see SI of Hertwich et al 2015) ThirdTHEMIS already includes recent industry road maps in its prospec-tive matrices (see section 31) eg concerning the shift of PVmarket shares from cristalline silicon modules towards moreefficient cadmium telluride (CdTe) or CIGS modules Overall460

the data from THEMIS seems most accurate concerning ma-terials metallurgy and energy sectors and further improve-ments should probably focus on other sectors like transportor services

4 Implications of a Decreasing EROI on Prices and GDP465

The forecast of declining EROIs made in the previous sec-tion calls for an assessment of its economic implications Themain channel through which a decrease in EROI could affectthe economy is arguably a rise in energy price (and correla-tively in energy expenditures) In this section I review the lit-470

erature on the relation between EROI and the price of energyestimate it empirically and extend a result from Herendeen(2015) to characterize this relation As in previous work aninverse relation is documented empirically Yet theoreticalanalysis shows that EROI and price might decrease together475

This theoretical result tempers the view that a decreasing EROInecessarily leads to a contraction of GDP

41 Inverse Relation Proposed in First Studies

King amp Hall (2011) point both theoretically and empiricallythat the price of a unit of energy pt and the EROI of a technol-480

ogy t are inversely related Defining the monetary return on

investment MROI (ie the financial yield $out$investment

) they de-rive the formula

pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(11)

Heun amp de Wit (2012) find an equivalent formula Theydesignate MROI as the mark-up mt consider production costs485

per gross output ct =$investmentEout+Ein

and use their own notion ofEROI

Table 3 Predicted average global price of electricity (in euroMWh)

year 2010 2030 2050scenario all BL BM ADV BL BM ADV

price 27 28 30 30 28 30 32

EROIHt =

Eout+EinEin

= EROIt +1 so that equation (11) rewrites

pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt

(12)

The problem with these formulas is that all variables movetogether when EROI varies so does the cost of production 490

so that we cannot predict the future price taking this cost asfixed Heun amp de Wit (2012) acknowledge this and thus studythe empirical link between EROI and price

42 Empirical Relation Between EROI and Price

Using US data on oil and EROI from (Cleveland 2005) 495

Heun amp de Wit (2012) regress pt on the EROI9 They obtain agood fit even in their simplest regression (R2

= 08) and findpoil =β0 middotEROIminus14

oil This result is interesting and documents a negative rela-

tionship between price and EROI which is close to an inverse 500

one As the authors do not regress price on the inverse ofEROI one cannot compare whether an inverse specificationwould provide as good a fit as a log-log one To undertake thiscomparison I run these two regressions using all estimates ofEROI computed using THEMIS one for each combination of 505

scenario year region and sector To obtain the price corre-sponding to each EROI which I take before taxes and subsi-dies on production I assemble from the columns compensa-

tion of employees and operating surplus of the characteriza-tion matrix of THEMIS a row vector v of value-added per unit 510

of each sector Indeed the vector of prices excluding tax p

can be seen as emerging from value-added according to

p = v middot (I minus A)minus1⊘E S (13)

because the price of energy in sector s ps is the sum ofthe value-added of inputs embodied in s v middot (I minus A)minus1

middot1s di-vided by the energy supplied by one unit of s E S

s To the ex- 515

tent that the physical constituents and processes of a giventechnology will not change in an unexpected way and as THEMISmodels technical progress but not behaviors nor general equi-librium effects prices forecast using the above formula seemless reliable than EROI estimates For this reason I report 520

only the global average electricity prices of the main scenar-ios (see Table 3) but I do not detail the substantial variationsbetween regions or sectors10

9Although they claim that their explained and explanatory variables arerespectively the cost of production and their notion of EROI EROIH the for-mer is indeed the producer price and the latter our notion of EROI accordingto the source of their data Cleveland (2005)

10The results are on-line and available on demand

8

100 renewable one The magnitude of the decline is sub-stantial an expected halving of global EROI may prove to bea challenge for the success of an energy transition to renew-440

ablesOne may wonder whether our results are driven by con-

servative forecasts concerning the progress in renewable tech-nologies or any other hypothesis concerning the evolution ofthe technology matrix Of course the quality of input-output445

data is never perfect and making predictions is notoriouslydifficult as was recently proven by the unexpected fall in theprice of photovoltaic (PV) modules However there are sev-eral reasons to be more confident into future EROIs estimatesfrom THEMIS than into past predictions on prices from other450

sources First technical coefficients are more stable than pricesSecond THEMIS accounts for materials and energy efficiencygains for electricity technologies and uses ldquofairly favorableassumptions regarding wind conditions insolation and re-sulting load factorsrdquo which if anything would bias EROIs of455

renewables upward (see SI of Hertwich et al 2015) ThirdTHEMIS already includes recent industry road maps in its prospec-tive matrices (see section 31) eg concerning the shift of PVmarket shares from cristalline silicon modules towards moreefficient cadmium telluride (CdTe) or CIGS modules Overall460

the data from THEMIS seems most accurate concerning ma-terials metallurgy and energy sectors and further improve-ments should probably focus on other sectors like transportor services

4 Implications of a Decreasing EROI on Prices and GDP465

The forecast of declining EROIs made in the previous sec-tion calls for an assessment of its economic implications Themain channel through which a decrease in EROI could affectthe economy is arguably a rise in energy price (and correla-tively in energy expenditures) In this section I review the lit-470

erature on the relation between EROI and the price of energyestimate it empirically and extend a result from Herendeen(2015) to characterize this relation As in previous work aninverse relation is documented empirically Yet theoreticalanalysis shows that EROI and price might decrease together475

This theoretical result tempers the view that a decreasing EROInecessarily leads to a contraction of GDP

41 Inverse Relation Proposed in First Studies

King amp Hall (2011) point both theoretically and empiricallythat the price of a unit of energy pt and the EROI of a technol-480

ogy t are inversely related Defining the monetary return on

investment MROI (ie the financial yield $out$investment

) they de-rive the formula

pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(11)

Heun amp de Wit (2012) find an equivalent formula Theydesignate MROI as the mark-up mt consider production costs485

per gross output ct =$investmentEout+Ein

and use their own notion ofEROI

Table 3 Predicted average global price of electricity (in euroMWh)

year 2010 2030 2050scenario all BL BM ADV BL BM ADV

price 27 28 30 30 28 30 32

EROIHt =

Eout+EinEin

= EROIt +1 so that equation (11) rewrites

pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt

(12)

The problem with these formulas is that all variables movetogether when EROI varies so does the cost of production 490

so that we cannot predict the future price taking this cost asfixed Heun amp de Wit (2012) acknowledge this and thus studythe empirical link between EROI and price

42 Empirical Relation Between EROI and Price

Using US data on oil and EROI from (Cleveland 2005) 495

Heun amp de Wit (2012) regress pt on the EROI9 They obtain agood fit even in their simplest regression (R2

= 08) and findpoil =β0 middotEROIminus14

oil This result is interesting and documents a negative rela-

tionship between price and EROI which is close to an inverse 500

one As the authors do not regress price on the inverse ofEROI one cannot compare whether an inverse specificationwould provide as good a fit as a log-log one To undertake thiscomparison I run these two regressions using all estimates ofEROI computed using THEMIS one for each combination of 505

scenario year region and sector To obtain the price corre-sponding to each EROI which I take before taxes and subsi-dies on production I assemble from the columns compensa-

tion of employees and operating surplus of the characteriza-tion matrix of THEMIS a row vector v of value-added per unit 510

of each sector Indeed the vector of prices excluding tax p

can be seen as emerging from value-added according to

p = v middot (I minus A)minus1⊘E S (13)

because the price of energy in sector s ps is the sum ofthe value-added of inputs embodied in s v middot (I minus A)minus1

middot1s di-vided by the energy supplied by one unit of s E S

s To the ex- 515

tent that the physical constituents and processes of a giventechnology will not change in an unexpected way and as THEMISmodels technical progress but not behaviors nor general equi-librium effects prices forecast using the above formula seemless reliable than EROI estimates For this reason I report 520

only the global average electricity prices of the main scenar-ios (see Table 3) but I do not detail the substantial variationsbetween regions or sectors10

9Although they claim that their explained and explanatory variables arerespectively the cost of production and their notion of EROI EROIH the for-mer is indeed the producer price and the latter our notion of EROI accordingto the source of their data Cleveland (2005)

10The results are on-line and available on demand

8

Table 4 Regressions of price on EROI (both estimated using THEMIS)All coefficients are significant at the 1h level

Obs N SpecificationCoefficients

R2a

a b

All 2079p =

aEROI

+b85 18 055

2010 104 72 21 054All 2079

log(

p)

= a middot log (EROI)+bminus057 20 058

2010 104 minus046 19 062

aThe R2 given for log-log fits is not the original one cf text

Table 4 reports the results of both the log-log and the in-verse fits I ran each model twice first on all 2079 positive525

observations available and then on the 104 observations foryear 2010 To make the R2 of the log-log fit comparable tothat of the inverse fit I compute it as the sum of squared er-rors between ldquoobservedrdquo prices and predicted prices (insteadof their respective logarithms) As all R2 are between 054 and530

062 the inverse fit is almost as accurate than the log-log fitMoreover although the elasticity of price on EROI estimatedhere is different from that found by Heun amp de Wit (2012) foroil (around minus05 as compared to minus14) both figures are closeto 1 Empirical findings confirm an inverse relation between535

price and EROI However Figure 5 shows that a significantshare of the variance in price remains unexplained by EROIeven more so for values of EROI around the global averages of6-12 where the fit is almost flat and the errors substantial Inaddition theoretical analysis rejects the existence of a map-540

ping between price and EROI

Figure 5 Regressions of price on EROI (all observations from THEMIS)

43 A Case Against Any Simple Relation

Herendeen (2015) shed new light on the theoretical rela-tion by treating the question from its matrix form and intro-ducing the concept of value-added Herendeen showed how545

to express rigorously the price in function of the EROI whenthe economy is constituted of two sectors (energy and ma-terials) and explained the limits of such exercise Hereafter

I extend the results of Herendeen to an arbitrary number ofsectors n His approach relies on the concept of energy in- 550

tensityTo deliver one unit of energy technology t the production

mobilized is (I minus A)minus1middot1t while the energy mobilized called

the energy intensity of t writes εt = 1TE middot (I minus A)minus1

middot1t whereE is the set of all energies11 εt is in fact the gross energy em- 555

bodied in t ie the sum of the delivered and the net embod-ied energy Hence the EROI of t is a simple function of εt

EROIt =1

TEmiddot1t

1TEmiddot((IminusA)minus1minusI)middot1t

=1

εtminus1

In Appendix D I show that the price of a technology t is acertain function of the coefficients of A12 and that each coef- 560

ficient of A can be expressed as a function of EROI Compos-ing two such functions we obtain that the price is inverselyrelated to EROI However the relation is not unique (as it de-pends on the coefficient of A chosen to make the connection)and the other parameters in the relation are not constant 565

This leads to the following Proposition

Proposition 1 (Generalization of Herendeen 2015) Assum-

ing that all coefficients of the transformation matrix A are con-

stant except one noted x = ai0 j0 and that EROI varies with x

the price of t can be expressed as a linear function of its energy 570

intensity εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β (14)

Proof See Appendix D

Remark With the terminology of Heun amp de Wit (2012) or Herendeen(2015) the relation above would write

pt = αEROIH

t

EROIHt minus1

+ γ with γ = βminusα This is because in their 575

definition of EROI the numerator is εt instead of 1

In the general case we cannot obtain a better result iea formula that still holds when letting more than one coeffi-cient vary Indeed denotingωi t the coefficient (i t) of (I minus A)minus1the Laplace expansion of I minus A gives us 580

ωi t =(minus1)i+ j

det(IminusA) det

(

(I minus A)j k)

jisinJ1nKi

kisinJ1nKt

Hence we have

εt =sum

eisinE

(

(I minus A)minus1)

et =sum

eisinE ωet and

pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =sumn

i=1 viωi t =sum

eisinE veωet+sum

inotinE viωi t

Denoting v =

sum

eisinE veωetsum

eisinE ωetand r = v +

sum

inotinE viωi t we obtain

pt = vεt +sum

inotinE

viωi t =v

EROIt+ r (15)

However one has to keep in mind that r v and EROIt

all depend on the coefficients of A and vary together whenA changes If there is only one type of energy (E = e) or if 585

value-added is equal for all types of energy (foralle isinE ve = v) v

11I assume here that the unit of an output of an energy sector e isin E henceof 1T

E is an energy unit like TWh

12More precisely a function field of a certain algebraic variety

9

does not depend on the coefficients of A anymore and we ob-tain a formula close to that of King amp Hall (2011) pt =

ve

EROIt+

r Still when the EROI varies because more than one coeffi-cient of A changes r varies concomitantly and the EROI can-590

not be used as a sufficient statistic to infer the price For thisreason one cannot identify empirically a linear relation be-tween price and the inverse of EROI without strong assump-tion on the steadiness of A

Actually the theoretical relation between EROI and price595

is so fragile that one cannot even conclude that it is a decreas-ing relation I provide in Appendix B a numerical exampleshowing that EROI and price can both increase at the sametime when more than one coefficient varies Such acknowl-edgment dissuades from predicting long run prices by simply600

looking at estimations of future EROIsDoes this mean that EROI is unrelated to any economic

concept Fizaine amp Court (2016) argue that there is a mini-mum EROI below which the US economy enters a recessionThey first show that energy expenditure Granger causes growth605

in the US then determine a threshold of energy expenditureabove which the US enters in a recession (consistent with thatof Bashmakov 2007) and finally use a modified version ofequation (11) to relate this to a minimum non-recessionaryEROI However they misleadingly replace the inverse of the610

energy intensity of energy investment $i nvestment

Einby that of the

whole economy GDPEout

This prevents them from noticing thatcost reductions in energy production could compensate theeffect of a decreasing EROI on energy prices and expendi-tures As we have seen EROI price and energy expenditure615

may all decrease at the same time which undermines the ideathat a recession caused by a surge in energy expenditure isineluctable as soon as EROI goes below some threshold Inaddition an energy price increase should have an expansion-ary effect on net exporters of energy at odds with the mech-620

anism extrapolated by Fizaine and Court from the case of theUnited States which has been historically a net energy im-porter Overall the analysis of this section indicates that theeconomic consequences of a change in EROI are ambiguousand that this physical notion cannot be used to predict future625

prices or GDP without empirical evidence

5 Concluding Remarks

This work includes a first attempt at estimating future EROIsin a decarbonized electricity system By examining a broadrange of scenarios it concludes that the system-wide EROI630

of the power sector should decrease until 2050 from 122 to107 in a business-as-usual scenario 80 in a partial transitionaway from fossil fuels or 58 in a scenario with 100 renew-able electricity Even though the EROI of each technology isexpected to remain well above 1 which was questioned theo-635

retically our results cast doubts on the energetic efficiency ofrenewable electricity

As an inverse relationship between EROIs and energy pricesis consistently found empirically a declining EROI could meanhigher energy prices However theoretical analysis of this re-640

lation showed that a declining EROI might also coincide withdecreasing energy prices and does not necessarily lead to arecession

Finally this paper assessed scenarios of transition in theelectricity sector but further research is still needed to esti- 645

mate future EROIs in complete energy transitions which in-clude a mutation of the transportation system agriculture andindustry Unfortunately this could not been done using thecurrent version of THEMIS and the question remains openfor future research 650

10

References

A Arvesen amp E G Hertwich More caution is needed when using life cy-cle assessment to determine energy return on investment (EROI) Energy

Policy 2015A Arvesen G Luderer M Pehl B L Bodirsky amp E G Hertwich Deriving655

life cycle assessment coefficients for application in integrated assessmentmodelling Environmental Modelling amp Software 2018

I Bashmakov Three laws of energy transitions Energy Policy 2007P Berrill A Arvesen Y Scholz H C Gils amp E G Hertwich Environmental

impacts of high penetration renewable energy scenarios for Europe En-660

vironmental Research Letters 2016L I Brand-Correa P E Brockway C L Copeland T J Foxon A Owen amp

P G Taylor Developing an Input-Output Based Method to Estimate aNational-Level Energy Return on Investment (EROI) Energies 2017

A R Brandt How Does Energy Resource Depletion Affect Prosperity Math-665

ematics of a Minimum Energy Return on Investment (EROI) BioPhysical

Economics and Resource Quality 2017A R Brandt amp M Dale A General Mathematical Framework for Calculat-

ing Systems-Scale Efficiency of Energy Extraction and Conversion EnergyReturn on Investment (EROI) and Other Energy Return Ratios Energies670

2011C J Cleveland Net energy from the extraction of oil and gas in the United

States Energy 2005V Court amp F Fizaine Long-Term Estimates of the Energy-Return-on-

Investment (EROI) of Coal Oil and Gas Global Productions Ecological675

Economics 2017M Dale S Krumdieck amp P Bodger Net energy yield from production of

conventional oil Energy Policy 2011M A J Dale Global Energy Modelling A Biophysical Approach (GEMBA)

2010680

Eurostat editor Eurostat Manual of supply use and input-output tables Amtfuumlr amtliche Veroumlffentlichungen der Europaumlischen Gemeinschaften Lux-embourg 2008 edition edition 2008 ISBN 978-92-79-04735-0

F Fizaine amp V Court Energy expenditure economic growth and the mini-mum EROI of society Energy Policy 2016685

R Frischknecht F Wyss S B Knoumlpfel T Luumltzkendorf amp M Balouktsi Cu-mulative energy demand in LCA the energy harvested approach The In-

ternational Journal of Life Cycle Assessment 2015T Gibon R Wood A Arvesen J D Bergesen S Suh amp E G Hertwich A

Methodology for Integrated Multiregional Life Cycle Assessment Scenar-690

ios under Large-Scale Technological Change Environmental Science amp

Technology 2015C A S Hall Introduction to Special Issue on New Studies in EROI (Energy

Return on Investment) Sustainability 2011C A S Hall S Balogh amp D J R Murphy What is the Minimum EROI that a695

Sustainable Society Must Have Energies 2009C A S Hall J G Lambert amp S B Balogh EROI of different fuels and the

implications for society Energy Policy 2014R A Herendeen Connecting net energy with the price of energy and other

goods and services Ecological Economics 2015700

E G Hertwich T Gibon E A Bouman A Arvesen S Suh G A Heath J DBergesen A Ramirez M I Vega amp L Shi Integrated life-cycle assess-ment of electricity-supply scenarios confirms global environmental ben-efit of low-carbon technologies Proceedings of the National Academy of

Sciences 2015705

M K Heun amp M de Wit Energy return on (energy) invested (EROI) oil pricesand energy transitions Energy Policy 2012

IEA Energy Technology Perspectives 2010T Junius amp J Oosterhaven The Solution of Updating or Regionalizing a Ma-

trix with both Positive and Negative Entries Economic Systems Research710

2003C W King Matrix method for comparing system and individual energy re-

turn ratios when considering an energy transition Energy 2014C W King amp C A S Hall Relating Financial and Energy Return on Invest-

ment Sustainability 2011715

O Koskinen amp C Breyer Energy Storage in Global and Transcontinental En-ergy Scenarios A Critical Review Energy Procedia 2016

J G Lambert amp G P Lambert Predicting the Psychological Response of theAmerican People to Oil Depletion and Declining Energy Return on Invest-ment (EROI) Sustainability 2011720

J G Lambert C A Hall S Balogh A Gupta amp M Arnold Energy EROI andquality of life Energy Policy 2014

W Leontief Input-Output Economics Oxford University Press USA 2 edi-tion 1986 ISBN 978-0-19-503527-8

R E Miller amp P D Blair Input-Output Analysis Foundations and Extensions 725

Cambridge University Press 2009 ISBN 978-0-521-51713-3E Muumlller L M Hilty R Widmer M Schluep amp M Faulstich Modeling Metal

Stocks and Flows A Review of Dynamic Material Flow Analysis MethodsEnvironmental Science amp Technology 2014

D J Murphy C A S Hall M Dale amp C Cleveland Order from Chaos A 730

Preliminary Protocol for Determining the EROI of Fuels Sustainability2011

NEEDS LCA of Background Processes Technical report New Energy Exter-nalities Developments for Sustainability 2009

M Pehl A Arvesen F Humpenoumlder A Popp E G Hertwich amp G Luderer 735

Understanding future emissions from low-carbon power systems by inte-gration of life-cycle assessment and integrated energy modelling Nature

Energy 2017A Poisson C Hall A Poisson amp C A S Hall Time Series EROI for Canadian

Oil and Gas Energies 2013 740

M Raugei Net energy analysis must not compare apples and oranges Na-

ture Energy 2019Y Scholz H C Gils amp R C Pietzcker Application of a high-detail energy sys-

tem model to derive power sector characteristics at high wind and solarshares Energy Economics 2017 745

S Suh Functions commodities and environmental impacts in an ecologi-calndasheconomic model Ecological Economics 2004

S Teske T Pregger S Simon amp T Naegler Energy [R]evolution Technicalreport Greenpeace 2015

G Tverberg The ldquoWind and Solar Will Save Usrdquo Delusion 2017 750

D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein En-ergy intensities EROIs (energy returned on invested) and energy paybacktimes of electricity generating power plants Energy 2013

R Wood K Stadler T Bulavskaya S Lutter S Giljum A de Koning J Kue-nen H Schuumltz J Acosta-Fernaacutendez A Usubiaga M Simas O Ivanova 755

J Weinzettel J H Schmidt S Merciai amp A Tukker Global SustainabilityAccountingmdashDeveloping EXIOBASE for Multi-Regional Footprint Analy-sis Sustainability 2014

11

Appendix A Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are read-760

ily available in THEMIS but these matrices have to be up-dated to the new electricity mix for the Greenpeace scenariosTo do this I exploit the fact that both THEMIS and Green-peace use the world regions of the IEA and I modify the elec-tricity input of each sector by the regional mix given by Green-765

peace The most accurate algorithm to update an input-outputmatrix is known as GRAS (Junius amp Oosterhaven 2003) Al-though I implemented this algorithm in pymrio I could notuse it because this algorithm uses the new sums of rows andcolumns to balance the matrix and the vector of final de-770

mand y or the vector of production x is necessary to knowthem As THEMIS does not include such vectors I had to usea simpler method which relies on the assumption that theelectricity mix of inputs is the same across sectors for a givenregion Given the perfect substitutability between electric-775

ity produced by different technologies and the uniqueness ofelectric grids this assumption seems justified

There are two different updates to make First I modifythe vector of second energy demand (used to infer the finaldemand of technology t yt ) so that it perfectly matches the780

demand of the scenario Second I modify the submatrix D ofA containing the rows of electricity sectors To convert D inenergy units I multiply each row t of D by the correspondingenergy supplied per unit of technology t E S

t I call the resultE the coefficient Ei s of E gives the electricity from sector i re-785

quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premultiplyE by a block diagonal matrix with R blocks of size T lowastT con-taining only ones (where R = 9 and T = 15 are the number ofTHEMIS regions and electricity sectors respectively) to ob-790

tain a matrix B Each row of B gives the total electricity froma given region r required to produced each output E tot

r andeach row E tot

r is replicated T times

B =

B1

BR

Br =

E totr

E totr

Next each row of B is multiplied by the share of a tech-nology t in the mix of the corresponding region which de-795

fines a matrix E Each coefficient Ei s of E gives the electricityfrom sector i required to make one unit of sector srsquo outputaccording to the new mix (by construction for all electricitysector j = i (t r ) the share of technology j in the regional mix

E j ssum

t Ei (t r )s is the same across all sectors s) Eventually I obtain800

the new submatrix D by converting each row of E to the origi-nal units of A (by dividing each row by the appropriate unitaryenergy supplied E S

t )A last update is needed for Greenpeace scenarios to ac-

count for the extra capacity needed when intermittent sources805

fail to deliver energy the ratio of capacity (in GW) over pro-duction (in TWh) is somewhat higher in Greenpeace scenar-ios than in IEATHEMIS ones Thus I multiply each columnof an energy sector (representing all inputs required for one

unit of output of this sector) by the ratio of the capacity-over- 810

production ratios of Greenpeace and IEATHEMIS Doing sorelies on the fact that the energy required to operate a powerplant is negligible in front of the energy required to build it(see eg Arvesen et al (2018))

Appendix B Example of Non-Decreasing Relation Between 815

EROI and Price

Herendeen (2015) proposes a calibration on US energydata of his toy model with 2 sectors (materials and energy)which yields as realistic results as a two-by-two model canyield I start from a slightly modified version of his calibration 820

(called base) in the sense that the figures are rounded and Ishow how a deviation of two coefficients (in the new calibra-tion) leads to an increase of both EROI and price of energyThis proves that in general nothing can be said of the rela-tion between EROI and price not even that it is a decreasing 825

relationFor this I use the formulas for EROI and price given by

Herendeen (2015) (where I convert the price to $gal usingthe conversion factor 1Btu = 114000gal)

EROI =1

Aee +Aem Ame

1minusAmm

(B1)

p =ve (1minus Amm )+ vm Ame

(1minus Aee ) (1minus Amm)minus Aem Amemiddot114000

Table B5 Example of sets of coefficients exhibiting a non-decreasingrelation between EROI and price in a two sectors model (seedesmoscomcalculatorne4oqunhsm)

base new

vm 05ve 5 middot10minus6

Aem 1700Ame 4 middot10minus6

Amm 05 09Aee 03 01

EROI 32 60price 15 34

Appendix C Complementary Results 830

Results without quality adjustment for IEATHEMIS sce-narios are provided in section 33 those for Greenpeacersquos sce-narios are in Table C6 Quality-adjusted results follows in Ta-ble C7 (IEATHEMIS) and C8 (Greenpeace) The quality ad-justment consists in separating each energy in the formula of 835

the EROI according to its origin (electric or thermal) and toweight electricity by a factor 26 For example the quality-adjusted (gross) embodied energy for a unit of technology t

writes

embodiedqual adjt = E S

⊙ (26 middot1electric +1thermal) middot (I minus A)minus1middot1t

(C1)

12

does not depend on the coefficients of A anymore and we ob-tain a formula close to that of King amp Hall (2011) pt =

ve

EROIt+

r Still when the EROI varies because more than one coeffi-cient of A changes r varies concomitantly and the EROI can-590

not be used as a sufficient statistic to infer the price For thisreason one cannot identify empirically a linear relation be-tween price and the inverse of EROI without strong assump-tion on the steadiness of A

Actually the theoretical relation between EROI and price595

is so fragile that one cannot even conclude that it is a decreas-ing relation I provide in Appendix B a numerical exampleshowing that EROI and price can both increase at the sametime when more than one coefficient varies Such acknowl-edgment dissuades from predicting long run prices by simply600

looking at estimations of future EROIsDoes this mean that EROI is unrelated to any economic

concept Fizaine amp Court (2016) argue that there is a mini-mum EROI below which the US economy enters a recessionThey first show that energy expenditure Granger causes growth605

in the US then determine a threshold of energy expenditureabove which the US enters in a recession (consistent with thatof Bashmakov 2007) and finally use a modified version ofequation (11) to relate this to a minimum non-recessionaryEROI However they misleadingly replace the inverse of the610

energy intensity of energy investment $i nvestment

Einby that of the

whole economy GDPEout

This prevents them from noticing thatcost reductions in energy production could compensate theeffect of a decreasing EROI on energy prices and expendi-tures As we have seen EROI price and energy expenditure615

may all decrease at the same time which undermines the ideathat a recession caused by a surge in energy expenditure isineluctable as soon as EROI goes below some threshold Inaddition an energy price increase should have an expansion-ary effect on net exporters of energy at odds with the mech-620

anism extrapolated by Fizaine and Court from the case of theUnited States which has been historically a net energy im-porter Overall the analysis of this section indicates that theeconomic consequences of a change in EROI are ambiguousand that this physical notion cannot be used to predict future625

prices or GDP without empirical evidence

5 Concluding Remarks

This work includes a first attempt at estimating future EROIsin a decarbonized electricity system By examining a broadrange of scenarios it concludes that the system-wide EROI630

of the power sector should decrease until 2050 from 122 to107 in a business-as-usual scenario 80 in a partial transitionaway from fossil fuels or 58 in a scenario with 100 renew-able electricity Even though the EROI of each technology isexpected to remain well above 1 which was questioned theo-635

retically our results cast doubts on the energetic efficiency ofrenewable electricity

As an inverse relationship between EROIs and energy pricesis consistently found empirically a declining EROI could meanhigher energy prices However theoretical analysis of this re-640

lation showed that a declining EROI might also coincide withdecreasing energy prices and does not necessarily lead to arecession

Finally this paper assessed scenarios of transition in theelectricity sector but further research is still needed to esti- 645

mate future EROIs in complete energy transitions which in-clude a mutation of the transportation system agriculture andindustry Unfortunately this could not been done using thecurrent version of THEMIS and the question remains openfor future research 650

10

References

A Arvesen amp E G Hertwich More caution is needed when using life cy-cle assessment to determine energy return on investment (EROI) Energy

Policy 2015A Arvesen G Luderer M Pehl B L Bodirsky amp E G Hertwich Deriving655

life cycle assessment coefficients for application in integrated assessmentmodelling Environmental Modelling amp Software 2018

I Bashmakov Three laws of energy transitions Energy Policy 2007P Berrill A Arvesen Y Scholz H C Gils amp E G Hertwich Environmental

impacts of high penetration renewable energy scenarios for Europe En-660

vironmental Research Letters 2016L I Brand-Correa P E Brockway C L Copeland T J Foxon A Owen amp

P G Taylor Developing an Input-Output Based Method to Estimate aNational-Level Energy Return on Investment (EROI) Energies 2017

A R Brandt How Does Energy Resource Depletion Affect Prosperity Math-665

ematics of a Minimum Energy Return on Investment (EROI) BioPhysical

Economics and Resource Quality 2017A R Brandt amp M Dale A General Mathematical Framework for Calculat-

ing Systems-Scale Efficiency of Energy Extraction and Conversion EnergyReturn on Investment (EROI) and Other Energy Return Ratios Energies670

2011C J Cleveland Net energy from the extraction of oil and gas in the United

States Energy 2005V Court amp F Fizaine Long-Term Estimates of the Energy-Return-on-

Investment (EROI) of Coal Oil and Gas Global Productions Ecological675

Economics 2017M Dale S Krumdieck amp P Bodger Net energy yield from production of

conventional oil Energy Policy 2011M A J Dale Global Energy Modelling A Biophysical Approach (GEMBA)

2010680

Eurostat editor Eurostat Manual of supply use and input-output tables Amtfuumlr amtliche Veroumlffentlichungen der Europaumlischen Gemeinschaften Lux-embourg 2008 edition edition 2008 ISBN 978-92-79-04735-0

F Fizaine amp V Court Energy expenditure economic growth and the mini-mum EROI of society Energy Policy 2016685

R Frischknecht F Wyss S B Knoumlpfel T Luumltzkendorf amp M Balouktsi Cu-mulative energy demand in LCA the energy harvested approach The In-

ternational Journal of Life Cycle Assessment 2015T Gibon R Wood A Arvesen J D Bergesen S Suh amp E G Hertwich A

Methodology for Integrated Multiregional Life Cycle Assessment Scenar-690

ios under Large-Scale Technological Change Environmental Science amp

Technology 2015C A S Hall Introduction to Special Issue on New Studies in EROI (Energy

Return on Investment) Sustainability 2011C A S Hall S Balogh amp D J R Murphy What is the Minimum EROI that a695

Sustainable Society Must Have Energies 2009C A S Hall J G Lambert amp S B Balogh EROI of different fuels and the

implications for society Energy Policy 2014R A Herendeen Connecting net energy with the price of energy and other

goods and services Ecological Economics 2015700

E G Hertwich T Gibon E A Bouman A Arvesen S Suh G A Heath J DBergesen A Ramirez M I Vega amp L Shi Integrated life-cycle assess-ment of electricity-supply scenarios confirms global environmental ben-efit of low-carbon technologies Proceedings of the National Academy of

Sciences 2015705

M K Heun amp M de Wit Energy return on (energy) invested (EROI) oil pricesand energy transitions Energy Policy 2012

IEA Energy Technology Perspectives 2010T Junius amp J Oosterhaven The Solution of Updating or Regionalizing a Ma-

trix with both Positive and Negative Entries Economic Systems Research710

2003C W King Matrix method for comparing system and individual energy re-

turn ratios when considering an energy transition Energy 2014C W King amp C A S Hall Relating Financial and Energy Return on Invest-

ment Sustainability 2011715

O Koskinen amp C Breyer Energy Storage in Global and Transcontinental En-ergy Scenarios A Critical Review Energy Procedia 2016

J G Lambert amp G P Lambert Predicting the Psychological Response of theAmerican People to Oil Depletion and Declining Energy Return on Invest-ment (EROI) Sustainability 2011720

J G Lambert C A Hall S Balogh A Gupta amp M Arnold Energy EROI andquality of life Energy Policy 2014

W Leontief Input-Output Economics Oxford University Press USA 2 edi-tion 1986 ISBN 978-0-19-503527-8

R E Miller amp P D Blair Input-Output Analysis Foundations and Extensions 725

Cambridge University Press 2009 ISBN 978-0-521-51713-3E Muumlller L M Hilty R Widmer M Schluep amp M Faulstich Modeling Metal

Stocks and Flows A Review of Dynamic Material Flow Analysis MethodsEnvironmental Science amp Technology 2014

D J Murphy C A S Hall M Dale amp C Cleveland Order from Chaos A 730

Preliminary Protocol for Determining the EROI of Fuels Sustainability2011

NEEDS LCA of Background Processes Technical report New Energy Exter-nalities Developments for Sustainability 2009

M Pehl A Arvesen F Humpenoumlder A Popp E G Hertwich amp G Luderer 735

Understanding future emissions from low-carbon power systems by inte-gration of life-cycle assessment and integrated energy modelling Nature

Energy 2017A Poisson C Hall A Poisson amp C A S Hall Time Series EROI for Canadian

Oil and Gas Energies 2013 740

M Raugei Net energy analysis must not compare apples and oranges Na-

ture Energy 2019Y Scholz H C Gils amp R C Pietzcker Application of a high-detail energy sys-

tem model to derive power sector characteristics at high wind and solarshares Energy Economics 2017 745

S Suh Functions commodities and environmental impacts in an ecologi-calndasheconomic model Ecological Economics 2004

S Teske T Pregger S Simon amp T Naegler Energy [R]evolution Technicalreport Greenpeace 2015

G Tverberg The ldquoWind and Solar Will Save Usrdquo Delusion 2017 750

D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein En-ergy intensities EROIs (energy returned on invested) and energy paybacktimes of electricity generating power plants Energy 2013

R Wood K Stadler T Bulavskaya S Lutter S Giljum A de Koning J Kue-nen H Schuumltz J Acosta-Fernaacutendez A Usubiaga M Simas O Ivanova 755

J Weinzettel J H Schmidt S Merciai amp A Tukker Global SustainabilityAccountingmdashDeveloping EXIOBASE for Multi-Regional Footprint Analy-sis Sustainability 2014

11

Appendix A Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are read-760

ily available in THEMIS but these matrices have to be up-dated to the new electricity mix for the Greenpeace scenariosTo do this I exploit the fact that both THEMIS and Green-peace use the world regions of the IEA and I modify the elec-tricity input of each sector by the regional mix given by Green-765

peace The most accurate algorithm to update an input-outputmatrix is known as GRAS (Junius amp Oosterhaven 2003) Al-though I implemented this algorithm in pymrio I could notuse it because this algorithm uses the new sums of rows andcolumns to balance the matrix and the vector of final de-770

mand y or the vector of production x is necessary to knowthem As THEMIS does not include such vectors I had to usea simpler method which relies on the assumption that theelectricity mix of inputs is the same across sectors for a givenregion Given the perfect substitutability between electric-775

ity produced by different technologies and the uniqueness ofelectric grids this assumption seems justified

There are two different updates to make First I modifythe vector of second energy demand (used to infer the finaldemand of technology t yt ) so that it perfectly matches the780

demand of the scenario Second I modify the submatrix D ofA containing the rows of electricity sectors To convert D inenergy units I multiply each row t of D by the correspondingenergy supplied per unit of technology t E S

t I call the resultE the coefficient Ei s of E gives the electricity from sector i re-785

quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premultiplyE by a block diagonal matrix with R blocks of size T lowastT con-taining only ones (where R = 9 and T = 15 are the number ofTHEMIS regions and electricity sectors respectively) to ob-790

tain a matrix B Each row of B gives the total electricity froma given region r required to produced each output E tot

r andeach row E tot

r is replicated T times

B =

B1

BR

Br =

E totr

E totr

Next each row of B is multiplied by the share of a tech-nology t in the mix of the corresponding region which de-795

fines a matrix E Each coefficient Ei s of E gives the electricityfrom sector i required to make one unit of sector srsquo outputaccording to the new mix (by construction for all electricitysector j = i (t r ) the share of technology j in the regional mix

E j ssum

t Ei (t r )s is the same across all sectors s) Eventually I obtain800

the new submatrix D by converting each row of E to the origi-nal units of A (by dividing each row by the appropriate unitaryenergy supplied E S

t )A last update is needed for Greenpeace scenarios to ac-

count for the extra capacity needed when intermittent sources805

fail to deliver energy the ratio of capacity (in GW) over pro-duction (in TWh) is somewhat higher in Greenpeace scenar-ios than in IEATHEMIS ones Thus I multiply each columnof an energy sector (representing all inputs required for one

unit of output of this sector) by the ratio of the capacity-over- 810

production ratios of Greenpeace and IEATHEMIS Doing sorelies on the fact that the energy required to operate a powerplant is negligible in front of the energy required to build it(see eg Arvesen et al (2018))

Appendix B Example of Non-Decreasing Relation Between 815

EROI and Price

Herendeen (2015) proposes a calibration on US energydata of his toy model with 2 sectors (materials and energy)which yields as realistic results as a two-by-two model canyield I start from a slightly modified version of his calibration 820

(called base) in the sense that the figures are rounded and Ishow how a deviation of two coefficients (in the new calibra-tion) leads to an increase of both EROI and price of energyThis proves that in general nothing can be said of the rela-tion between EROI and price not even that it is a decreasing 825

relationFor this I use the formulas for EROI and price given by

Herendeen (2015) (where I convert the price to $gal usingthe conversion factor 1Btu = 114000gal)

EROI =1

Aee +Aem Ame

1minusAmm

(B1)

p =ve (1minus Amm )+ vm Ame

(1minus Aee ) (1minus Amm)minus Aem Amemiddot114000

Table B5 Example of sets of coefficients exhibiting a non-decreasingrelation between EROI and price in a two sectors model (seedesmoscomcalculatorne4oqunhsm)

base new

vm 05ve 5 middot10minus6

Aem 1700Ame 4 middot10minus6

Amm 05 09Aee 03 01

EROI 32 60price 15 34

Appendix C Complementary Results 830

Results without quality adjustment for IEATHEMIS sce-narios are provided in section 33 those for Greenpeacersquos sce-narios are in Table C6 Quality-adjusted results follows in Ta-ble C7 (IEATHEMIS) and C8 (Greenpeace) The quality ad-justment consists in separating each energy in the formula of 835

the EROI according to its origin (electric or thermal) and toweight electricity by a factor 26 For example the quality-adjusted (gross) embodied energy for a unit of technology t

writes

embodiedqual adjt = E S

⊙ (26 middot1electric +1thermal) middot (I minus A)minus1middot1t

(C1)

12

References

A Arvesen amp E G Hertwich More caution is needed when using life cy-cle assessment to determine energy return on investment (EROI) Energy

Policy 2015A Arvesen G Luderer M Pehl B L Bodirsky amp E G Hertwich Deriving655

life cycle assessment coefficients for application in integrated assessmentmodelling Environmental Modelling amp Software 2018

I Bashmakov Three laws of energy transitions Energy Policy 2007P Berrill A Arvesen Y Scholz H C Gils amp E G Hertwich Environmental

impacts of high penetration renewable energy scenarios for Europe En-660

vironmental Research Letters 2016L I Brand-Correa P E Brockway C L Copeland T J Foxon A Owen amp

P G Taylor Developing an Input-Output Based Method to Estimate aNational-Level Energy Return on Investment (EROI) Energies 2017

A R Brandt How Does Energy Resource Depletion Affect Prosperity Math-665

ematics of a Minimum Energy Return on Investment (EROI) BioPhysical

Economics and Resource Quality 2017A R Brandt amp M Dale A General Mathematical Framework for Calculat-

ing Systems-Scale Efficiency of Energy Extraction and Conversion EnergyReturn on Investment (EROI) and Other Energy Return Ratios Energies670

2011C J Cleveland Net energy from the extraction of oil and gas in the United

States Energy 2005V Court amp F Fizaine Long-Term Estimates of the Energy-Return-on-

Investment (EROI) of Coal Oil and Gas Global Productions Ecological675

Economics 2017M Dale S Krumdieck amp P Bodger Net energy yield from production of

conventional oil Energy Policy 2011M A J Dale Global Energy Modelling A Biophysical Approach (GEMBA)

2010680

Eurostat editor Eurostat Manual of supply use and input-output tables Amtfuumlr amtliche Veroumlffentlichungen der Europaumlischen Gemeinschaften Lux-embourg 2008 edition edition 2008 ISBN 978-92-79-04735-0

F Fizaine amp V Court Energy expenditure economic growth and the mini-mum EROI of society Energy Policy 2016685

R Frischknecht F Wyss S B Knoumlpfel T Luumltzkendorf amp M Balouktsi Cu-mulative energy demand in LCA the energy harvested approach The In-

ternational Journal of Life Cycle Assessment 2015T Gibon R Wood A Arvesen J D Bergesen S Suh amp E G Hertwich A

Methodology for Integrated Multiregional Life Cycle Assessment Scenar-690

ios under Large-Scale Technological Change Environmental Science amp

Technology 2015C A S Hall Introduction to Special Issue on New Studies in EROI (Energy

Return on Investment) Sustainability 2011C A S Hall S Balogh amp D J R Murphy What is the Minimum EROI that a695

Sustainable Society Must Have Energies 2009C A S Hall J G Lambert amp S B Balogh EROI of different fuels and the

implications for society Energy Policy 2014R A Herendeen Connecting net energy with the price of energy and other

goods and services Ecological Economics 2015700

E G Hertwich T Gibon E A Bouman A Arvesen S Suh G A Heath J DBergesen A Ramirez M I Vega amp L Shi Integrated life-cycle assess-ment of electricity-supply scenarios confirms global environmental ben-efit of low-carbon technologies Proceedings of the National Academy of

Sciences 2015705

M K Heun amp M de Wit Energy return on (energy) invested (EROI) oil pricesand energy transitions Energy Policy 2012

IEA Energy Technology Perspectives 2010T Junius amp J Oosterhaven The Solution of Updating or Regionalizing a Ma-

trix with both Positive and Negative Entries Economic Systems Research710

2003C W King Matrix method for comparing system and individual energy re-

turn ratios when considering an energy transition Energy 2014C W King amp C A S Hall Relating Financial and Energy Return on Invest-

ment Sustainability 2011715

O Koskinen amp C Breyer Energy Storage in Global and Transcontinental En-ergy Scenarios A Critical Review Energy Procedia 2016

J G Lambert amp G P Lambert Predicting the Psychological Response of theAmerican People to Oil Depletion and Declining Energy Return on Invest-ment (EROI) Sustainability 2011720

J G Lambert C A Hall S Balogh A Gupta amp M Arnold Energy EROI andquality of life Energy Policy 2014

W Leontief Input-Output Economics Oxford University Press USA 2 edi-tion 1986 ISBN 978-0-19-503527-8

R E Miller amp P D Blair Input-Output Analysis Foundations and Extensions 725

Cambridge University Press 2009 ISBN 978-0-521-51713-3E Muumlller L M Hilty R Widmer M Schluep amp M Faulstich Modeling Metal

Stocks and Flows A Review of Dynamic Material Flow Analysis MethodsEnvironmental Science amp Technology 2014

D J Murphy C A S Hall M Dale amp C Cleveland Order from Chaos A 730

Preliminary Protocol for Determining the EROI of Fuels Sustainability2011

NEEDS LCA of Background Processes Technical report New Energy Exter-nalities Developments for Sustainability 2009

M Pehl A Arvesen F Humpenoumlder A Popp E G Hertwich amp G Luderer 735

Understanding future emissions from low-carbon power systems by inte-gration of life-cycle assessment and integrated energy modelling Nature

Energy 2017A Poisson C Hall A Poisson amp C A S Hall Time Series EROI for Canadian

Oil and Gas Energies 2013 740

M Raugei Net energy analysis must not compare apples and oranges Na-

ture Energy 2019Y Scholz H C Gils amp R C Pietzcker Application of a high-detail energy sys-

tem model to derive power sector characteristics at high wind and solarshares Energy Economics 2017 745

S Suh Functions commodities and environmental impacts in an ecologi-calndasheconomic model Ecological Economics 2004

S Teske T Pregger S Simon amp T Naegler Energy [R]evolution Technicalreport Greenpeace 2015

G Tverberg The ldquoWind and Solar Will Save Usrdquo Delusion 2017 750

D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein En-ergy intensities EROIs (energy returned on invested) and energy paybacktimes of electricity generating power plants Energy 2013

R Wood K Stadler T Bulavskaya S Lutter S Giljum A de Koning J Kue-nen H Schuumltz J Acosta-Fernaacutendez A Usubiaga M Simas O Ivanova 755

J Weinzettel J H Schmidt S Merciai amp A Tukker Global SustainabilityAccountingmdashDeveloping EXIOBASE for Multi-Regional Footprint Analy-sis Sustainability 2014

11

Appendix A Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are read-760

ily available in THEMIS but these matrices have to be up-dated to the new electricity mix for the Greenpeace scenariosTo do this I exploit the fact that both THEMIS and Green-peace use the world regions of the IEA and I modify the elec-tricity input of each sector by the regional mix given by Green-765

peace The most accurate algorithm to update an input-outputmatrix is known as GRAS (Junius amp Oosterhaven 2003) Al-though I implemented this algorithm in pymrio I could notuse it because this algorithm uses the new sums of rows andcolumns to balance the matrix and the vector of final de-770

mand y or the vector of production x is necessary to knowthem As THEMIS does not include such vectors I had to usea simpler method which relies on the assumption that theelectricity mix of inputs is the same across sectors for a givenregion Given the perfect substitutability between electric-775

ity produced by different technologies and the uniqueness ofelectric grids this assumption seems justified

There are two different updates to make First I modifythe vector of second energy demand (used to infer the finaldemand of technology t yt ) so that it perfectly matches the780

demand of the scenario Second I modify the submatrix D ofA containing the rows of electricity sectors To convert D inenergy units I multiply each row t of D by the correspondingenergy supplied per unit of technology t E S

t I call the resultE the coefficient Ei s of E gives the electricity from sector i re-785

quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premultiplyE by a block diagonal matrix with R blocks of size T lowastT con-taining only ones (where R = 9 and T = 15 are the number ofTHEMIS regions and electricity sectors respectively) to ob-790

tain a matrix B Each row of B gives the total electricity froma given region r required to produced each output E tot

r andeach row E tot

r is replicated T times

B =

B1

BR

Br =

E totr

E totr

Next each row of B is multiplied by the share of a tech-nology t in the mix of the corresponding region which de-795

fines a matrix E Each coefficient Ei s of E gives the electricityfrom sector i required to make one unit of sector srsquo outputaccording to the new mix (by construction for all electricitysector j = i (t r ) the share of technology j in the regional mix

E j ssum

t Ei (t r )s is the same across all sectors s) Eventually I obtain800

the new submatrix D by converting each row of E to the origi-nal units of A (by dividing each row by the appropriate unitaryenergy supplied E S

t )A last update is needed for Greenpeace scenarios to ac-

count for the extra capacity needed when intermittent sources805

fail to deliver energy the ratio of capacity (in GW) over pro-duction (in TWh) is somewhat higher in Greenpeace scenar-ios than in IEATHEMIS ones Thus I multiply each columnof an energy sector (representing all inputs required for one

unit of output of this sector) by the ratio of the capacity-over- 810

production ratios of Greenpeace and IEATHEMIS Doing sorelies on the fact that the energy required to operate a powerplant is negligible in front of the energy required to build it(see eg Arvesen et al (2018))

Appendix B Example of Non-Decreasing Relation Between 815

EROI and Price

Herendeen (2015) proposes a calibration on US energydata of his toy model with 2 sectors (materials and energy)which yields as realistic results as a two-by-two model canyield I start from a slightly modified version of his calibration 820

(called base) in the sense that the figures are rounded and Ishow how a deviation of two coefficients (in the new calibra-tion) leads to an increase of both EROI and price of energyThis proves that in general nothing can be said of the rela-tion between EROI and price not even that it is a decreasing 825

relationFor this I use the formulas for EROI and price given by

Herendeen (2015) (where I convert the price to $gal usingthe conversion factor 1Btu = 114000gal)

EROI =1

Aee +Aem Ame

1minusAmm

(B1)

p =ve (1minus Amm )+ vm Ame

(1minus Aee ) (1minus Amm)minus Aem Amemiddot114000

Table B5 Example of sets of coefficients exhibiting a non-decreasingrelation between EROI and price in a two sectors model (seedesmoscomcalculatorne4oqunhsm)

base new

vm 05ve 5 middot10minus6

Aem 1700Ame 4 middot10minus6

Amm 05 09Aee 03 01

EROI 32 60price 15 34

Appendix C Complementary Results 830

Results without quality adjustment for IEATHEMIS sce-narios are provided in section 33 those for Greenpeacersquos sce-narios are in Table C6 Quality-adjusted results follows in Ta-ble C7 (IEATHEMIS) and C8 (Greenpeace) The quality ad-justment consists in separating each energy in the formula of 835

the EROI according to its origin (electric or thermal) and toweight electricity by a factor 26 For example the quality-adjusted (gross) embodied energy for a unit of technology t

writes

embodiedqual adjt = E S

⊙ (26 middot1electric +1thermal) middot (I minus A)minus1middot1t

(C1)

12

Appendix A Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are read-760

ily available in THEMIS but these matrices have to be up-dated to the new electricity mix for the Greenpeace scenariosTo do this I exploit the fact that both THEMIS and Green-peace use the world regions of the IEA and I modify the elec-tricity input of each sector by the regional mix given by Green-765

peace The most accurate algorithm to update an input-outputmatrix is known as GRAS (Junius amp Oosterhaven 2003) Al-though I implemented this algorithm in pymrio I could notuse it because this algorithm uses the new sums of rows andcolumns to balance the matrix and the vector of final de-770

mand y or the vector of production x is necessary to knowthem As THEMIS does not include such vectors I had to usea simpler method which relies on the assumption that theelectricity mix of inputs is the same across sectors for a givenregion Given the perfect substitutability between electric-775

ity produced by different technologies and the uniqueness ofelectric grids this assumption seems justified

There are two different updates to make First I modifythe vector of second energy demand (used to infer the finaldemand of technology t yt ) so that it perfectly matches the780

demand of the scenario Second I modify the submatrix D ofA containing the rows of electricity sectors To convert D inenergy units I multiply each row t of D by the correspondingenergy supplied per unit of technology t E S

t I call the resultE the coefficient Ei s of E gives the electricity from sector i re-785

quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premultiplyE by a block diagonal matrix with R blocks of size T lowastT con-taining only ones (where R = 9 and T = 15 are the number ofTHEMIS regions and electricity sectors respectively) to ob-790

tain a matrix B Each row of B gives the total electricity froma given region r required to produced each output E tot

r andeach row E tot

r is replicated T times

B =

B1

BR

Br =

E totr

E totr

Next each row of B is multiplied by the share of a tech-nology t in the mix of the corresponding region which de-795

fines a matrix E Each coefficient Ei s of E gives the electricityfrom sector i required to make one unit of sector srsquo outputaccording to the new mix (by construction for all electricitysector j = i (t r ) the share of technology j in the regional mix

E j ssum

t Ei (t r )s is the same across all sectors s) Eventually I obtain800

the new submatrix D by converting each row of E to the origi-nal units of A (by dividing each row by the appropriate unitaryenergy supplied E S

t )A last update is needed for Greenpeace scenarios to ac-

count for the extra capacity needed when intermittent sources805

fail to deliver energy the ratio of capacity (in GW) over pro-duction (in TWh) is somewhat higher in Greenpeace scenar-ios than in IEATHEMIS ones Thus I multiply each columnof an energy sector (representing all inputs required for one

unit of output of this sector) by the ratio of the capacity-over- 810

production ratios of Greenpeace and IEATHEMIS Doing sorelies on the fact that the energy required to operate a powerplant is negligible in front of the energy required to build it(see eg Arvesen et al (2018))

Appendix B Example of Non-Decreasing Relation Between 815

EROI and Price

Herendeen (2015) proposes a calibration on US energydata of his toy model with 2 sectors (materials and energy)which yields as realistic results as a two-by-two model canyield I start from a slightly modified version of his calibration 820

(called base) in the sense that the figures are rounded and Ishow how a deviation of two coefficients (in the new calibra-tion) leads to an increase of both EROI and price of energyThis proves that in general nothing can be said of the rela-tion between EROI and price not even that it is a decreasing 825

relationFor this I use the formulas for EROI and price given by

Herendeen (2015) (where I convert the price to $gal usingthe conversion factor 1Btu = 114000gal)

EROI =1

Aee +Aem Ame

1minusAmm

(B1)

p =ve (1minus Amm )+ vm Ame

(1minus Aee ) (1minus Amm)minus Aem Amemiddot114000

Table B5 Example of sets of coefficients exhibiting a non-decreasingrelation between EROI and price in a two sectors model (seedesmoscomcalculatorne4oqunhsm)

base new

vm 05ve 5 middot10minus6

Aem 1700Ame 4 middot10minus6

Amm 05 09Aee 03 01

EROI 32 60price 15 34

Appendix C Complementary Results 830

Results without quality adjustment for IEATHEMIS sce-narios are provided in section 33 those for Greenpeacersquos sce-narios are in Table C6 Quality-adjusted results follows in Ta-ble C7 (IEATHEMIS) and C8 (Greenpeace) The quality ad-justment consists in separating each energy in the formula of 835

the EROI according to its origin (electric or thermal) and toweight electricity by a factor 26 For example the quality-adjusted (gross) embodied energy for a unit of technology t

writes

embodiedqual adjt = E S

⊙ (26 middot1electric +1thermal) middot (I minus A)minus1middot1t

(C1)

12

Table C6 EROIs and share in electricity mix of electric technologies in the model THEMIS for the Greenpeace scenariosThe bottom line in columns mix gives the total secondary energy demand in PWha

Scenario all REF ER (+2degC no CCS no nuclear) ADV (100 renewable)Year 2012 2030 2050 2030 2050 2030 2050

Variable EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix

biomass w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000biomassampWaste 85 002 64 003 50 003 53 006 47 006 52 005 46 005

ocean 47 000 20 000 25 000 43 001 45 003 48 001 49 003geothermal 56 000 38 001 25 001 36 003 37 007 38 003 39 007

solar CSP 355 000 93 000 80 001 85 005 77 017 93 007 78 022solar PV 137 000 70 002 53 002 56 011 44 020 54 014 47 021

wind offshore 91 000 86 001 78 001 56 003 59 008 65 004 64 010wind onshore 97 002 91 005 72 005 72 015 60 022 72 017 58 024

hydro 122 016 114 014 112 013 110 014 111 010 110 013 109 008nuclear 122 011 73 010 71 008 83 002 ndash 000 83 002 ndash 000

gas w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000coal w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000

oil 84 005 111 002 114 001 100 001 92 000 100 001 ndash 000gas 149 023 153 023 156 025 166 021 172 006 165 018 ndash 000coal 118 040 113 040 113 039 107 019 108 001 104 016 115 000Total 121 2260 107 3626 101 5011 84 3360 59 4920 81 3674 58 6404

Table C7 Quality-adjusted EROIs (with a factor of 26 for electricity) and share in electricity mix of electric technologies for IEATHEMIS scenariosThe bottom line in columns mix gives the total secondary energy demand in PWha

Scenario Baseline (BL) Blue Map (BM +2deg)

Year 2010 2030 2050 2030 2050

Variable EROI mix EROI mix EROI mix EROI mix EROI mix

biomass w CCS ndash 000 ndash 000 ndash 000 95 000 85 001biomassampWaste 208 001 126 002 117 003 116 006 111 005

ocean 91 000 44 000 55 000 70 000 113 000geothermal 115 000 103 001 102 001 106 001 114 002

solar CSP 446 000 177 000 182 001 173 002 169 006solar PV 175 000 148 001 145 001 133 002 129 006

wind offshore 196 000 212 001 205 001 154 003 130 004wind onshore 184 001 181 004 158 004 145 008 154 008

hydro 256 016 230 014 230 012 252 018 263 014nuclear 192 014 134 011 130 010 136 019 139 024

gas w CCS ndash 000 ndash 000 144 000 162 001 188 005coal w CCS ndash 000 ndash 000 117 000 137 005 143 012

oil 129 006 156 002 160 001 155 003 118 001gas 225 021 246 021 244 023 290 014 335 011coal 202 042 182 045 182 045 184 018 196 001Total 204 1976 187 3429 184 4597 170 2801 160 4022

13

Table C8 Quality-adjusted EROIs (with a factor of 26 for electricity) and share in electricity mix of electric technologies for Greenpeace scenariosThe bottom line in columns mix gives the total secondary energy demand in PWha

Scenario all REF ER (+2degC no CCS no nuclear) ADV (100 renewable)Year 2012 2030 2050 2030 2050 2030 2050

Variable EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix

biomass w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000biomassampWaste 157 002 126 003 98 003 104 006 92 006 103 005 91 005

ocean 78 000 36 000 46 000 84 001 91 003 93 001 97 003geothermal 121 000 76 001 50 001 72 003 73 007 76 003 77 007

solar CSP 732 000 191 000 161 001 176 005 161 017 190 007 162 022solar PV 258 000 137 002 105 002 115 011 92 020 113 014 99 021

wind offshore 173 000 160 001 151 001 116 003 122 008 133 004 132 010wind onshore 186 002 176 005 141 005 149 015 124 022 148 017 121 024

hydro 235 016 221 014 219 013 214 014 213 010 214 013 211 008nuclear 228 011 136 010 133 008 164 002 ndash 000 164 002 ndash 000

gas w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000coal w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000

oil 128 005 178 002 182 001 159 001 133 000 158 001 ndash 000gas 238 023 250 023 258 025 271 021 282 006 272 018 ndash 000coal 186 040 178 040 180 039 166 019 166 001 162 016 153 000

Total (PWha) 201 2260 185 3626 177 5011 159 3360 120 4920 155 3674 119 6404

Figure C6 Evolution of global quality-adjusted EROIs (with a factor 26 for electricity) and mixes of electricity for different scenarios

14

Appendix D Proof of Proposition 1840

The demonstration starts with a lemma

Lemma 1 Let A be an invertible matrix and let x be a coeffi-

cient of A Then

(i) the determinant of A is a linear function of x denoted

D A 845

(ii) each coefficient (ij) of the adjugate of A is a linear func-

tion of x denoted P Ai j

(iii) each coefficient (ij) of Aminus1 is a rational function in x

of degree 1 which writes(

Aminus1)

i j =P A

i j (x)

D A (x)

Proof Let A =(

ai j

)

1lei jlenisinGLn (R) an invertible matrix and850

let(

i0 j0)

isin J1nK2 so that without loss of generality x = ai0 j0 (i) From its definition by the Leibniz formula the determi-nant of A writes det (A) =

sum

σisinSnsgn(σ)

prodni=1 ai σ(i) where sgn(σ)

is the signature of permutation σ and Sn the set of all per-mutations of n elements In this linear combination each855

term is a product containing x at most once it is thus a lin-ear function of x (ii) A minor being the determinant of a sub-matrix of A we know from (i) that it is a linear function of x

(which reduces to a constant for submatrices that do not con-tain x) Each coefficient of the adjugate of A is (plus or minus)860

a minor of A hence a linear function of x (iii) Using (i) and

(ii) and the Laplace expansion of A Aminus1=

adj(A)det(A) we reckon

(

Aminus1)

i j =P A

i j(x)

D A (x)

Proof (Proposition 1) Defining R (x) = D IminusA(

δi0 j0 minus x)

lemma

1 yields that for all (e t)isin J1nK2 there is a unique linear func-865

tion P IminusAet such that

(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

R(x) where δi j isthe Kronecker delta As a linear combination of compositionsof linear functions the functionsQ (x) =

sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)

are themselves linear By definition we have870

εt =sum

eisinE

(

(I minus A)minus1)

et so that Q (x) = εt R (x) As P Q and R

are linear and as εt varies with x it is easy to show that thereare unique real numbers α and γ such that P (x) = αQ (x)+γR (x) Finally observing that pt =

sumni=1 vi

(

(I minus A)minus1)

i t =P (x)R(x)

we have875

pt =αQ(x)+γR(x)

R(x) =αεt +γ

15

• Introduction
• The EROI of a Technology Is Not Intrinsic
• • A Simple Model With A Unique Energy Technology
  • A Simple Model With A Mix of Two Energy Technologies
  • • Estimation of Current and Future EROIs Using THEMIS
    • • Definitions and Setting
      • Data Sources and Method
      • Main Results
      • • Implications of a Decreasing EROI on Prices and GDP
        • • Inverse Relation Proposed in First Studies
          • Empirical Relation Between EROI and Price
          • A Case Against Any Simple Relation
          • • Concluding Remarks
            • Updating a Matrix A To a New Given Mix
            • Example of Non-Decreasing Relation Between EROI and Price
            • Complementary Results
            • Proof of Proposition 1

Table C8 Quality-adjusted EROIs (with a factor of 26 for electricity) and share in electricity mix of electric technologies for Greenpeace scenariosThe bottom line in columns mix gives the total secondary energy demand in PWha

Scenario all REF ER (+2degC no CCS no nuclear) ADV (100 renewable)Year 2012 2030 2050 2030 2050 2030 2050

Variable EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix EROI mix

biomass w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000biomassampWaste 157 002 126 003 98 003 104 006 92 006 103 005 91 005

ocean 78 000 36 000 46 000 84 001 91 003 93 001 97 003geothermal 121 000 76 001 50 001 72 003 73 007 76 003 77 007

solar CSP 732 000 191 000 161 001 176 005 161 017 190 007 162 022solar PV 258 000 137 002 105 002 115 011 92 020 113 014 99 021

wind offshore 173 000 160 001 151 001 116 003 122 008 133 004 132 010wind onshore 186 002 176 005 141 005 149 015 124 022 148 017 121 024

hydro 235 016 221 014 219 013 214 014 213 010 214 013 211 008nuclear 228 011 136 010 133 008 164 002 ndash 000 164 002 ndash 000

gas w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000coal w CCS ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000 ndash 000

oil 128 005 178 002 182 001 159 001 133 000 158 001 ndash 000gas 238 023 250 023 258 025 271 021 282 006 272 018 ndash 000coal 186 040 178 040 180 039 166 019 166 001 162 016 153 000

Total (PWha) 201 2260 185 3626 177 5011 159 3360 120 4920 155 3674 119 6404

Figure C6 Evolution of global quality-adjusted EROIs (with a factor 26 for electricity) and mixes of electricity for different scenarios

14

Appendix D Proof of Proposition 1840

The demonstration starts with a lemma

Lemma 1 Let A be an invertible matrix and let x be a coeffi-

cient of A Then

(i) the determinant of A is a linear function of x denoted

D A 845

(ii) each coefficient (ij) of the adjugate of A is a linear func-

tion of x denoted P Ai j

(iii) each coefficient (ij) of Aminus1 is a rational function in x

of degree 1 which writes(

Aminus1)

i j =P A

i j (x)

D A (x)

Proof Let A =(

ai j

)

1lei jlenisinGLn (R) an invertible matrix and850

let(

i0 j0)

isin J1nK2 so that without loss of generality x = ai0 j0 (i) From its definition by the Leibniz formula the determi-nant of A writes det (A) =

sum

σisinSnsgn(σ)

prodni=1 ai σ(i) where sgn(σ)

is the signature of permutation σ and Sn the set of all per-mutations of n elements In this linear combination each855

term is a product containing x at most once it is thus a lin-ear function of x (ii) A minor being the determinant of a sub-matrix of A we know from (i) that it is a linear function of x

(which reduces to a constant for submatrices that do not con-tain x) Each coefficient of the adjugate of A is (plus or minus)860

a minor of A hence a linear function of x (iii) Using (i) and

(ii) and the Laplace expansion of A Aminus1=

adj(A)det(A) we reckon

(

Aminus1)

i j =P A

i j(x)

D A (x)

Proof (Proposition 1) Defining R (x) = D IminusA(

δi0 j0 minus x)

lemma

1 yields that for all (e t)isin J1nK2 there is a unique linear func-865

tion P IminusAet such that

(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

R(x) where δi j isthe Kronecker delta As a linear combination of compositionsof linear functions the functionsQ (x) =

sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)

are themselves linear By definition we have870

εt =sum

eisinE

(

(I minus A)minus1)

et so that Q (x) = εt R (x) As P Q and R

are linear and as εt varies with x it is easy to show that thereare unique real numbers α and γ such that P (x) = αQ (x)+γR (x) Finally observing that pt =

sumni=1 vi

(

(I minus A)minus1)

i t =P (x)R(x)

we have875

pt =αQ(x)+γR(x)

R(x) =αεt +γ

15

• Introduction
• The EROI of a Technology Is Not Intrinsic
• • A Simple Model With A Unique Energy Technology
  • A Simple Model With A Mix of Two Energy Technologies
  • • Estimation of Current and Future EROIs Using THEMIS
    • • Definitions and Setting
      • Data Sources and Method
      • Main Results
      • • Implications of a Decreasing EROI on Prices and GDP
        • • Inverse Relation Proposed in First Studies
          • Empirical Relation Between EROI and Price
          • A Case Against Any Simple Relation
          • • Concluding Remarks
            • Updating a Matrix A To a New Given Mix
            • Example of Non-Decreasing Relation Between EROI and Price
            • Complementary Results
            • Proof of Proposition 1

Appendix D Proof of Proposition 1840

The demonstration starts with a lemma

Lemma 1 Let A be an invertible matrix and let x be a coeffi-

cient of A Then

(i) the determinant of A is a linear function of x denoted

D A 845

(ii) each coefficient (ij) of the adjugate of A is a linear func-

tion of x denoted P Ai j

(iii) each coefficient (ij) of Aminus1 is a rational function in x

of degree 1 which writes(

Aminus1)

i j =P A

i j (x)

D A (x)

Proof Let A =(

ai j

)

1lei jlenisinGLn (R) an invertible matrix and850

let(

i0 j0)

isin J1nK2 so that without loss of generality x = ai0 j0 (i) From its definition by the Leibniz formula the determi-nant of A writes det (A) =

sum

σisinSnsgn(σ)

prodni=1 ai σ(i) where sgn(σ)

is the signature of permutation σ and Sn the set of all per-mutations of n elements In this linear combination each855

term is a product containing x at most once it is thus a lin-ear function of x (ii) A minor being the determinant of a sub-matrix of A we know from (i) that it is a linear function of x

(which reduces to a constant for submatrices that do not con-tain x) Each coefficient of the adjugate of A is (plus or minus)860

a minor of A hence a linear function of x (iii) Using (i) and

(ii) and the Laplace expansion of A Aminus1=

adj(A)det(A) we reckon

(

Aminus1)

i j =P A

i j(x)

D A (x)

Proof (Proposition 1) Defining R (x) = D IminusA(

δi0 j0 minus x)

lemma

1 yields that for all (e t)isin J1nK2 there is a unique linear func-865

tion P IminusAet such that

(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

R(x) where δi j isthe Kronecker delta As a linear combination of compositionsof linear functions the functionsQ (x) =

sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)

are themselves linear By definition we have870

εt =sum

eisinE

(

(I minus A)minus1)

et so that Q (x) = εt R (x) As P Q and R

are linear and as εt varies with x it is easy to show that thereare unique real numbers α and γ such that P (x) = αQ (x)+γR (x) Finally observing that pt =

sumni=1 vi

(

(I minus A)minus1)

i t =P (x)R(x)

we have875

pt =αQ(x)+γR(x)

R(x) =αεt +γ

15

• Introduction
• The EROI of a Technology Is Not Intrinsic
• • A Simple Model With A Unique Energy Technology
  • A Simple Model With A Mix of Two Energy Technologies
  • • Estimation of Current and Future EROIs Using THEMIS
    • • Definitions and Setting
      • Data Sources and Method
      • Main Results
      • • Implications of a Decreasing EROI on Prices and GDP
        • • Inverse Relation Proposed in First Studies
          • Empirical Relation Between EROI and Price
          • A Case Against Any Simple Relation
          • • Concluding Remarks
            • Updating a Matrix A To a New Given Mix
            • Example of Non-Decreasing Relation Between EROI and Price
            • Complementary Results
            • Proof of Proposition 1