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Policy Papers

Evolution of EROIs of Electricity Until 2050 estimation Using the Input-Output Model

THEMIS

Adrien FABRE

PP 2018-09

Suggested citation AFabre (2018) Evolution of EROIs of Electricity Until 2050 estimation Using the Input-Output

Model THEMIS FAERE Policy Paper 2018-09

wwwfaerefr

Evolution of EROIs of Electricity Until 2050Estimation Using the Input-Output Model THEMIS

Adrien Fabre1 PRELIMINARY VERSION

Abstract

The EROI ndashfor Energy Returned On Investedndash of an energy technology measures its ability to provide energy efficientlyPrevious studies draw a link between the affluence of a society and the EROI of its energy system and show that EROIs ofrenewables are lower than those of conventional fossil fuels Logically concerns have been expressed that system-wide EROImay decrease during a renewable energy transition First I explain theoretically that the EROIs of renewables themselves couldthen decrease as energy-efficient fossil fuels would be replaced by less energy-efficient renewables in the chain of productionThen using the multiregional input-output model THEMIS I estimate the evolution of EROIs and prices of electric technologiesfrom 2010 to 2050 for the baseline and the Blue Map scenarios of the International Energy Agency and for the 100 renewableGreenpeacersquos electricity [r]evolution scenario Global EROI of electricity is predicted to remain quite stable going from 8 in 2010to 6 or 7 in 2050 depending on the scenario Finally I study the economic implication of a declining EROI through its relationwith price I show that in theory both quantities can decrease at the same time This suggests that the inverse relation foundempirically represents an average tendency which should not overshadow the high unexplained variability and the theoreticalfinding that ldquoanything can happenrdquo

Keywords EROI input-output THEMIS MRIO sustainability energy transition

Acknowledgments I am grateful to Thomas Pregger and histeam at DLR for graciously providing me the tables of futureenergy consumption in the Greenpeace [R]evolution scenar-ios I am indebted to Thomas Gibon Anders Aversen and theNorwegian University of Science and Technology (NTNU) forproviding me the data of THEMIS and helping me using itI am thankful Arjan de Koning for answering my questionsrelated to Cecilia 2050 I hail the work of Konstantin Stadleron pymrio an invaluable open-source python library whendealing with Multi-Regional Input-Output Tables (MRIO) Iam grateful to Cyril Franccedilois for his thoughtful comments onthis project I am thankful to the Isterre for providing me anoffice at Grenoble to conduct this research I am grateful toOlivier Vidal and Mouez Fodha for their support

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

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

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

1 Introduction

As the harmful impacts of climate change call for a promptenergy transition away from fossil fuels mdashnot to mention theirdepletion that shall ultimately make this transition unavoid-able concerns have been expressed that in a decarbonizedenergy system the lower efficiency of renewable energy mightnot allow to sustain advanced standards of living [26 34]2 Wemeasure the energy efficiency of a technology or energy sys-tem using the Energy Returned On Invested (EROI) which isthe ratio between the energy it delivers throughout its lifetimeand the energy required to build operate and dismantle it Aminimal requirement for a technology or energy system to beenergetically sustainable is to have an EROI above 1 meaningthat it provides more energy than it requires

One issue to assess future energy systems is that the futureEROI of a given technology cannot be readily deduced fromcurrent estimates Indeed as King [22] remarked the EROI ofa technology is not intrinsic but depends on the whole tech-nological structure of the economy To see this let us sup-pose that the plants where solar panels are built employedrenewable electricity instead of electricity from coal as theirsources of energy Then provided that the EROI of renew-able electricity is lower than that from fossils the energy re-quired to build solar panels will increase and their EROI will

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

decrease Surprisingly it seems that no study has aimed atestimating future EROIs while taking this system dependencyinto account although some have called for such studies [4]3

Yet granted 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 motivations 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 Dale [9] reviews all EROI estimates until 2010 while Hallet al [16] aggregate the estimates of the literature in a meta-analysis I choose to present the results of Weiszligbach et al[35] (see Figure 1) because they compute the EROIs of differ-ent technologies in a comparable manner In addition thebuffered EROIs of Weiszligbach et al [35] take into account thesupplementary capacity grid and storage required for the de-ployment of renewable technologies which yields lower butpresumably more accurate estimates for their EROIs As an-ticipated the EROIs of renewable electricity sectors they findare significantly lower than those of electricity from fossil fu-els except for hydro

Figure 1 Estimates of EROIs of different electricity technologies fromWeiszligbach et al [35] where supplementary capacity and storage required forthe deployment of these technologies is accounted for

Furthermore some authors argue that the value of EROI isof primary relevance as they draw a link between the system-wide EROI and affluence of a society [15 25 26 11] Here ishow Hall [14] summarizes the 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 [15] examined theEROI required to actually run a truck and foundthat if the energy included was enough to build

3Admittedly King [22] provided a numerical application of the system de-pendency of EROI (see his Table 4) but his computations had a purely illus-trative purpose and his input values were not supposed to be accurate thisis why he did not even comment on the result that the EROI fell incidentallybelow 1 in his 100 renewable mix

and maintain the truck and the roads and bridgesrequired to use it (ie depreciation) one wouldneed at least a 31 EROI at the wellhead Now ifyou wanted to put something in the truck saysome grain and deliver it that would require 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 logicHallrsquos argument relies on two questionable assumptions thatfactors of production (and especially the labor force) are usedat their full capacity and that technical and organizationalprogress will not be sufficient to sustain current level of pros-perity with significantly less labor (or other factors of produc-tion in limited supply) In rejection of these assumptionsone can imagine a sustained level of prosperity with a lowersystem-wide EROI provided that a higher share of factors ofproduction be devoted to the energy sector That being saidgiven that current system-wide EROI is already declining dueto the decline in fossil fuels quality [8 31 7] 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 stay largelyabove 1 and not decrease too much for prosperous standardsof living to be sustained Admittedly a system-wide EROIclose to the theoretical lower bound of 1 might fuel an indus-trial civilization in the very long run as long as the resourcesrequired for the massive deployment of energy systems areavailable (this is not guaranteed as human labor land andmaterials are in limited supply and may be required for otheruse) However in the medium term and with more sensiblescenarios a diminishing EROI would probably imply that asubstantial share of the labor force will shift their occupationto the energy sector

In view of the potential implications of a declining EROIthis paper provides an assessment of the EROI of differentelectricity technologies in various prospective scenarios whichincludes a 100 renewable electricity system To this endI employ input-output analysis and I rely on a prospectiveseries of multi-regional Input-Output Tables (IOT) THEMIS[13] which models two scenarios from the International En-ergy Agency [20] Baseline and Blue Map Then I furthermodify THEMISrsquo IOTs to embed two decarbonized scenarioof power generation Greenpeacersquos Energy [R]evolution (ER)and Advanced Energy [R]evolution (ADV) [33] Although Pehlet al [30] and Arvesen et al [2] computed energy require-ments similar to mine in some respects this work is the firstto estimate EROIs in a scenario with 100 renewable electric-ity (the ADV scenario)

Then I analyze the economic implications of a declin-ing EROI through its relation with price Admittedly previ-

2

ous studies suggest an inverse relation between EROIs andenergy prices and such an average relation is retrieved em-pirically using prices observed and predicted from THEMISYet such a simple rule does not withstand a careful theoret-ical analysis Indeed while explaining to what extent EROIand price are related I will show that they do not necessarilymove in opposite directions This calls for taking prices pre-dictions from input-output analysis with more caution thanEROI estimates because IOT is better suited to handle physi-cal notions than economic ones Finally the economic anal-ysis weakens the view that a decrease in EROI would neces-sarily lead to a surge in energy expenditures and hence to acontraction of GDP

Section 2 explains theoretically why the EROI of a tech-nology is not an intrinsic property section 3 presents the method-ology and the results section 4 studies the implications of de-clining EROIs on prices and GDP section 5 concludes

2 The EROI of a Technology Is Not Intrinsic

21 A Simple Model With A Unique Energy Technology

The element ai j of the technology matrix A representsthe quantity of input i required to produce one unit of outputj Below is an illustrative technology matrix with three inputs(and the same three outputs) an energy technology materi-als and energy me denotes the quantity of materials (m) re-quired to produce one unit of energy technology (e) and thisnotation extends naturally to all elements of A The numericalvalues of the coefficients have a purely pedagogical purposeand have been arbitrarily chosen taking other figures wouldnot change qualitatively the results

A =

0 0 1me mm 0Ee Em 0

=

0 0 1me 02 001 05 0

energy technomaterials

energy

The system-wide EROI or Energy Returned On Investedis the ratio between the energy delivered by the system andthe energy required to build operate maintain and disman-tle it In other words it is the inverse of the amount of energyrequired to produce one unit of energy when the series of allembodied inputs are taken into account

The embodied inputs x required for a final demand y canbe calculated using the well known formula [27 10 28]

x(

y)

= (I minus A)minus1middot y

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 x

1TE 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

)

After some calculations we find

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

Ee (mm minus1)minusEmme

=072minus05me

008+05me

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 devel-oping 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 requir-ing no energy to access them) Such a world is physically im-possible

22 A Simple Model With A Mix of Two 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 numbers

3

are completely made up but they respect the fact that PV ismore material intensive than gas [18] Here is our new tech-nology matrix where p represents the share of PV in the en-ergy (or electricity) mix

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 we obtain

EROI =067minus03p

013+03p

This corresponds to the system-wide EROI But now thatwe have two technologies we can compute the EROI of eachof them4

EROIPV = 1558minus0698p

EROIg as = 5154minus2308p

Logically the EROI of PV is lower as compared to gas be-cause of its higher material intensity But it is worth noticingthat both EROIs depend on the energy mix p the EROI of atechnology is not an intrinsic property Indeed it dependson the whole economic system or more precisely of all tech-nologies used in their chain of production5 Here the higherthe share of PV in the mix the more the lower EROI of PV con-taminates each technology and the lower the EROI of bothtechnologies

One can see on Figure 3 that for highest penetration of PVthe EROI falls below unity In other words a renewable energymix with 100 PV is not sustainable in this example Evenmore worryingly if one computes the EROI of PV in an en-ergy mix relying mostly on gas one would find a high-enoughEROI for PV (meaning above 1) Hence one cannot concludethat a technology is sufficiently efficient (or sustainable) justby computing its EROI in the current energy mix Yet EROIscomputations have always been done from actual data of oureconomy and could falsely represent the efficiencies of en-ergy technologies in another energy mix say a 100 renew-able one This uncertainty concerning the sustainability of adecarbonized energy system motivates the core of this paperthe estimation of EROIs after a global energy transition

4Similarly 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

5Chain of production recursive or embodied inputs are synonyms theiranalysis 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

3 Estimation of Current and Future EROIs Using THEMIS

31 Setting and Data

Different notions of EROIs have been used in the litera-ture and some papers clarify them all (eg 5 29) The mostrelevant notion for this research is defined by Brandt amp Dale[5] as the Gross Energy Ratio (GER) and called by King [22] thenet external energy ratio6 The GER measures the ratio of en-ergy delivered over energy embodied in inputs net of the en-ergy of the fuels used in the process Thus for example thedenominator of the GER does not take into account the en-ergy provided by gas in a gas powered plant The term ldquogrossrdquois used because all energy output is taken into account onthe contrary Net Energy Ratios subtract from the numeratorall ldquoself-userdquo output that is used in the pathway of produc-tion of the technology7 A related indicator that is sometimesused to compute EROI as it is already included in many input-output databases is the Cumulated Energy Demand (CED) Ido not use it because Arvesen amp Hertwich [1] have shown thatit is erroneous to use the CED directly for EROI computationswithout making adjustments

In most cases EROIs (or energy ratios) are defined us-ing quantities of primary energy However I adopt a differ-ent approach in this paper and use only secondary energiesin my computations Indeed as Arvesen amp Hertwich [1] putit ldquoEROI does not need to measure primary energy per sethe crucial point is to measure energy diverted from societyin a unit of equivalencerdquo Also the choice of secondary en-ergy carriers is consistent with an energy system relying on

6The terminologies of these two papers are not compatible I followBrandt amp Dale [5] as their paper aims at harmonising the terminology

7For King rsquogrossrsquo energy is the total energy diverted from Nature whilersquonetrsquo is the output of energy from the technology what Brandt and Dale callrsquogrossrsquo Furthermore King would qualify rsquoexternalrsquo any notion that subtractdirect fuel inputs from the denominator (rather than just subtracting the out-put) while Brandt and Dale always take this as a base case and employ rsquoex-ternalrsquo when indirect self-use inputs are also subtracted it mirrors their no-tion of rsquonetrsquo for the denominator

4

renewable electricity while for such systems the definition ofprimary energy is not harmonized and this can lead to incon-sistencies Frischknecht et al [12] spot for example a factor6 between the cumulative (primary) energy demand for solarphotovoltaic computed according to different methods Al-though the sectors bringing energy are not the same in thetwo approaches (the primary approach uses crude oil whenthe secondary approaches uses gasoline for example) bothapproaches are equally valid

Furthermore practitioners often use a factor of conver-sion (around 3) to account for the higher quality of electric-ity as compared to fossil fuels I follow the recommendationof Murphy et al [29] by undertaking my computations with-out and with a quality-adjustment factor of 26 However Iprefer not to bring to the fore the quality-adjusted computa-tions provided in Appendix C and I focus instead on non-quality adjusted EROIs The reason for this is that the factorof conversion is not well established it represents the inverseof the yield of a thermal power station (about 38) but 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 disproportion-ately used for these applications for which they have a higheryield therefore the difference in quality between fossils andelectricity may be smaller than usually assumed

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

t Let usrecall that y is the vector of final demand given by the sce-nario and A is the technology matrix (or input-output table)E S is the 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 t8

supplyt = E Smiddot yt

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

⊙1secondary is the vector of unitarysecondary energy supply The main term at the denominatorof the GER is the secondary energy embodied in inputs netof the energy supplied by the technology

net secondary embodiedt = E S⊙1secondarymiddot

(

(I minus A)minus1middot yt minus yt

)

To this term we also need to subtract the energy suppliedby the secondary fuels in the last step of the process9 (logi-cally this term is nil for renewables) Indeed such energy is

8In practice y is obtained from the scenario of energy demand from theIEA

yt =(

demandt ⊘ES

)

⊙1t

9Technically we would need to subtract secondary fuels to all similar stepsin the embodied inputs for each embodied input that uses a thermal powerplant However I neglect these terms as this would complicate the presenta-tion for no significant difference in the result a variation in the value of fuel

input below 01 for all technologies considered

not used to build or maintain the energy system rather it isan energy transformed and delivered by the energy technol-ogy

fuel inputt = E S⊙1secondary fuel middot A middot yt

Finally we have

GER2ndt =

supplyt

net secondary embodiedt minus fuel inputt

I apply these formulas to the IOTs (ie technology ma-trices A) and the vectors of unitary energy supply E S fromTHEMIS [13] THEMIS contains hybrid input-output tablesprecise data on electricity units (the foreground) is completedwith data on other sectors that originates from life cycle in-ventories and national accounts (the background) Gibon et al[13] have compiled various life cycle inventories into the 609sectors of the foreground The background contains data inphysical units for 4087 sectors from the life cycle inventoryecoinvent and data in monetary units for 203 sectors fromthe input-output database Exiobase 2 [36] The 44 Exiobaseregions are aggregated into 9 macro-regions that coincideswith those of the International Energy Agency (IEA) so thatthe number of rows and columns in each IOT is 9 times thenumber of sectors 44046 Starting from data of the 2010 IOTthe 2030 and 2050 IOTs of THEMIS embed expected techno-logical efficiency improvements of key sectors Furthermoreit is worth noting that THEMIS IOTs are constructed as if thewhole economy were at a steady-state contrarily to nationalaccounts which give the flows between sectors for a givenyear This matches perfectly our purpose because there is noneed to adjust the EROI computations for the growth of somesector or for the lifetimes of some technologies

The two scenarios native in THEMIS are the baseline (BL)and the Blue Map (BM) scenarios of the IEA [20] While theformer posits an almost constant electricity mix the latter iscompatible with a 50 probability to contain the global meantemperature anomaly to +2degC in 2100 As Blue Map still re-lies at 30 on fossil fuels based electricity in 2050 mdashincluding17 with Carbon Capture and Storage (CCS) it does not al-low to assess more decarbonized scenarios Hence I com-bined with THEMIS the scenarios from Greenpeacersquos Energy[R]evolution report [33] Greenpeace proposes a business asusual scenario (REF) close to baseline as well as two scenar-ios compatible with the 2degC target Both exclude CCS andphase out from nuclear between 2012 and 205010 The firstGreenpeace scenario Energy [R]evolution (ER) comprises 93

10The 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 model Berrillet al [3] minimize the cost of European electricity generation under differentcarbon prices Interestingly an outcome of the model was to phase nuclearout but to select coal with CCS This indicates that the choices of Greenpeacewere not solely motivated by a minimization of costs but also by expert judg-ment and ethical considerations

5

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

of electricity from renewable sources in 2050 while the sec-ond one Advanced Energy [R]evolution (ADV) attains 100renewable As the difference is small between these two sce-narios I focus on the 100 renewable one I describe mymethodology for embedding the regional electricity mixes ofGreenpeacersquos scenarios into THEMIS in Appendix A

In the literature most EROIs estimations follow a bottom-up approach that use data from life cycle inventories Bottom-up studies describe in details the power facilities and the mostdirect inputs to the energy technologies mdashthe foreground butthey do not cover the background indirect inputs such asclerical work or RampD On the contrary the input-output methodallows to encompass all embodied inputs exhaustively Asa consequence of this more comprehensive account of em-bodied energy than usual we expect estimates of EROIs lowerthan the average of the literature That being said it is not aconcern if our estimates are not directly comparable to thoseof the literature as we are mainly interested in comparingthem internally among the different years and scenarios andto scrutinize whether they vary substantially or not

Because renewable sources are intermittent and dispersedthe capacity grid extension and storage they require do notincrease linearly with the electricity delivered Hence as Green-peace scenarios are not native in THEMIS they need furtheradjustments to account for these non-linearities I explain inAppendix A how the need for surcapacity is addressed Con-cerning transmission and storage however the requirementsare not given by the Greenpeace report [33] so they have notbeen taken into account Even if the report does not pre-cise any plan relative to storage hydrogen produced from re-newables seems to play a substantial role in Greenpeace sce-narios as its share in the electricity mix is 5 in ADV 2050However as the sector rsquoElectricity from hydrogenrsquo is absentfrom THEMIS hydrogen has been excluded from this anal-ysis These limitations should be addressed in future worktogether with the study of an energy transition in the trans-portation sector (which also relies on hydrogen) Meanwhileother references can provide information on orders of mag-

nitude of storage and transmission [3 24 32] Applying thesame optimization model that is used in the report Scholzet al [32] show that the cost of storage and transmission com-bined is 46 of total cost in a business-as-usual scenario and106 in a 100 renewable one The adjustment needed forthe cost around 6 gives a rough estimate of the upwardbias of unadjusted EROI estimates (see section 42 on the re-lation between price and EROI)

Finally data for Concentrated Solar Panels (CSP) had tobe adjusted because the original data mistakenly containedan energy supplied by unit of solar CSP of 0 (leading to ab-normally low EROIs around 2) Backed by Thomas Giboncore developer of THEMIS I corrected this error by settingthe unitary energy supplied for solar CSP in all regions to itsvalue in OECD North America (still letting the value dependon the scenario and the year)

32 Main Results

Main results are shown in Figure 4 and in Table 1 whilecomplementary results can be found in Appendix C notablyfor quality-adjusted EROIs Some EROIs are missing becausenot all technologies already existed on an industrial scale in2010 and some technologies are discarded in the future bysome scenarios One can notice that as expected PV andwind panels have a lower EROI than electricity from fossil fu-els The EROIs of renewables decrease as anticipated in theprevious section However they remain largely above 1 sug-gesting that renewables are truly sustainable energetically Thesystem-wide EROI for the entire electricity sector is given atthe bottom line It is currently 80 it decreases slightly until74plusmn02 in 2030 and 2050 in both IEATHEMIS scenarios Un-surprisingly the decrease is a little more pronounced in thescenario with 100 renewable electricity at 69 in 2030 and58 in 2050

While EROIs of renewable technologies are lower in sce-narios with higher shares of renewable which was expectedone may be surprised that global EROI is slightly lower in theBaseline scenario (72 in 2050) than in Blue Map (76) driven

6

Table 1 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 0 ndash 0 ndash 0 46 0 4 001 ndash 0 ndash 0biomassampWaste 113 001 62 002 58 003 55 006 52 005 52 005 46 005

ocean 55 0 24 0 29 0 37 0 58 0 48 001 49 003geothermal 53 0 51 001 5 001 52 001 54 002 38 003 39 007

solar CSP 215 0 88 0 91 001 82 002 79 006 92 007 78 022solar PV 92 0 74 001 71 001 63 002 6 006 54 014 47 021

wind offshore 93 0 109 001 105 001 76 003 62 004 65 004 64 01wind onshore 94 001 92 004 8 004 7 008 73 008 71 017 58 024

hydro 131 016 118 014 118 012 127 018 131 014 11 013 109 008nuclear 104 014 72 011 7 01 73 019 74 024 83 002 ndash 0

gas w CCS ndash 0 ndash 0 74 0 79 001 91 005 ndash 0 ndash 0coal w CCS ndash 0 ndash 0 61 0 71 005 71 012 ndash 0 ndash 0

oil 82 006 95 002 96 001 94 003 73 001 99 001 ndash 0gas 137 021 148 021 146 023 171 014 196 011 164 018 ndash 0coal 126 042 113 045 113 045 114 018 124 001 103 016 115 0

Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404

by lower EROIs for gas (146 in BL 2050 vs 196 in BM 2050)and coal (113 vs 124) Although available in the on-line code(or on demand) regional estimates are not reported here Theyshow that EROIs of renewables gas and coal are lower in BMas compared to BL in OECD Europe and OECD Pacific butare almost all higher in the other regions Unsurprisingly thesign of the difference between both scenarios is the same forrenewables and hydrocarbon in almost all cases suggestingthat the EROIs of gas and coal drive those of renewables Hencethe higher EROIs of BM are probably due to the early shut-down of the least efficient thermal power plant in this sce-nario Another unexpected result is the slight decrease of EROIsin the BL scenario from 8 in 2010 to 72 in 2050 The ex-amination of regional estimates reveals that this decrease isdriven by the increasing share of regions with lower EROIs inthe global mix the EROIs in each region remaining quite sta-ble

4 Implications of a Decreasing EROI on Prices and GDP

The forecast of declining EROIs made in the previous sec-tion calls for an assessment of its economic implications Themain channel through which a decrease in EROI could affectthe economy is arguably a rise in energy price (and correla-tively in energy expenditures) In this section I review the lit-erature on the relation between EROI and the price of energyestimate it empirically and extend a result from Herendeen[17] to characterize this relation Although an inverse relationholds in special cases and has some explanatory power on ob-servations the theoretical analysis shows that ldquoanything canhappenrdquo This theoretical result contrasts with the view thatdecreasing a EROI necessarily leads to a recession

41 Inverse Relation Proposed in First Studies

King amp Hall [23] point both theoretically and empiricallythat the price of a unit of energy pt and the EROI of a tech-nology t are inversely related Defining the monetary return

on investment MROI (ie the financial yield $out$investment

) theyderive the formula

pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)

Heun amp de Wit [19] find an equivalent formula They des-ignate MROI as the mark-up mt consider production costs

per gross output ct =$investmentEout+Ein

and use their own notion ofEROIEROIH

t =Eout+Ein

Ein= EROIt +1 so that equation (1) rewrites

pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt

The problem with these formulas is that all variables movetogether when EROI varies so does the cost of productionso that we cannot predict the future price taking this cost asfixed Heun amp de Wit [19] acknowledge this and thus studythe empirical link between EROI and price

42 Empirical Relation Between EROI and Price

Using US data on oil and EROI from [6] Heun amp de Wit[19] regress pt on the EROI11 They obtain a good fit even intheir simplest regression (R2

= 08) and find

11Although 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 more standard notion ofEROI according to the source of their data Cleveland [6]

7

Table 2 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

poil =β0 middotEROIminus14oil

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 such ldquoinverse fitrdquo with their ldquolog-log fitrdquo To undertake this comparison I run these two regres-sions using all estimates of EROI computed using THEMISone for each combination of scenario year region and sec-tor To obtain the price corresponding to each EROI I use thevector v of value-added per unit of each sector provided byTHEMIS Indeed prices can be seen as emerging from value-added according to

p = v middot (I minus A)minus1

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

middot1s To the extent thatthe physical constituents and processes of a given technologywill not change in an unexpected way and as THEMIS modelstechnical progress but not behaviors nor general equilibriumeffects the prices forecast using the above formula seem lessreliable than the EROI estimates For this reason I report onlythe global average electricity prices of the main scenarios (seeTable 2) but I do not detail the substantial variations betweenregions or sectors12

Table 3 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 tothat of the inverse fit I compute it as the sum of squared er-rors between ldquoobservedrdquo prices and predicted prices (insteadof their respective logarithms) As all R2 are between 053 and057 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 [19] foroil (around minus05 as compared to minus14) both figures are closeto 1 At first sight empirical findings appear to confirm aninverse relation between price and EROI However Figure 5shows that a high share of the variance in price remains un-explained by EROI even more so for values of EROI aroundthe global averages of 6-8 where the fit is almost flat and theerrors substantial In addition theoretical analysis rejects theexistence of a mapping between price and EROI

43 The Case Against Any Simple Relation

Herendeen [17] shed new light on the theoretical relationby treating the question from its matrix form and introduc-ing the concept of value-added Herendeen showed how to

12The results are on-line and available on demand

Table 3 Regressions of price on EROI (both estimated using THEMIS)All coefficients are significant at the 1h level

Obs N SpecificationCoefficients

R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)

= a middot log(EROI)+bminus057 20 057

2010 104 minus044 19 057

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

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

express rigorously the price in function of the EROI when theeconomy is constituted of two sectors (energy and materials)and explained the limits of such exercise Hereafter I extendthe results of Herendeen to an arbitrary number of sectors nFirst I need to introduce one concept he uses energy inten-sity

To deliver one unit of energy technology t the productionmobilized is (I minus A)minus1

middot1t while the energy mobilized calledthe energy intensity of t writes εt = 1

TE middot (I minus A)minus1

middot1t whereE is the set of all energies13 εt is in fact the gross energy em-bodied in t ie the sum of the delivered and the net embod-ied energy Hence the EROI of t is a simple function of εt

EROIt =1

TEmiddot1t

1TEmiddot((IminusA)minus1minusI)middot1t

=1

εtminus1

In the following I show that the price a technology t is acertain function of the coefficients of A14 and that each coef-ficient of A can be expressed as a function of EROI Compos-ing two such functions we obtain that the price is inverselyrelated to EROI However the relation is not unique (as it de-pends on the coefficient of A chosen to make the connection)and the other parameters in the relation are not constant Thedemonstration starts with a lemma

13I assume here that the unit of an output of an energy sector e isin E henceof 1T

E is an energy unit like TWh

14More precisely a function field of a certain algebraic variety

8

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 func-

tion of x denoted P Ai j

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

of degree 1 which writes(

Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)

1lei jlenisin GLn (R) and let

(

i0 j0)

isin J1nK2

so that without loss of generality x = ai0 j0 (i) From its def-inition by the Leibniz formula the determinant of A writesdet (A) =

sum

σisinSnsgn(σ)

prodni=1 ai σ(i) In this linear combination

each term is a product containing x at most once it is thus alinear 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)

Proposition 1 (Generalization of 17) Assuming that all co-

efficients 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

Remark With the terminology of Heun amp de Wit [19] or Heren-

deen [17] the relation above would write pt =αEROIH

t

EROIHt minus1

+γ

with γ=βminusα This is because in their definition of EROI thenumerator is εt instead of 1

Proof From lemma 1 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)

D IminusA(

δi0 j0minusx)

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 definition we haveεt =

sum

eisinE

(

(I minus A)minus1)

et so that Q (x) = εt D IminusA(

δi0 j0 minus x)

As

P Q and x 7rarr D IminusA(

δi0 j0 minus x)

are linear and as εt varies withx it is easy to show that there are unique real numbers α andγ such that P (x) =αQ (x)+γD IminusA

(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(

δi0 j0minusx) we have

pt =αQ(x)+γD IminusA

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx) =αεt +γ

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

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

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

all depend on the coefficients of A and vary together whenA changes If there is only one type of energy (E = e) or ifvalue-added is equal for all types of energy (foralle isinE ve = v) v

does not depend on the coefficients of A anymore and we ob-tain a formula close to that of King amp Hall [23] pt =

ve

EROIt+ r

Still when the EROI varies because more than one coefficientof A changes r varies concomitantly and the EROI cannot beused as a sufficient statistic to infer the price For this rea-son one cannot identify empirically a linear relation betweenprice and the inverse of EROI without strong assumption onthe steadiness of A

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

Does this mean that EROI is unrelated to any economicconcept Fizaine amp Court [11] argue that there is a minimumEROI below which the US economy enters a recession Theyfirst show that energy expenditure Granger causes growth inthe US then determine a threshold of energy expenditure abovewhich the US enters in a recession and finally use a modi-fied version of equation (1) to relate this to a minimum non-recessionary EROI However they misleadingly replace the

inverse of the energy intensity of investment $i nvestment

Einby that

of the whole economy GDPEout

This prevents them from notic-ing that cost reductions in the energy production could com-pensate the effect of a decreasing EROI on prices As we haveseen EROI price and energy expenditure can all decreaseat the same time which undermines the idea that a reces-sion caused by a surge in energy expenditure is ineluctableas soon as EROI goes below some threshold In addition anenergy price increase should have an expansionary effect onnet exporters of energy at odds with the mechanism extrapo-lated by Fizaine and Court from the case of the United Stateswhich is historically a net energy importer Overall the anal-ysis of this section tempers the idea that EROI is relevant ineconomic issues and suggests that this notion is best suitedfor the physical study of energy systems

5 Concluding Remarks

This work includes a first attempt at estimating future EROIsin a decarbonized electricity system By examining a broad

9

range of scenarios it concludes that the system-wide EROI ofthe power sector should slightly decrease until 2050 from 8 to72 in a business-as-usual scenario or 58 in a scenario with100 renewable electricity As the EROI of each technologiesis expected to remain well above 1 our results restore con-fidence about the energetic sustainability of renewable elec-tricity which was questioned theoretically

Even though an inverse relationship between EROIs andenergy prices was found empirically a large share of pricevariability remains unexplained Furthermore a theoreticalanalysis of this relation showed that a declining EROI doesnot automatically imply increasing energy prices and doesnot necessarily lead 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 agriculture andindustry

References

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

ergy Policy 2015[2] A Arvesen G Luderer M Pehl B L Bodirsky amp E G Hertwich De-

riving life cycle assessment coefficients for application in integrated as-sessment modelling Environmental Modelling amp Software 2018

[3] P Berrill A Arvesen Y Scholz H C Gils amp E G Hertwich Environ-mental impacts of high penetration renewable energy scenarios for Eu-rope Environmental Research Letters 2016

[4] A R Brandt How Does Energy Resource Depletion Affect ProsperityMathematics of a Minimum Energy Return on Investment (EROI) Bio-

Physical Economics and Resource Quality 2017[5] A R Brandt amp M Dale A General Mathematical Framework for Cal-

culating Systems-Scale Efficiency of Energy Extraction and ConversionEnergy Return on Investment (EROI) and Other Energy Return RatiosEnergies 2011

[6] C J Cleveland Net energy from the extraction of oil and gas in theUnited States Energy 2005

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

Economics 2017[8] M Dale S Krumdieck amp P Bodger Net energy yield from production

of conventional oil Energy Policy 2011[9] M A J Dale Global Energy Modelling A Biophysical Approach

(GEMBA) 2010[10] Eurostat editor Eurostat Manual of supply use and input-output ta-

bles Amt fuumlr amtliche Veroumlffentlichungen der Europaumlischen Gemein-schaften Luxembourg 2008 edition edition 2008 ISBN 978-92-79-04735-0

[11] F Fizaine amp V Court Energy expenditure economic growth and theminimum EROI of society Energy Policy 2016

[12] R Frischknecht F Wyss S B Knoumlpfel T Luumltzkendorf amp M Balouk-tsi Cumulative energy demand in LCA the energy harvested approachThe International Journal of Life Cycle Assessment 2015

[13] T Gibon R Wood A Arvesen J D Bergesen S Suh amp E G Hertwich AMethodology for Integrated Multiregional Life Cycle Assessment Sce-narios under Large-Scale Technological Change Environmental Sci-

ence amp Technology 2015[14] C A S Hall Introduction to Special Issue on New Studies in EROI (En-

ergy Return on Investment) Sustainability 2011[15] C A S Hall S Balogh amp D J R Murphy What is the Minimum EROI

that a Sustainable Society Must Have Energies 2009[16] C A S Hall J G Lambert amp S B Balogh EROI of different fuels and

the implications for society Energy Policy 2014[17] R A Herendeen Connecting net energy with the price of energy and

other goods and services Ecological Economics 2015[18] E 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 as-sessment of electricity-supply scenarios confirms global environmen-tal benefit of low-carbon technologies Proceedings of the National

Academy of Sciences 2015[19] M K Heun amp M de Wit Energy return on (energy) invested (EROI) oil

prices and energy transitions Energy Policy 2012[20] IEA Energy Technology Perspectives 2010[21] T Junius amp J Oosterhaven The Solution of Updating or Regionalizing a

Matrix with both Positive and Negative Entries Economic Systems Re-

search 2003[22] C W King Matrix method for comparing system and individual energy

return ratios when considering an energy transition Energy 2014[23] C W King amp C A S Hall Relating Financial and Energy Return on In-

vestment Sustainability 2011[24] O Koskinen amp C Breyer Energy Storage in Global and Transcontinental

Energy Scenarios A Critical Review Energy Procedia 2016[25] J G Lambert amp G P Lambert Predicting the Psychological Response of

the American People to Oil Depletion and Declining Energy Return onInvestment (EROI) Sustainability 2011

[26] J G Lambert C A Hall S Balogh A Gupta amp M Arnold Energy EROIand quality of life Energy Policy 2014

[27] W Leontief Input-Output Economics Oxford University Press USA 2edition 1986 ISBN 978-0-19-503527-8

10

[28] R E Miller amp P D Blair Input-Output Analysis Foundations and Ex-

tensions Cambridge University Press 2009 ISBN 978-0-521-51713-3[29] D J Murphy C A S Hall M Dale amp C Cleveland Order from Chaos A

Preliminary Protocol for Determining the EROI of Fuels Sustainability2011

[30] M Pehl A Arvesen F Humpenoumlder A Popp E G Hertwich amp G Lud-erer Understanding future emissions from low-carbon power systemsby integration of life-cycle assessment and integrated energy mod-elling Nature Energy 2017

[31] A Poisson C Hall A Poisson amp C A S Hall Time Series EROI forCanadian Oil and Gas Energies 2013

[32] Y Scholz H C Gils amp R C Pietzcker Application of a high-detail en-ergy system model to derive power sector characteristics at high windand solar shares Energy Economics 2017

[33] S Teske T Pregger S Simon amp T Naegler Energy [R]evolution Tech-nical report Greenpeace 2015

[34] G Tverberg The ldquoWind and Solar Will Save Usrdquo Delusion 2017[35] D Weiszligbach G Ruprecht A Huke K Czerski S Gottlieb amp A Hus-

sein Energy intensities EROIs (energy returned on invested) and en-ergy payback times of electricity generating power plants Energy 2013

[36] R Wood K Stadler T Bulavskaya S Lutter S Giljum A de Kon-ing J Kuenen H Schuumltz J Acosta-Fernaacutendez A Usubiaga M SimasO Ivanova J Weinzettel J H Schmidt S Merciai amp A Tukker GlobalSustainability AccountingmdashDeveloping EXIOBASE for Multi-RegionalFootprint Analysis Sustainability 2014

Appendix A Updating a Matrix A To a New Given Mix

The technology matrices A for the IEA scenarios are read-ily available in THEMIS but these matrices have to be up-dated to the new electricity mix for the Greenpeace scenariosTo do this I exploit the fact that both THEMIS and Green-peace use the world regions of the IEA and I modify the elec-tricity input of each sector by the regional mix given by Green-peace The most accurate algorithm to update an input-outputmatrix is known as GRAS [21] Although I implemented thisalgorithm in pymrio I could not use it because this algorithmuses the new sums of rows and columns to balance the ma-trix and the vector of final demand y or the vector of pro-duction x is necessary to know them As THEMIS doesnrsquot in-clude such vectors I had to use a simpler method which re-lies on the assumption that the electricity mix of inputs is thesame across sectors for a given region Given the perfect sub-stitutability between electricity produced by different tech-nologies and the uniqueness of electric grids this assumptionseems 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 premultiplyE by a block diagonal matrix with R blocks of size T lowastT con-taining only ones (where R = 9 and T = 15 are the number ofTHEMIS regions and electricity sectors respectively) to ob-tain a matrix B Each row of B gives the total electricity froma given region r required to produced each output E tot

r andeach row E tot

r is replicated T times

B =

B1

BR

Br =

E totr

E totr

Next each row of B is multiplied by the share of a tech-nology t in the mix of the corresponding region which de-fines a matrix E Each coefficient Ei s of E gives the electricityfrom sector i required to make one unit of sector srsquo outputaccording to the new mix (by construction for all electricitysector j = i (t r ) the share of technology j in the regional mix

E j ssum

t Ei (t r )s is the same across all sectors s) Eventually I 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 sourcesfail to deliver energy the ratio of capacity (in GW) over pro-duction (in TWh) is somewhat higher in Greenpeace scenar-ios than in IEATHEMIS ones Thus I multiply each columnof an energy sector (representing all inputs required for oneunit of output of this sector) by the ratio of the capacity-over-production ratios of Greenpeace and IEATHEMIS Doing sorelies on the fact that the energy required to operate a powerplant is negligible in front of the energy required to build it(see eg Arvesen et al [2])

Appendix B Example of Non-Decreasing Relation Between

EROI and Price

Herendeen [17] proposes a calibration on US energy dataof his toy model with 2 sectors (materials and energy) whichyields as realistic results as a two-by-two model can yield Istart from a slightly modified version of his calibration (calledbase) in the sense that the figures are rounded and I showhow a deviation of two coefficients (in the new calibration)leads to an increase of both EROI and price of energy Thisproves that in general nothing can be said of the relation be-tween EROI and price not even that it is a decreasing relation

For this I use the formulas for EROI and price given byHerendeen [17] (where I convert the price to $gal using theconversion factor 1Btu = 114000gal)

EROI =1

Aee +Aem Ame

1minusAmm

p =ve (1minus Amm )+ vm Ame

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

11

Table B4 Example of sets of coefficients exhibiting a non-decreasing rela-tion between EROI and price in a two sectors model

base new

vm 05ve 5 middot10minus6

Aem 1700Ame 4 middot10minus6

Amm 05 06Aee 03 025

EROI 32 37price 15 16

Appendix C Complete Results

Results without quality adjustment for IEATHEMIS sce-narios are provided in section 32 those for Greenpeacersquos sce-narios are in Table C5 Quality-adjusted results follows in Ta-ble C6 (IEATHEMIS) and C7 (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

embodiedqual adjt = E S

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

12

Table C5 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


biomass w CCS ndash 0 ndash 0 ndash 0 ndash 0 ndash 0 ndash 0 ndash 0biomassampWaste 84 002 64 003 5 003 53 006 47 006 52 005 46 005




hydro 121 016 113 014 111 013 11 014 11 01 11 013 109 008nuclear 12 011 72 01 7 008 83 002 ndash 0 83 002 ndash 0

gas w CCS ndash 0 ndash 0 ndash 0 ndash 0 ndash 0 ndash 0 ndash 0coal w CCS ndash 0 ndash 0 ndash 0 ndash 0 ndash 0 ndash 0 ndash 0


Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404

Table C6 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 206 001 125 002 117 003 115 006 110 005

ocean 91 000 44 000 55 000 70 000 113 000geothermal 114 000 102 001 101 001 105 001 113 002

solar CSP 445 000 176 000 181 001 172 002 169 006solar PV 174 000 147 001 144 001 132 002 128 006

wind offshore 195 000 211 001 203 001 153 003 130 004wind onshore 183 001 18 004 156 004 145 008 153 008

hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024

gas w CCS ndash 000 ndash 000 143 000 161 001 188 005coal w CCS ndash 000 ndash 000 116 000 137 005 142 012

oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13

Table C7 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










Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404

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

14

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 Technologies

Estimation of Current and Future EROIs Using THEMIS

Setting and Data

Main Results

Implications of a Decreasing EROI on Prices and GDP

Inverse Relation Proposed in First Studies

Empirical Relation Between EROI and Price

The 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

Complete Results

Evolution of EROIs of Electricity Until 2050Estimation Using the Input-Output Model THEMIS

Adrien Fabre1 PRELIMINARY VERSION

Abstract

The EROI ndashfor Energy Returned On Investedndash of an energy technology measures its ability to provide energy efficientlyPrevious studies draw a link between the affluence of a society and the EROI of its energy system and show that EROIs ofrenewables are lower than those of conventional fossil fuels Logically concerns have been expressed that system-wide EROImay decrease during a renewable energy transition First I explain theoretically that the EROIs of renewables themselves couldthen decrease as energy-efficient fossil fuels would be replaced by less energy-efficient renewables in the chain of productionThen using the multiregional input-output model THEMIS I estimate the evolution of EROIs and prices of electric technologiesfrom 2010 to 2050 for the baseline and the Blue Map scenarios of the International Energy Agency and for the 100 renewableGreenpeacersquos electricity [r]evolution scenario Global EROI of electricity is predicted to remain quite stable going from 8 in 2010to 6 or 7 in 2050 depending on the scenario Finally I study the economic implication of a declining EROI through its relationwith price I show that in theory both quantities can decrease at the same time This suggests that the inverse relation foundempirically represents an average tendency which should not overshadow the high unexplained variability and the theoreticalfinding that ldquoanything can happenrdquo

Keywords EROI input-output THEMIS MRIO sustainability energy transition

Acknowledgments I am grateful to Thomas Pregger and histeam at DLR for graciously providing me the tables of futureenergy consumption in the Greenpeace [R]evolution scenar-ios I am indebted to Thomas Gibon Anders Aversen and theNorwegian University of Science and Technology (NTNU) forproviding me the data of THEMIS and helping me using itI am thankful Arjan de Koning for answering my questionsrelated to Cecilia 2050 I hail the work of Konstantin Stadleron pymrio an invaluable open-source python library whendealing with Multi-Regional Input-Output Tables (MRIO) Iam grateful to Cyril Franccedilois for his thoughtful comments onthis project I am thankful to the Isterre for providing me anoffice at Grenoble to conduct this research I am grateful toOlivier Vidal and Mouez Fodha for their support

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

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

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

1 Introduction

As the harmful impacts of climate change call for a promptenergy transition away from fossil fuels mdashnot to mention theirdepletion that shall ultimately make this transition unavoid-able concerns have been expressed that in a decarbonizedenergy system the lower efficiency of renewable energy mightnot allow to sustain advanced standards of living [26 34]2 Wemeasure the energy efficiency of a technology or energy sys-tem using the Energy Returned On Invested (EROI) which isthe ratio between the energy it delivers throughout its lifetimeand the energy required to build operate and dismantle it Aminimal requirement for a technology or energy system to beenergetically sustainable is to have an EROI above 1 meaningthat it provides more energy than it requires

One issue to assess future energy systems is that the futureEROI of a given technology cannot be readily deduced fromcurrent estimates Indeed as King [22] remarked the EROI ofa technology is not intrinsic but depends on the whole tech-nological structure of the economy To see this let us sup-pose that the plants where solar panels are built employedrenewable electricity instead of electricity from coal as theirsources of energy Then provided that the EROI of renew-able electricity is lower than that from fossils the energy re-quired to build solar panels will increase and their EROI will

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












2






A =

0 0 1me mm 0Ee Em 0

=

0 0 1me 02 001 05 0


energy



x(

y)




001






net embodied energy

=1

1TEmiddot(


)




=072minus05me

008+05me









3


A =

0 0 0 p

0 0 0 1minusp


=

0 0 0 p

0 0 0 1minusp

07 01 02 001 01 05 0

PVgas

materialsenergy


EROI =067minus03p

013+03p










31 Setting and Data





4











(


)



yt =(

demandt ⊘ES

)

⊙1t





Finally we have

GER2ndt =

supplyt





5







32 Main Results



6



Year 2010 2030 2050 2030 2050 2030 2050









Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404








pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)




t =Eout+Ein


pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt




= 08) and find


7



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results












2






A =

0 0 1me mm 0Ee Em 0

=

0 0 1me 02 001 05 0


energy



x(

y)




001






net embodied energy

=1

1TEmiddot(


)




=072minus05me

008+05me









3


A =

0 0 0 p

0 0 0 1minusp


=

0 0 0 p

0 0 0 1minusp

07 01 02 001 01 05 0

PVgas

materialsenergy


EROI =067minus03p

013+03p










31 Setting and Data





4











(


)



yt =(

demandt ⊘ES

)

⊙1t





Finally we have

GER2ndt =

supplyt





5







32 Main Results



6



Year 2010 2030 2050 2030 2050 2030 2050









Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404








pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)




t =Eout+Ein


pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt




= 08) and find


7



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results






A =

0 0 1me mm 0Ee Em 0

=

0 0 1me 02 001 05 0


energy



x(

y)




001






net embodied energy

=1

1TEmiddot(


)




=072minus05me

008+05me









3


A =

0 0 0 p

0 0 0 1minusp


=

0 0 0 p

0 0 0 1minusp

07 01 02 001 01 05 0

PVgas

materialsenergy


EROI =067minus03p

013+03p










31 Setting and Data





4











(


)



yt =(

demandt ⊘ES

)

⊙1t





Finally we have

GER2ndt =

supplyt





5







32 Main Results



6



Year 2010 2030 2050 2030 2050 2030 2050









Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404








pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)




t =Eout+Ein


pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt




= 08) and find


7



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results


A =

0 0 0 p

0 0 0 1minusp


=

0 0 0 p

0 0 0 1minusp

07 01 02 001 01 05 0

PVgas

materialsenergy


EROI =067minus03p

013+03p










31 Setting and Data





4











(


)



yt =(

demandt ⊘ES

)

⊙1t





Finally we have

GER2ndt =

supplyt





5







32 Main Results



6



Year 2010 2030 2050 2030 2050 2030 2050









Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404








pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)




t =Eout+Ein


pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt




= 08) and find


7



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results











(


)



yt =(

demandt ⊘ES

)

⊙1t





Finally we have

GER2ndt =

supplyt





5







32 Main Results



6



Year 2010 2030 2050 2030 2050 2030 2050









Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404








pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)




t =Eout+Ein


pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt




= 08) and find


7



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results







32 Main Results



6



Year 2010 2030 2050 2030 2050 2030 2050









Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404








pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)




t =Eout+Ein


pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt




= 08) and find


7



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results



Year 2010 2030 2050 2030 2050 2030 2050









Total (PWha) 8 1976 74 3429 72 4597 75 2801 76 4022 69 3674 58 6404








pt =$out

Eout=

MROIt

EROItmiddot

$investment

Ein(1)




t =Eout+Ein


pt =mt

EROIHt minus1

middot$investment

Eout +Einmiddot

Eout +Ein

Ein=

mt middotct

1minus1EROIHt




= 08) and find


7



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results



price 27 28 30 30 28 30 32












R2a

a b

All 2079p =

aEROI

+b85 18 054

2010 104 71 20 053All 2079

log(

p)


2010 104 minus044 19 057








EROIt =1

TEmiddot1t


=1

εtminus1





8


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results


cient of A Then


D A





Aminus1)

i j =P A

i j(x)

D A (x)

Proof Let A =(

ai j

)


(

i0 j0)

isin J1nK2


sum

σisinSnsgn(σ)




adj(A)det(A) we

reckon(

Aminus1)

i j =P A

i j(x)

D A (x)





εt = 1+ 1EROIt

so that

exist(

αβ)

isinR2 pt =

α

EROIt+β



t

EROIHt minus1

+γ




(

(I minus A)minus1)

et =P IminusA

et

(

δi0 j0minusx)

D IminusA(

δi0 j0minusx)


sum

eisinE P IminusAet

(

δi0 j0 minus x)

and P (x) =sumn

i=1 vi P IminusAi t

(

δi0 j0 minus x)


sum

eisinE

(

(I minus A)minus1)


δi0 j0 minus x)

As


δi0 j0 minus x)


(

δi0 j0 minus x)

Finally observ-

ing that pt =sumn

i=1 vi

(

(I minus A)minus1)

i t =P (x)

D IminusA(



(

δi0 j0minusx)

D IminusA(



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


sum


pt = vεt +sum

inotinE

viωi t =v

EROIt+ r




ve

EROIt+ r





Einby that





9




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results




References

































10















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results















r andeach row E tot


B =

B1

BR

Br =

E totr

E totr


E j ssum






EROI and Price



EROI =1

Aee +Aem Ame

1minusAmm



11


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results


base new



Amm 05 06Aee 03 025




writes



12











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results











Total (PWha) 78 226 74 3626 71 5011 69 336 57 492 69 3674 58 6404



Year 2010 2030 2050 2030 2050






hydro 254 016 229 014 228 012 251 018 263 014nuclear 191 014 133 011 129 010 135 019 139 024


oil 127 006 154 002 158 001 154 003 118 001gas 223 021 244 021 241 023 289 014 334 011coal 20 042 18 045 18 045 183 018 196 001

Total (PWha) 152 1976 142 3429 14 4597 148 2801 153 4022

13











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results











Total (PWha) 149 226 143 3626 137 5011 138 336 118 492 137 3674 119 6404


14

Introduction





Setting and Data

Main Results





Concluding Remarks



Complete Results

Policy Papersfaere.fr/pub/PolicyPapers/Fabre_FAERE_PP2018.09.pdf · with price. I show that in...

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