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 renewables arelower than those of fossil fuels Logically concerns have been expressed that system-wide EROI may decrease during a rene-wable energy transition First I explain theoretically that the EROIs of renewables themselves could then decrease as energy-efficient fossil fuels would be replaced by less energy-efficient renewables in the supply-chain Then using the multiregionalinput-output model THEMIS I estimate the evolution of EROIs and prices of electric technologies from 2010 to 2050 for diffe-rent 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 revie-wers 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 provi-ding 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 aprompt energy transition away from fossil fuels mdashnot to men-tion their depletion that shall ultimately make this transitionunavoidable concerns have been expressed that in a de-carbonized energy system the lower efficiency of renewableenergy might not allow to sustain advanced standards of li-ving (Lambert et al 2014 Tverberg 2017)2 We measure theenergy efficiency of a technology or energy system using theEnergy Returned On Invested (EROI) which is the ratio be-tween the energy it delivers throughout its lifetime and theenergy required to build operate and dismantle it A minimalrequirement for a technology or energy system to be energe-tically sustainable is to have an EROI above 1 meaning that itprovides more energy than it requires

One issue to assess future energy systems is that the fu-ture EROI of a given technology cannot be readily deducedfrom current estimates Indeed as King (2014) remarkedthe EROI of a technology is not intrinsic but depends on thewhole technological structure of the economy Indeed sup-pose that solar panels have a lower EROI than thermal po-wer plants so they require more energy to supply the sameamount of energy Then a plant producing solar panels willrequire more energy if the electricity it uses is produced bysolar panels rather than by thermal plants Ultimately solarpanels built using electricity from solar panels rather fossilswill require more energy and have a lower EROI Some have

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

called to compute the evolution of EROIs during a renewa-ble energy transition (Brandt 2017) and this study aims todo so while accounting for their system dependency Indeedprovided that EROIs of renewables are lower than EROIs offossils and that decreasing EROIs jeopardize prosperity theevolution of EROIs during the energy transition is of criticalimportance let us review these two hypotheses in turn

Many estimations of EROIs have been made and amongthe various different figures derived from diverse data setsand methodologies none stands out as singularly authorita-tive as shown by the controversy between Raugei (2013) andWeiszligbach et al (2014) Still Dale (2010) reviews all EROI es-timates until 2010 while Hall et al (2014) aggregate the esti-mates of the literature in a meta-analysis and King amp Bergh(2018) provide the likely ranges of electricity EROIs I chooseto present the results of Weiszligbach et al (2013) (see Figure1) because they compute the EROIs of different technolo-gies in a comparable manner In addition the buffered EROIsof Weiszligbach et al (2013) take into account the supplemen-tary capacity grid and storage required for the deploymentof renewable technologies which yields lower but presuma-bly more accurate estimates for their EROIs As anticipatedthe EROIs of renewable electricity sectors they find are signi-ficantly lower than those of electricity from fossil fuels exceptfor 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 pri-mary concern as they draw a link between the system-wide EROI and affluence of a society (Hall et al 2009Hall 2011 Lambert amp Lambert 2011 Lambert et al 2014Fizaine amp Court 2016) Here is how Hall (2011) summarizesthe argument

Think of a society dependent upon one resourceits 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 buildand maintain the truck and the roads and bridges

required 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 anEROI 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

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 lo-gic Hallrsquos argument relies on two questionable assumptionsthat factors of production (and especially the labor force) areused at their full capacity and that technical and organiza-tional progress will not be sufficient to sustain current levelof prosperity with significantly less labor (or other factors ofproduction in limited supply) If one rejects these assump-tions one can imagine a sustained level of prosperity with alower system-wide EROI provided that a higher share of fac-tors of production be devoted to the energy sector for ex-ample unemployed people could be mobilized to sustain theenergy surplus available to the rest of society In parallel to ashift in the labor force Raugei (2019) explains that an increa-sed efficiency of energy use may also counteract the decreasein energy services implied by a declining EROI That beingsaid given that current system-wide EROI is already decli-ning due to the decline in fossil fuels quality (Dale et al 2011Poisson et al 2013 Court amp Fizaine 2017) and that technicalprogress is incremental the aforementioned analyses shouldnot be neglected Under the current system of productionwhich will persist in the short term EROI should not decreasetoo much for prosperous standards of living to be sustained

In view of the potential implications of a declining EROIthis paper provides an assessment of the EROI of diffe-rent electricity technologies in various prospective scena-rios which includes a 100 renewable electricity systemTo this end I employ input-output analysis and I rely ona prospective series of multi-regional Input-Output Tables(IOT) THEMIS (Gibon et al 2015) which models two scena-rios from the International Energy Agency (IEA 2010) Ba-seline and Blue Map In addition I modify THEMISrsquo IOTsto embed two decarbonized scenarios of power generationGreenpeacersquos Energy [R]evolution (ER) and Advanced Energy[R]evolution (ADV) (Teske et al 2015) Although Pehl et al(2017) and Arvesen et al (2018) already computed energy re-quirements of electricity technologies for prospective scena-rios they focused on life-cycle assessment coefficients suchas future CO2 emissions and did not provide results in termsof EROI let alone system-wide EROI Furthermore they didnot study a scenario with 100 renewable electricity I intendto fill this gap

Then I analyze the economic implications of a decliningEROI through its relation with price Previous studies sug-gest an inverse relation between EROIs and energy prices

2

and such an average relation is retrieved empirically usingprices observed and predicted from THEMIS However theo-retical analysis tempers this finding Indeed while explainingto what extent EROI and price are related I show that theydo not necessarily move in opposite directions This callsfor taking prices predictions from input-output analysis withmore caution than EROI estimates because IOT is better sui-ted to handle physical notions than economic ones Finallythe economic analysis weakens the view that a decrease inEROI would necessarily lead to a surge in energy expenditu-res and hence to a contraction of GDP

Section 2 explains theoretically why the EROI of a techno-logy is not an intrinsic property section 3 presents the met-hodology and the results section 4 studies the implicationsof declining 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 outputj 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-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 Inves-ted is the ratio between the energy delivered by the systemand the energy required to build operate maintain and dis-mantle it In other words it is the inverse of the amount ofenergy required to produce one unit of energy when the se-ries of all embodied inputs are taken into account

The embodied inputs x required for a final demand y

can be calculated using the Leontief inverse matrix (Leontief1986 Eurostat 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 the

last input of our list 1E =

001

and the gross embodied energy

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

TE middot (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 technologybecause 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 deve-loping because (in net) it consumes energy rather than pro-viding it Such a system is not sustainable (and not realistic)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 oneunit 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 requi-ring no energy to access them) Such a world is physicallyimpossible

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-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

3

and such an average relation is retrieved empirically usingprices observed and predicted from THEMIS However theo-retical analysis tempers this finding Indeed while explainingto what extent EROI and price are related I show that theydo not necessarily move in opposite directions This callsfor taking prices predictions from input-output analysis withmore caution than EROI estimates because IOT is better sui-ted to handle physical notions than economic ones Finallythe economic analysis weakens the view that a decrease inEROI would necessarily lead to a surge in energy expenditu-res and hence to a contraction of GDP

Section 2 explains theoretically why the EROI of a techno-logy is not an intrinsic property section 3 presents the met-hodology and the results section 4 studies the implicationsof declining 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 outputj 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-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 Inves-ted is the ratio between the energy delivered by the systemand the energy required to build operate maintain and dis-mantle it In other words it is the inverse of the amount ofenergy required to produce one unit of energy when the se-ries of all embodied inputs are taken into account

The embodied inputs x required for a final demand y

can be calculated using the Leontief inverse matrix (Leontief1986 Eurostat 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 the

last input of our list 1E =

001

and the gross embodied energy

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

TE middot (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 technologybecause 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 deve-loping because (in net) it consumes energy rather than pro-viding it Such a system is not sustainable (and not realistic)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 oneunit 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 requi-ring no energy to access them) Such a world is physicallyimpossible

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-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

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 depends onthe whole economic system or more precisely of all techno-logies used in their chain of production4 Here the higher theshare 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 ofPV the EROI falls below unity In other words a renewableenergy mix with 100 PV is not sustainable in this exampleEven more worryingly if one computes the EROI of PV inan energy mix relying mostly on gas one would find a high-enough EROI for PV (meaning above 1) Hence one cannotconclude that a technology is sufficiently efficient (or sustai-nable) just by computing its EROI in the current energy mixYet EROIs computations have always been done from actualdata of our economy and could falsely represent the efficien-cies of energy technologies in another energy mix say a 100renewable one This uncertainty concerning the sustainabi-lity of a decarbonized energy system motivates the core ofthis paper the estimation of EROIs after a global energy tran-sition

3 Estimation of Current and Future EROIs Using THEMIS

31 Definitions and Setting

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 forthis research is defined by Brandt amp Dale (2011) as the GrossEnergy Ratio (GER) The GER measures the ratio of energydelivered over energy embodied in inputs net of the energyof the fuels transformed in the process Thus for examplethe denominator of the GER does not take into account theenergy provided by gas in a gas powered plant The termldquogrossrdquo is used because all energy output is taken into ac-count on the contrary Net Energy Ratios subtract from thenumerator all ldquoself-userdquo output that is used in the pathwayof production of the technology5 A related indicator that issometimes used to compute EROI (as it is already includedin many input-output databases) is the Cumulated EnergyDemand (CED) I do not use it because Arvesen amp Hertwich(2015) have shown that it is erroneous to use the CED directlyfor EROI computations without making adjustments

In most cases EROIs (or energy ratios) are defined usingquantities of primary energy However I adopt a different ap-proach in this paper and use only secondary energies in mycomputations Indeed as Arvesen amp Hertwich (2015) put itldquoEROI does not need to measure primary energy per se thecrucial point is to measure energy diverted from society ina unit of equivalencerdquo Also the choice of secondary energycarriers is consistent with an energy system relying on rene-wable electricity while for such systems the definition of pri-mary energy is not harmonized and this can lead to incon-sistencies Frischknecht et al (2015) spot for example a fac-tor 6 between the cumulative (primary) energy demand for

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

solar photovoltaic computed according to different methodsAlthough 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 valid

Furthermore practitioners often use a factor of conver-sion (around 3) to account for the higher quality of electri-city as compared to fossil fuels I follow the recommendationof Murphy et al (2011) by undertaking my computations wit-hout and with a quality-adjustment factor of 26 HoweverI prefer not to bring to the fore the quality-adjusted compu-tations provided in AppendixC 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 thisyield 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 disproportiona-tely used for these applications for which they have a higheryield therefore the difference in quality between fossils andelectricity 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 EROIs

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

t Let usrecall that y is the vector of final demand given by the scena-rio and A is the technology matrix (or input-output table) E S

is 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 se-

condary 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 elec-tricity somewhere in the supply-chain including at the laststage Indeed such energy is not used to build or maintainthe energy system rather it is an energy transformed anddelivered by the electricity technology so including it wouldamount to double-counting This term is especially impor-tant when t is 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

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 fore-

ground) is completed with data on other sectors that origi-nates from life cycle inventories and national accounts (thebackground) Gibon et al (2015) have compiled various lifecycle inventories into the 609 sectors of the foreground in-cluding original and up-to-date life cycle inventories for elec-tricity sectors Hertwich et al 2015 and its SupplementaryInformation (SI) detail sources and values retained for theevolution of crucial parameters of electricity technologiessuch as energy efficiency and market shares of different pho-tovoltaic modules The background contains data in physicalunits for 4087 sectors from the life cycle inventory ecoinventand data in monetary units for 203 sectors from the input-output database Exiobase (Wood et al 2014) The 44 Exio-base regions are aggregated into 9 macro-regions that coin-cide with those of the International Energy Agency (IEA) sothat the number of rows and columns in each IOT is 9 ti-mes the number of sectors 44046 Starting from data of the2010 IOT the 2030 and 2050 IOTs of THEMIS embed expectedtechnological efficiency improvements of key backgroundsectors produced by the New Energy Externalities Deve-lopment for Sustainability project (NEEDS 2009) NEEDSrsquorealistic-optimistic scenario was identified as the closest ma-tch to the Blue Map and Greenpeacersquos scenarios assumpti-ons namely the deployment of best available techniques andreasonable efficiency trends while the realistic-pessimisticscenario matched the Baseline assumptions Besides im-provements in foreground processes are modeled using (1)industry road maps (2) technology learning curves and (3)expert opinion (see SI of Hertwich et al (2015) for more de-tails) Furthermore it is worth noting that THEMIS IOTs areconstructed as if the whole economy were at a steady-statecontrarily to national accounts which give the flows betweensectors for a given year This matches perfectly our purposebecause there is no need to adjust the EROI computations forthe growth of some sector or for the lifetimes of some techno-logies Finally as THEMIS is multiregional EROIs are givenin total rather than internal terms meaning that embodiedenergy contains energy embodied in importsThe two scena-rios native in THEMIS are the baseline (BL) and the Blue Map

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

(BM) scenarios of the IEA (IEA 2010) While the former po-sits an almost constant electricity mix the latter is compati-ble with a 50 probability to contain the global mean tempe-rature anomaly to +2degC in 2100 As Blue Map still relies at 30on fossil fuels based electricity in 2050 mdashincluding 17 withCarbon Capture and Storage (CCS) it does not allow to assessmore decarbonized scenarios Hence I combine with THE-MIS the scenarios from Greenpeacersquos Energy [R]evolution re-port (Teske et al 2015) Greenpeace proposes a business asusual scenario (REF) close to baseline as well as two sce-narios compatible with the 2degC target Both exclude CCSand phase out from nuclear between 2012 and 20507 Thefirst Greenpeace scenario Energy [R]evolution (ER) compri-ses 93 of electricity from renewable sources in 2050 whilethe second one Advanced Energy [R]evolution (ADV) attains100 renewable As the difference is small between these twoscenarios I focus on the 100 renewable one I describe mymethodology for embedding the regional electricity mixes ofGreenpeacersquos scenarios into THEMIS in AppendixA

In the literature most EROIs estimations follow a 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 choi-ces of Greenpeace were not solely motivated by a minimization of costs butalso by expert judgment and ethical considerations

up approach that use data from life cycle inventoriesBottom-up studies describe in details the power facilities andthe most direct inputs to the energy technologies but theydo not cover the entire economy indirect inputs such as cle-rical work or RampD are often beyond their system boundaries(Suh 2004) On the contrary the input-output method allowsto encompass all embodied inputs exhaustively As a con-sequence of this more comprehensive account of embodiedenergy than usual we expect estimates of EROIs lower thanthe average of the literature That being said it is not a con-cern if our estimates are not directly comparable to those ofthe literature as we are mainly interested in comparing theminternally among the different years and scenarios and toscrutinize whether they vary substantially or not

Because renewable sources are intermittent and disper-sed the capacity grid extension and storage they require donot increase linearly with the electricity delivered Hence asGreenpeace scenarios are not native in THEMIS they needfurther adjustments to account for these non-linearities Iexplain in AppendixA how the need for overcapacity is ad-dressed Concerning transmission and storage howeverthe requirements are not given by the Greenpeace report(Teske et al 2015) so they have not been taken into accountEven if the report does not precise any plan relative to storagehydrogen produced from renewables seems to play a sub-stantial role in Greenpeace scenarios as its share in the elec-tricity mix is 5 in ADV 2050 However as the sector lsquoElec-tricity from hydrogenrsquo is absent from THEMIS hydrogen has

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

been excluded from this analysis These limitations shouldbe addressed in future work together with the study of anenergy transition in the transportation sector (which alsopartly relies on hydrogen) Such extension will not be easyas the transportation sectors are still not sufficiently disag-gregated in THEMIS to study a change in their technologyMeanwhile other references can provide information on or-ders of magnitude of storage and transmission (Berrill et al2016 Koskinen amp Breyer 2016 Scholz et al 2017) ApplyingREMix the same optimization model that is used in theGreenpeace report Scholz et al (2017) show that the cost ofstorage and transmission combined is 46 of total cost in abusiness-as-usual scenario and 106 in a 100 renewableone The adjustment needed for the cost around 6 givesa rough estimate of the upward bias of unadjusted EROI es-timates (see section 42 on the relation between price andEROI)

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 regions(leading to abnormally low EROIs around 2) Backed by Tho-mas Gibon core developer of THEMIS I corrected this errorby setting the unitary energy supplied for solar CSP in all regi-ons to its value in OECD North America (still letting the valuedepend on the scenario and the year)

33 Main Results

Main results are shown in Figure 4 and in Table 2 Com-plementary results for quality-adjusted EROIs and all scena-rios can be found in AppendixC Complete results are provi-ded in the Supplementary Information spreadsheet they in-clude eg regional estimates and a decomposition of EROIsrsquo

denominators between direct and indirect energy SomeEROIs are missing because not all technologies already exis-ted on an industrial scale in 2010 and some technologies arediscarded in the future by some scenarios Conversely someEROIs are given for apparent shares of production of 0 this isthe case when the share is rounded to 0 but not 0

One can notice that as expected PV and wind panelshave a lower EROI than electricity from fossil fuels TheEROIs of renewables decrease as anticipated in the previ-ous section However they remain largely above 1 sugges-ting that renewables are energetically sustainable Recall thatthis was not evident as in theory nothing guarantees thatEROIs stay above 1 when the energy mix changes (see section22) Values for current EROIs range from 8 to 22 Thisrange is in-line with that from Hall et al (2014) but not withWeiszligbach et al (2013) who find more contrasts between re-newables and fossils Such discrepancy is common in theEROI literature may be due to differences in the methodology(Weiszligbach uses bottom-up data from specific locations) anddoes not affect this paperrsquos results on the evolution of EROIs

The system-wide EROI for the entire electricity sector isgiven 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(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-

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

nounced when the penetration of renewable is higher downto 80 in 2050 in the Blue Map scenario and even 58 in the100 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 renewa-bles

One may wonder whether our results are driven by con-servative forecasts concerning the progress in renewabletechnologies or any other hypothesis concerning the evolu-tion of the technology matrix Of course the quality of input-output data is never perfect and making predictions is no-toriously difficult as was recently proven by the unexpectedfall in the price of photovoltaic (PV) modules However thereare several reasons to be more confident into future EROIsestimates from THEMIS than into past predictions on pricesfrom other sources First technical coefficients are more sta-ble than prices Second THEMIS accounts for materials andenergy efficiency gains for electricity technologies and usesldquofairly favorable assumptions regarding wind conditions in-solation and resulting load factorsrdquo which if anything wouldbias EROIs of renewables upward (see SI of Hertwich et al2015) Third THEMIS already includes recent industry roadmaps in its prospective matrices (see section 31) eg con-cerning the shift of PV market shares from cristalline siliconmodules towards more efficient cadmium telluride (CdTe) orCIGS modules Overall the data from THEMIS seems mostaccurate concerning materials metallurgy and energy sec-tors and further improvements should probably focus on ot-her sectors like transport or services

4 Implications of a Decreasing EROI on Prices and GDP

The forecast of declining EROIs made in the previoussection calls for an assessment of its economic implicationsThe main channel through which a decrease in EROI couldaffect the economy is arguably a rise in energy price (andcorrelatively in energy expenditures) In this section I re-view the literature on the relation between EROI and the priceof energy estimate it empirically and extend a result fromHerendeen (2015) to characterize this relation As in previ-ous work an inverse relation is documented empirically Yettheoretical analysis shows that EROI and price might decre-ase together This theoretical result tempers the view that adecreasing EROI necessarily leads to a contraction of GDP

41 Inverse Relation Proposed in First Studies

King amp Hall (2011) point both theoretically and empiri-cally that the price of a unit of energy pt and the EROI of atechnology t are inversely related Defining the monetary re-turn on investment MROI (ie the financial yield $out

$investment)

they derive 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

costs per gross output ct =$investmentEout+Ein

and use their own notionof EROIEROIH

t =Eout+Ein

Ein= 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 variablesmove together when EROI varies so does the cost of pro-duction so that we cannot predict the future price taking thiscost as fixed Heun amp de Wit (2012) acknowledge this andthus study the empirical link between EROI and price

42 Empirical Relation Between EROI and Price

Using US data on oil and EROI from (Cleveland 2005)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 inverseone 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 undertakethis comparison I run these two regressions using all esti-mates of EROI computed using THEMIS one for each com-bination of scenario year region and sector To obtain theprice corresponding to each EROI which I take before taxesand subsidies on production I assemble from the columnscompensation of employees and operating surplus of the cha-racterization matrix of THEMIS a row vector v of value-addedper unit of each sector Indeed the vector of prices excludingtax 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 theextent that the physical constituents and processes of a gi-ven technology will not change in an unexpected way and asTHEMIS models technical progress but not behaviors nor ge-neral equilibrium effects prices forecast using the above for-mula seem less reliable than EROI estimates For this reasonI report only the global average electricity prices of the mainscenarios (see Table 3) but I do not detail the substantial va-riations between regions or sectors10

Table 4 reports the results of both the log-log and the in-verse fits I ran each model twice first on all 2079 positiveobservations available and then on the 104 observations foryear 2010 To make the R2 of the log-log fit comparable to

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

been excluded from this analysis These limitations shouldbe addressed in future work together with the study of anenergy transition in the transportation sector (which alsopartly relies on hydrogen) Such extension will not be easyas the transportation sectors are still not sufficiently disag-gregated in THEMIS to study a change in their technologyMeanwhile other references can provide information on or-ders of magnitude of storage and transmission (Berrill et al2016 Koskinen amp Breyer 2016 Scholz et al 2017) ApplyingREMix the same optimization model that is used in theGreenpeace report Scholz et al (2017) show that the cost ofstorage and transmission combined is 46 of total cost in abusiness-as-usual scenario and 106 in a 100 renewableone The adjustment needed for the cost around 6 givesa rough estimate of the upward bias of unadjusted EROI es-timates (see section 42 on the relation between price andEROI)

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 regions(leading to abnormally low EROIs around 2) Backed by Tho-mas Gibon core developer of THEMIS I corrected this errorby setting the unitary energy supplied for solar CSP in all regi-ons to its value in OECD North America (still letting the valuedepend on the scenario and the year)

33 Main Results

Main results are shown in Figure 4 and in Table 2 Com-plementary results for quality-adjusted EROIs and all scena-rios can be found in AppendixC Complete results are provi-ded in the Supplementary Information spreadsheet they in-clude eg regional estimates and a decomposition of EROIsrsquo

denominators between direct and indirect energy SomeEROIs are missing because not all technologies already exis-ted on an industrial scale in 2010 and some technologies arediscarded in the future by some scenarios Conversely someEROIs are given for apparent shares of production of 0 this isthe case when the share is rounded to 0 but not 0

One can notice that as expected PV and wind panelshave a lower EROI than electricity from fossil fuels TheEROIs of renewables decrease as anticipated in the previ-ous section However they remain largely above 1 sugges-ting that renewables are energetically sustainable Recall thatthis was not evident as in theory nothing guarantees thatEROIs stay above 1 when the energy mix changes (see section22) Values for current EROIs range from 8 to 22 Thisrange is in-line with that from Hall et al (2014) but not withWeiszligbach et al (2013) who find more contrasts between re-newables and fossils Such discrepancy is common in theEROI literature may be due to differences in the methodology(Weiszligbach uses bottom-up data from specific locations) anddoes not affect this paperrsquos results on the evolution of EROIs

The system-wide EROI for the entire electricity sector isgiven 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(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-

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

nounced when the penetration of renewable is higher downto 80 in 2050 in the Blue Map scenario and even 58 in the100 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 renewa-bles

One may wonder whether our results are driven by con-servative forecasts concerning the progress in renewabletechnologies or any other hypothesis concerning the evolu-tion of the technology matrix Of course the quality of input-output data is never perfect and making predictions is no-toriously difficult as was recently proven by the unexpectedfall in the price of photovoltaic (PV) modules However thereare several reasons to be more confident into future EROIsestimates from THEMIS than into past predictions on pricesfrom other sources First technical coefficients are more sta-ble than prices Second THEMIS accounts for materials andenergy efficiency gains for electricity technologies and usesldquofairly favorable assumptions regarding wind conditions in-solation and resulting load factorsrdquo which if anything wouldbias EROIs of renewables upward (see SI of Hertwich et al2015) Third THEMIS already includes recent industry roadmaps in its prospective matrices (see section 31) eg con-cerning the shift of PV market shares from cristalline siliconmodules towards more efficient cadmium telluride (CdTe) orCIGS modules Overall the data from THEMIS seems mostaccurate concerning materials metallurgy and energy sec-tors and further improvements should probably focus on ot-her sectors like transport or services

4 Implications of a Decreasing EROI on Prices and GDP

The forecast of declining EROIs made in the previoussection calls for an assessment of its economic implicationsThe main channel through which a decrease in EROI couldaffect the economy is arguably a rise in energy price (andcorrelatively in energy expenditures) In this section I re-view the literature on the relation between EROI and the priceof energy estimate it empirically and extend a result fromHerendeen (2015) to characterize this relation As in previ-ous work an inverse relation is documented empirically Yettheoretical analysis shows that EROI and price might decre-ase together This theoretical result tempers the view that adecreasing EROI necessarily leads to a contraction of GDP

41 Inverse Relation Proposed in First Studies

King amp Hall (2011) point both theoretically and empiri-cally that the price of a unit of energy pt and the EROI of atechnology t are inversely related Defining the monetary re-turn on investment MROI (ie the financial yield $out

$investment)

they derive 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

costs per gross output ct =$investmentEout+Ein

and use their own notionof EROIEROIH

t =Eout+Ein

Ein= 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 variablesmove together when EROI varies so does the cost of pro-duction so that we cannot predict the future price taking thiscost as fixed Heun amp de Wit (2012) acknowledge this andthus study the empirical link between EROI and price

42 Empirical Relation Between EROI and Price

Using US data on oil and EROI from (Cleveland 2005)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 inverseone 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 undertakethis comparison I run these two regressions using all esti-mates of EROI computed using THEMIS one for each com-bination of scenario year region and sector To obtain theprice corresponding to each EROI which I take before taxesand subsidies on production I assemble from the columnscompensation of employees and operating surplus of the cha-racterization matrix of THEMIS a row vector v of value-addedper unit of each sector Indeed the vector of prices excludingtax 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 theextent that the physical constituents and processes of a gi-ven technology will not change in an unexpected way and asTHEMIS models technical progress but not behaviors nor ge-neral equilibrium effects prices forecast using the above for-mula seem less reliable than EROI estimates For this reasonI report only the global average electricity prices of the mainscenarios (see Table 3) but I do not detail the substantial va-riations between regions or sectors10

Table 4 reports the results of both the log-log and the in-verse fits I ran each model twice first on all 2079 positiveobservations available and then on the 104 observations foryear 2010 To make the R2 of the log-log fit comparable to

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

nounced when the penetration of renewable is higher downto 80 in 2050 in the Blue Map scenario and even 58 in the100 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 renewa-bles

One may wonder whether our results are driven by con-servative forecasts concerning the progress in renewabletechnologies or any other hypothesis concerning the evolu-tion of the technology matrix Of course the quality of input-output data is never perfect and making predictions is no-toriously difficult as was recently proven by the unexpectedfall in the price of photovoltaic (PV) modules However thereare several reasons to be more confident into future EROIsestimates from THEMIS than into past predictions on pricesfrom other sources First technical coefficients are more sta-ble than prices Second THEMIS accounts for materials andenergy efficiency gains for electricity technologies and usesldquofairly favorable assumptions regarding wind conditions in-solation and resulting load factorsrdquo which if anything wouldbias EROIs of renewables upward (see SI of Hertwich et al2015) Third THEMIS already includes recent industry roadmaps in its prospective matrices (see section 31) eg con-cerning the shift of PV market shares from cristalline siliconmodules towards more efficient cadmium telluride (CdTe) orCIGS modules Overall the data from THEMIS seems mostaccurate concerning materials metallurgy and energy sec-tors and further improvements should probably focus on ot-her sectors like transport or services

4 Implications of a Decreasing EROI on Prices and GDP

The forecast of declining EROIs made in the previoussection calls for an assessment of its economic implicationsThe main channel through which a decrease in EROI couldaffect the economy is arguably a rise in energy price (andcorrelatively in energy expenditures) In this section I re-view the literature on the relation between EROI and the priceof energy estimate it empirically and extend a result fromHerendeen (2015) to characterize this relation As in previ-ous work an inverse relation is documented empirically Yettheoretical analysis shows that EROI and price might decre-ase together This theoretical result tempers the view that adecreasing EROI necessarily leads to a contraction of GDP

41 Inverse Relation Proposed in First Studies

King amp Hall (2011) point both theoretically and empiri-cally that the price of a unit of energy pt and the EROI of atechnology t are inversely related Defining the monetary re-turn on investment MROI (ie the financial yield $out

$investment)

they derive 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

costs per gross output ct =$investmentEout+Ein

and use their own notionof EROIEROIH

t =Eout+Ein

Ein= 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 variablesmove together when EROI varies so does the cost of pro-duction so that we cannot predict the future price taking thiscost as fixed Heun amp de Wit (2012) acknowledge this andthus study the empirical link between EROI and price

42 Empirical Relation Between EROI and Price

Using US data on oil and EROI from (Cleveland 2005)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 inverseone 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 undertakethis comparison I run these two regressions using all esti-mates of EROI computed using THEMIS one for each com-bination of scenario year region and sector To obtain theprice corresponding to each EROI which I take before taxesand subsidies on production I assemble from the columnscompensation of employees and operating surplus of the cha-racterization matrix of THEMIS a row vector v of value-addedper unit of each sector Indeed the vector of prices excludingtax 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 theextent that the physical constituents and processes of a gi-ven technology will not change in an unexpected way and asTHEMIS models technical progress but not behaviors nor ge-neral equilibrium effects prices forecast using the above for-mula seem less reliable than EROI estimates For this reasonI report only the global average electricity prices of the mainscenarios (see Table 3) but I do not detail the substantial va-riations between regions or sectors10

Table 4 reports the results of both the log-log and the in-verse fits I ran each model twice first on all 2079 positiveobservations available and then on the 104 observations foryear 2010 To make the R2 of the log-log fit comparable to

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 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

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

Obs N SpecificationCoefficients

R211

a b

All 2079p =

aEROI +b

85 18 0552010 104 72 21 054

All 2079log

(

p)

= a middot log (EROI)+bminus057 20 058

2010 104 minus046 19 062

that 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 and062 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 betweenprice 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-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 how

11The R2 given for log-log fits is not the original one cf text

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

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 energies12 εt is in fact the gross energy em-bodied in t ie the sum of the delivered and the net embo-died energy Hence the EROI of t is a simple function of εt

EROIt =1

TE middot1t

1TEmiddot((IminusA)minus1minusI)middot1t

=1

εtminus1

In AppendixD I show that the price of a technology t is acertain function of the coefficients of A13 and that each coef-ficient of A can be expressed as a function of EROI Compo-sing 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 constantThis leads to the following Proposition

Proposition 1 (Generalization of Herendeen 2015) Assu-

ming that all coefficients of the transformation matrix A are

constant 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 intensity εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β (14)

Proof See AppendixD

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

pt = αEROIH

t

EROIHt minus1

+ γ with γ = βminusα This is because in their

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 coef-ficient vary Indeed denoting ωi t the coefficient (i t) of(I minus A)minus1 the Laplace expansion of I minus A gives us

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 alldepend on the coefficients of A and vary together when A

12I assume here that the unit of an output of an energy sector e isin E henceof 1T

E is an energy unit like TWh

13More precisely a function field of a certain algebraic variety

9

changes If there is only one type of energy (E = e) or ifvalue-added is equal for all types of energy (foralle isin E ve = v)v does not depend on the coefficients of A anymore andwe obtain a formula close to that of King amp Hall (2011) pt =

ve

EROIt+ r Still when the EROI varies because more than one

coefficient of A changes r varies concomitantly and the EROIcannot be used as a sufficient statistic to infer the price Forthis reason one cannot identify empirically a linear relationbetween price and the inverse of EROI without strong as-sumption on the steadiness of A

Actually the theoretical relation between EROI and priceis so fragile that one cannot even conclude that it is a decre-asing relation I provide in AppendixB a numerical exampleshowing that EROI and price can both increase at the sametime when more than one coefficient varies Such acknow-ledgment dissuades from predicting long run prices by sim-ply looking at estimations of future EROIs

Does this mean that EROI is unrelated to any economicconcept 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 gro-wth in the US then determine a threshold of energy expendi-ture above which the US enters in a recession (consistent withthat of Bashmakov 2007) and finally use a modified versionof equation (11) to relate this to a minimum non-recessionaryEROI However they misleadingly replace the inverse of the

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 expenditu-res As we have seen EROI price and energy expenditure mayall decrease at the same time which undermines the idea thata recession caused by a surge in energy expenditure is ineluc-table as soon as EROI goes below some threshold In addi-tion an energy price increase should have an expansionaryeffect on net exporters of energy at odds with the mechanismextrapolated by Fizaine and Court from the case of the Uni-ted States which has been historically a net energy importerOverall the analysis of this section indicates that the econo-mic consequences of a change in EROI are ambiguous andthat this physical notion cannot be used to predict future pri-ces or GDP without empirical evidence

5 Concluding Remarks

This work includes a first attempt at estimating futureEROIs in a decarbonized electricity system By examininga broad range of scenarios it concludes that the system-wide EROI of the power sector should decrease until 2050from 122 to 107 in a business-as-usual scenario 80 in apartial transition away from fossil fuels or 58 in a scenariowith 100 renewable electricity Even though the EROI ofeach technology is expected to remain well above 1 whichwas questioned theoretically our results show that renewa-ble electricity is not as energetically efficient as previouslythought

As an inverse relationship between EROIs and energy pri-ces is consistently found empirically a declining EROI couldmean higher energy prices However theoretical analysis ofthis relation showed that a declining EROI might also coin-cide with decreasing energy prices and does not necessarilylead to a recession

Finally this paper assessed scenarios of transition in theelectricity sector but further research is still needed to esti-mate future EROIs in complete energy transitions which in-clude a mutation of the transportation system agricultureand industry Unfortunately this could not be done usingthe current version of THEMIS and the question remainsopen for future research Another goal for the field wouldbe to converge on the methods to compute EROIs of rene-wables notably on how to integrate buffering ie overcapa-city and storage requirements Indeed if EROIs of renewableswere higher than those of fossils mdashwhich is sometimes foundwhen buffering is not accounted for eg in Raugei amp Leccisi(2016)mdash then a decline in system-wide EROI would not evenbe a concern Hence defining common methodological prin-ciples would help to narrow the gap between estimates fromdifferent sources

To conclude as this article suggests that EROIs of renewa-bles are lower than those of fossils implying that global EROIwould decrease substantially in a renewable energy transi-tion it is worth emphasizing that the choice to undergo a re-newable transition ndash or not ndash should not be reduced to con-siderations of EROIs Indeed given the negative externalitiesand the scarcity of fossil fuels a renewable transition is welljustified

10

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

licy 2015A Arvesen G Luderer M Pehl B L Bodirsky amp E G Hertwich Deriving

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-

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 Mat-hematics of a Minimum Energy Return on Investment (EROI) BioPhysical

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

Systems-Scale Efficiency of Energy Extraction and Conversion EnergyReturn on Investment (EROI) and Other Energy Return Ratios Energies2011

C J Cleveland Net energy from the extraction of oil and gas in the UnitedStates Energy 2005

V Court amp F Fizaine Long-Term Estimates of the Energy-Return-on-Investment (EROI) of Coal Oil and Gas Global Productions Ecological

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)

2010Eurostat editor Eurostat Manual of supply use and input-output tables Amt

fuumlr 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 2016

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 Scena-rios under Large-Scale Technological Change Environmental Science amp

Technology 2015C A S Hall S Balogh amp D J R Murphy What is the Minimum EROI that a

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 2014C A Hall Introduction to Special Issue on New Studies in EROI (Energy

Return on Investment) Sustainability 2011R A Herendeen Connecting net energy with the price of energy and other

goods and services Ecological Economics 2015E G Hertwich T Gibon E A Bouman A Arvesen S Suh G A Heath J D

Bergesen A Ramirez M I Vega amp L Shi Integrated life-cycle assessmentof electricity-supply scenarios confirms global environmental benefit oflow-carbon technologies Proceedings of the National Academy of Scien-

ces 2015M K Heun amp M de Wit Energy return on (energy) invested (EROI) oil prices

and energy transitions Energy Policy 2012IEA 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 Research2003

C W King Matrix method for comparing system and individual energy re-turn ratios when considering an energy transition Energy 2014

C W King amp C A S Hall Relating Financial and Energy Return on Invest-ment Sustainability 2011

L C King amp J C J M v d Bergh Implications of net energy-return-on-investment for a low-carbon energy transition Nature Energy 2018

O Koskinen amp C Breyer Energy Storage in Global and TranscontinentalEnergy 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 Inves-tment (EROI) Sustainability 2011

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 ExtensionsCambridge University Press 2009 ISBN 978-0-521-51713-3

E Muumlller L M Hilty R Widmer M Schluep amp M Faulstich Modeling MetalStocks 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 APreliminary 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 LudererUnderstanding 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 2013M Raugei Comments on ldquoEnergy intensities EROIs (energy returned

on invested) and energy payback times of electricity generating powerplantsrdquomdashMaking clear of quite some confusion Energy 2013

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

ture Energy 2019M Raugei amp E Leccisi A comprehensive assessment of the energy perfor-

mance of the full range of electricity generation technologies deployed inthe United Kingdom Energy Policy 2016

Y Scholz H C Gils amp R C Pietzcker Application of a high-detail energy sy-stem model to derive power sector characteristics at high wind and solarshares Energy Economics 2017

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 2017D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein

Energy intensities EROIs (energy returned on invested) and energy pay-back times of electricity generating power plants Energy 2013

D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein Re-ply on ldquoComments on lsquoEnergy intensities EROIs (energy returned on in-vested) and energy payback times of electricity generating power plantsrsquondash Making clear of quite some confusionrdquo Energy 2014

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

11

AppendixA Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are re-adily available in THEMIS but these matrices have to beupdated to the new electricity mix for the Greenpeace sce-narios To do this I exploit the fact that both THEMISand Greenpeace use the world regions of the IEA and Imodify the electricity input of each sector by the regio-nal mix given by Greenpeace The most accurate algo-rithm to update an input-output matrix is known as GRAS(Junius amp Oosterhaven 2003) Although I implemented thisalgorithm in pymrio I could not use it because this algo-rithm uses the new sums of rows and columns to balance thematrix and the vector of final demand y or the vector of pro-duction x is necessary to know them As THEMIS does notinclude such vectors I had to use a simpler method whichrelies on the assumption that the electricity mix of inputs isthe same across sectors for a given region Given the per-fect substitutability between electricity produced by differenttechnologies and the uniqueness of electric grids this as-sumption 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 thedemand 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-quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premulti-ply E by a block diagonal matrix with R blocks of size T lowastT

containing only ones (where R = 9 and T = 15 are the num-ber of THEMIS regions and electricity sectors respectively)to obtain a matrix B Each row of B gives the total electricityfrom a given region r required to produced each output E tot

r and each 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 techno-logy t in the mix of the corresponding region which defines amatrix E Each coefficient Ei s of E gives the electricity fromsector i required to make one unit of sector srsquo output accor-ding to the new mix (by construction for all electricity sec-tor 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 obtain

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 sour-ces fail to deliver energy the ratio of capacity (in GW)over production (in TWh) is somewhat higher in Greenpe-ace scenarios than in IEATHEMIS ones Thus I multi-

ply each column of an energy sector (representing all in-puts required for one unit of output of this sector) by the ra-tio of the capacity-over-production ratios of Greenpeace andIEATHEMIS Doing so relies on the fact that the energy re-quired to operate a power plant is negligible in front of theenergy required to build it (see eg Arvesen et al (2018))

AppendixB Example of Non-Decreasing Relation Between

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(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 decreasingrelation

For this I use the formulas for EROI and price given byHerendeen (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

AppendixC Complementary Results

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 ofthe 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

12

changes If there is only one type of energy (E = e) or ifvalue-added is equal for all types of energy (foralle isin E ve = v)v does not depend on the coefficients of A anymore andwe obtain a formula close to that of King amp Hall (2011) pt =

ve

EROIt+ r Still when the EROI varies because more than one

coefficient of A changes r varies concomitantly and the EROIcannot be used as a sufficient statistic to infer the price Forthis reason one cannot identify empirically a linear relationbetween price and the inverse of EROI without strong as-sumption on the steadiness of A

Actually the theoretical relation between EROI and priceis so fragile that one cannot even conclude that it is a decre-asing relation I provide in AppendixB a numerical exampleshowing that EROI and price can both increase at the sametime when more than one coefficient varies Such acknow-ledgment dissuades from predicting long run prices by sim-ply looking at estimations of future EROIs

Does this mean that EROI is unrelated to any economicconcept 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 gro-wth in the US then determine a threshold of energy expendi-ture above which the US enters in a recession (consistent withthat of Bashmakov 2007) and finally use a modified versionof equation (11) to relate this to a minimum non-recessionaryEROI However they misleadingly replace the inverse of the

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 expenditu-res As we have seen EROI price and energy expenditure mayall decrease at the same time which undermines the idea thata recession caused by a surge in energy expenditure is ineluc-table as soon as EROI goes below some threshold In addi-tion an energy price increase should have an expansionaryeffect on net exporters of energy at odds with the mechanismextrapolated by Fizaine and Court from the case of the Uni-ted States which has been historically a net energy importerOverall the analysis of this section indicates that the econo-mic consequences of a change in EROI are ambiguous andthat this physical notion cannot be used to predict future pri-ces or GDP without empirical evidence

5 Concluding Remarks

This work includes a first attempt at estimating futureEROIs in a decarbonized electricity system By examininga broad range of scenarios it concludes that the system-wide EROI of the power sector should decrease until 2050from 122 to 107 in a business-as-usual scenario 80 in apartial transition away from fossil fuels or 58 in a scenariowith 100 renewable electricity Even though the EROI ofeach technology is expected to remain well above 1 whichwas questioned theoretically our results show that renewa-ble electricity is not as energetically efficient as previouslythought

As an inverse relationship between EROIs and energy pri-ces is consistently found empirically a declining EROI couldmean higher energy prices However theoretical analysis ofthis relation showed that a declining EROI might also coin-cide with decreasing energy prices and does not necessarilylead to a recession

Finally this paper assessed scenarios of transition in theelectricity sector but further research is still needed to esti-mate future EROIs in complete energy transitions which in-clude a mutation of the transportation system agricultureand industry Unfortunately this could not be done usingthe current version of THEMIS and the question remainsopen for future research Another goal for the field wouldbe to converge on the methods to compute EROIs of rene-wables notably on how to integrate buffering ie overcapa-city and storage requirements Indeed if EROIs of renewableswere higher than those of fossils mdashwhich is sometimes foundwhen buffering is not accounted for eg in Raugei amp Leccisi(2016)mdash then a decline in system-wide EROI would not evenbe a concern Hence defining common methodological prin-ciples would help to narrow the gap between estimates fromdifferent sources

To conclude as this article suggests that EROIs of renewa-bles are lower than those of fossils implying that global EROIwould decrease substantially in a renewable energy transi-tion it is worth emphasizing that the choice to undergo a re-newable transition ndash or not ndash should not be reduced to con-siderations of EROIs Indeed given the negative externalitiesand the scarcity of fossil fuels a renewable transition is welljustified

10

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

licy 2015A Arvesen G Luderer M Pehl B L Bodirsky amp E G Hertwich Deriving

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-

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 Mat-hematics of a Minimum Energy Return on Investment (EROI) BioPhysical

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

Systems-Scale Efficiency of Energy Extraction and Conversion EnergyReturn on Investment (EROI) and Other Energy Return Ratios Energies2011

C J Cleveland Net energy from the extraction of oil and gas in the UnitedStates Energy 2005

V Court amp F Fizaine Long-Term Estimates of the Energy-Return-on-Investment (EROI) of Coal Oil and Gas Global Productions Ecological

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)

2010Eurostat editor Eurostat Manual of supply use and input-output tables Amt

fuumlr 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 2016

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 Scena-rios under Large-Scale Technological Change Environmental Science amp

Technology 2015C A S Hall S Balogh amp D J R Murphy What is the Minimum EROI that a

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 2014C A Hall Introduction to Special Issue on New Studies in EROI (Energy

Return on Investment) Sustainability 2011R A Herendeen Connecting net energy with the price of energy and other

goods and services Ecological Economics 2015E G Hertwich T Gibon E A Bouman A Arvesen S Suh G A Heath J D

Bergesen A Ramirez M I Vega amp L Shi Integrated life-cycle assessmentof electricity-supply scenarios confirms global environmental benefit oflow-carbon technologies Proceedings of the National Academy of Scien-

ces 2015M K Heun amp M de Wit Energy return on (energy) invested (EROI) oil prices

and energy transitions Energy Policy 2012IEA 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 Research2003

C W King Matrix method for comparing system and individual energy re-turn ratios when considering an energy transition Energy 2014

C W King amp C A S Hall Relating Financial and Energy Return on Invest-ment Sustainability 2011

L C King amp J C J M v d Bergh Implications of net energy-return-on-investment for a low-carbon energy transition Nature Energy 2018

O Koskinen amp C Breyer Energy Storage in Global and TranscontinentalEnergy 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 Inves-tment (EROI) Sustainability 2011

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 ExtensionsCambridge University Press 2009 ISBN 978-0-521-51713-3

E Muumlller L M Hilty R Widmer M Schluep amp M Faulstich Modeling MetalStocks 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 APreliminary 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 LudererUnderstanding 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 2013M Raugei Comments on ldquoEnergy intensities EROIs (energy returned

on invested) and energy payback times of electricity generating powerplantsrdquomdashMaking clear of quite some confusion Energy 2013

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

ture Energy 2019M Raugei amp E Leccisi A comprehensive assessment of the energy perfor-

mance of the full range of electricity generation technologies deployed inthe United Kingdom Energy Policy 2016

Y Scholz H C Gils amp R C Pietzcker Application of a high-detail energy sy-stem model to derive power sector characteristics at high wind and solarshares Energy Economics 2017

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 2017D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein

Energy intensities EROIs (energy returned on invested) and energy pay-back times of electricity generating power plants Energy 2013

D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein Re-ply on ldquoComments on lsquoEnergy intensities EROIs (energy returned on in-vested) and energy payback times of electricity generating power plantsrsquondash Making clear of quite some confusionrdquo Energy 2014

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

11

AppendixA Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are re-adily available in THEMIS but these matrices have to beupdated to the new electricity mix for the Greenpeace sce-narios To do this I exploit the fact that both THEMISand Greenpeace use the world regions of the IEA and Imodify the electricity input of each sector by the regio-nal mix given by Greenpeace The most accurate algo-rithm to update an input-output matrix is known as GRAS(Junius amp Oosterhaven 2003) Although I implemented thisalgorithm in pymrio I could not use it because this algo-rithm uses the new sums of rows and columns to balance thematrix and the vector of final demand y or the vector of pro-duction x is necessary to know them As THEMIS does notinclude such vectors I had to use a simpler method whichrelies on the assumption that the electricity mix of inputs isthe same across sectors for a given region Given the per-fect substitutability between electricity produced by differenttechnologies and the uniqueness of electric grids this as-sumption 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 thedemand 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-quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premulti-ply E by a block diagonal matrix with R blocks of size T lowastT

containing only ones (where R = 9 and T = 15 are the num-ber of THEMIS regions and electricity sectors respectively)to obtain a matrix B Each row of B gives the total electricityfrom a given region r required to produced each output E tot

r and each 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 techno-logy t in the mix of the corresponding region which defines amatrix E Each coefficient Ei s of E gives the electricity fromsector i required to make one unit of sector srsquo output accor-ding to the new mix (by construction for all electricity sec-tor 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 obtain

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 sour-ces fail to deliver energy the ratio of capacity (in GW)over production (in TWh) is somewhat higher in Greenpe-ace scenarios than in IEATHEMIS ones Thus I multi-

ply each column of an energy sector (representing all in-puts required for one unit of output of this sector) by the ra-tio of the capacity-over-production ratios of Greenpeace andIEATHEMIS Doing so relies on the fact that the energy re-quired to operate a power plant is negligible in front of theenergy required to build it (see eg Arvesen et al (2018))

AppendixB Example of Non-Decreasing Relation Between

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(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 decreasingrelation

For this I use the formulas for EROI and price given byHerendeen (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

AppendixC Complementary Results

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 ofthe 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

12

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

licy 2015A Arvesen G Luderer M Pehl B L Bodirsky amp E G Hertwich Deriving

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-

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 Mat-hematics of a Minimum Energy Return on Investment (EROI) BioPhysical

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

Systems-Scale Efficiency of Energy Extraction and Conversion EnergyReturn on Investment (EROI) and Other Energy Return Ratios Energies2011

C J Cleveland Net energy from the extraction of oil and gas in the UnitedStates Energy 2005

V Court amp F Fizaine Long-Term Estimates of the Energy-Return-on-Investment (EROI) of Coal Oil and Gas Global Productions Ecological

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)

2010Eurostat editor Eurostat Manual of supply use and input-output tables Amt

fuumlr 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 2016

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 Scena-rios under Large-Scale Technological Change Environmental Science amp

Technology 2015C A S Hall S Balogh amp D J R Murphy What is the Minimum EROI that a

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 2014C A Hall Introduction to Special Issue on New Studies in EROI (Energy

Return on Investment) Sustainability 2011R A Herendeen Connecting net energy with the price of energy and other

goods and services Ecological Economics 2015E G Hertwich T Gibon E A Bouman A Arvesen S Suh G A Heath J D

Bergesen A Ramirez M I Vega amp L Shi Integrated life-cycle assessmentof electricity-supply scenarios confirms global environmental benefit oflow-carbon technologies Proceedings of the National Academy of Scien-

ces 2015M K Heun amp M de Wit Energy return on (energy) invested (EROI) oil prices

and energy transitions Energy Policy 2012IEA 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 Research2003

C W King Matrix method for comparing system and individual energy re-turn ratios when considering an energy transition Energy 2014

C W King amp C A S Hall Relating Financial and Energy Return on Invest-ment Sustainability 2011

L C King amp J C J M v d Bergh Implications of net energy-return-on-investment for a low-carbon energy transition Nature Energy 2018

O Koskinen amp C Breyer Energy Storage in Global and TranscontinentalEnergy 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 Inves-tment (EROI) Sustainability 2011

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 ExtensionsCambridge University Press 2009 ISBN 978-0-521-51713-3

E Muumlller L M Hilty R Widmer M Schluep amp M Faulstich Modeling MetalStocks 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 APreliminary 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 LudererUnderstanding 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 2013M Raugei Comments on ldquoEnergy intensities EROIs (energy returned

on invested) and energy payback times of electricity generating powerplantsrdquomdashMaking clear of quite some confusion Energy 2013

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

ture Energy 2019M Raugei amp E Leccisi A comprehensive assessment of the energy perfor-

mance of the full range of electricity generation technologies deployed inthe United Kingdom Energy Policy 2016

Y Scholz H C Gils amp R C Pietzcker Application of a high-detail energy sy-stem model to derive power sector characteristics at high wind and solarshares Energy Economics 2017

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 2017D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein

Energy intensities EROIs (energy returned on invested) and energy pay-back times of electricity generating power plants Energy 2013

D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hussein Re-ply on ldquoComments on lsquoEnergy intensities EROIs (energy returned on in-vested) and energy payback times of electricity generating power plantsrsquondash Making clear of quite some confusionrdquo Energy 2014

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

11

AppendixA Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are re-adily available in THEMIS but these matrices have to beupdated to the new electricity mix for the Greenpeace sce-narios To do this I exploit the fact that both THEMISand Greenpeace use the world regions of the IEA and Imodify the electricity input of each sector by the regio-nal mix given by Greenpeace The most accurate algo-rithm to update an input-output matrix is known as GRAS(Junius amp Oosterhaven 2003) Although I implemented thisalgorithm in pymrio I could not use it because this algo-rithm uses the new sums of rows and columns to balance thematrix and the vector of final demand y or the vector of pro-duction x is necessary to know them As THEMIS does notinclude such vectors I had to use a simpler method whichrelies on the assumption that the electricity mix of inputs isthe same across sectors for a given region Given the per-fect substitutability between electricity produced by differenttechnologies and the uniqueness of electric grids this as-sumption 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 thedemand 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-quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premulti-ply E by a block diagonal matrix with R blocks of size T lowastT

containing only ones (where R = 9 and T = 15 are the num-ber of THEMIS regions and electricity sectors respectively)to obtain a matrix B Each row of B gives the total electricityfrom a given region r required to produced each output E tot

r and each 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 techno-logy t in the mix of the corresponding region which defines amatrix E Each coefficient Ei s of E gives the electricity fromsector i required to make one unit of sector srsquo output accor-ding to the new mix (by construction for all electricity sec-tor 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 obtain

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 sour-ces fail to deliver energy the ratio of capacity (in GW)over production (in TWh) is somewhat higher in Greenpe-ace scenarios than in IEATHEMIS ones Thus I multi-

ply each column of an energy sector (representing all in-puts required for one unit of output of this sector) by the ra-tio of the capacity-over-production ratios of Greenpeace andIEATHEMIS Doing so relies on the fact that the energy re-quired to operate a power plant is negligible in front of theenergy required to build it (see eg Arvesen et al (2018))

AppendixB Example of Non-Decreasing Relation Between

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(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 decreasingrelation

For this I use the formulas for EROI and price given byHerendeen (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

AppendixC Complementary Results

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 ofthe 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

12

AppendixA Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are re-adily available in THEMIS but these matrices have to beupdated to the new electricity mix for the Greenpeace sce-narios To do this I exploit the fact that both THEMISand Greenpeace use the world regions of the IEA and Imodify the electricity input of each sector by the regio-nal mix given by Greenpeace The most accurate algo-rithm to update an input-output matrix is known as GRAS(Junius amp Oosterhaven 2003) Although I implemented thisalgorithm in pymrio I could not use it because this algo-rithm uses the new sums of rows and columns to balance thematrix and the vector of final demand y or the vector of pro-duction x is necessary to know them As THEMIS does notinclude such vectors I had to use a simpler method whichrelies on the assumption that the electricity mix of inputs isthe same across sectors for a given region Given the per-fect substitutability between electricity produced by differenttechnologies and the uniqueness of electric grids this as-sumption 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 thedemand 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-quired to make one unit of sector srsquo output where i = i (t r )corresponds to technology t in region r Then I premulti-ply E by a block diagonal matrix with R blocks of size T lowastT

containing only ones (where R = 9 and T = 15 are the num-ber of THEMIS regions and electricity sectors respectively)to obtain a matrix B Each row of B gives the total electricityfrom a given region r required to produced each output E tot

r and each 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 techno-logy t in the mix of the corresponding region which defines amatrix E Each coefficient Ei s of E gives the electricity fromsector i required to make one unit of sector srsquo output accor-ding to the new mix (by construction for all electricity sec-tor 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 obtain

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 sour-ces fail to deliver energy the ratio of capacity (in GW)over production (in TWh) is somewhat higher in Greenpe-ace scenarios than in IEATHEMIS ones Thus I multi-

ply each column of an energy sector (representing all in-puts required for one unit of output of this sector) by the ra-tio of the capacity-over-production ratios of Greenpeace andIEATHEMIS Doing so relies on the fact that the energy re-quired to operate a power plant is negligible in front of theenergy required to build it (see eg Arvesen et al (2018))

AppendixB Example of Non-Decreasing Relation Between

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(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 decreasingrelation

For this I use the formulas for EROI and price given byHerendeen (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

AppendixC Complementary Results

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 ofthe 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

12

embodiedqual adjt = E S

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

(C1)

AppendixD Proof of Proposition 1

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

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

function 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 and

let(

i0 j0)

isin J1nK2 so that without loss of generality x =

ai0 j0 (i) From its definition by the Leibniz formula the deter-minant 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 allpermutations of n elements In this linear combination eachterm is a product containing x at most once it is thus a li-near function of x (ii) A minor being the determinant of asubmatrix of A we know from (i) that it is a linear functionof x (which reduces to a constant for submatrices that do notcontain x) Each coefficient of the adjugate of A is (plus orminus) 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 function P IminusAet such that

(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

R(x) where δi j is the Kronecker delta As a linear combination ofcompositions of 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 defini-tion we haveεt =

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 have

pt =αQ(x)+γR(x)

R(x) =αεt +γ

13

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

14

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

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 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

14

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

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

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