What is the minimum energy return on energy invested (EROI) that a modern industrial society must have for its energy system for that society to survive? To allow a profitable business venture? To afford arts, culture, education, medical care? To grow? Is it the same as the minimum EROI that a fuel must have to make a meaningful contribution to a society's material well-being? And what is the price of energy at this minimum EROI? There has been remarkably little discussion of this issue in the last 50 years outside of our own previous papers on the subject [1] even though we believe that it might be a defining aspect of future societies. Many earlier authors, including anthropologist Leslie White [2], economist Kenneth Boulding [3], anthropologist/historian Joseph Tainter [4,5] and ecologist Howard Odum [6] have argued that for a society to have cultural, economic and educational richness it must have a large quantity of energy resources with sufficient net energy. That is to say complex societies need a high EROI built on a large primary energy base.

With the exception of the considerable discussion around whether corn-based ethanol is or is not a net energy yielder [7,8] there has been almost no contemporary discussion of the implications of changing EROI on industrial society. The lack of such studies seems curious as this will be a very important issue relating to our future, during which the mutual impacts of peak oil and declining EROI of fossil resources are likely to cause a very large overall decline in the net energy delivered to our industrial society. Furthermore, a lack of consistent and sufficient net energy comparisons among fossil fuels and renewable energy alternatives for liquid fuels and electricity prevents adequate understanding of our investments in alternative energy systems with different EROIs. This issue is exacerbated by the failure of the public at large, the media and even most of the scientific community to be able to see through the generally self-serving and shallowly analyzed pronouncements of various energy possibilities. For example, a wind farm and coal-fired power plant with equal EROI are not fully equal in terms of providing the same energy service until the wind farm is as dispatchable (on minute to daily time scales) as the coal plant. Additionally, a coal-fired power plant has more long-term uncertainty in EROI than a wind farm based upon the mining energy requirements. The wind farm long term certainty stems from the fact that the average wind speed will occur over the decadal life span of the turbines. Of course, environmental impacts and externalities (e.g., equivalent CO₂ emissions) also could play a major role, but we restrict the scope of this paper to the pure energy economic implications of changing EROI. If in the future environmental externalities are priced into the economic market, our general methodology would still hold, but will need to be updated.

There may already be very large impacts of declining EROI on our society, although this is difficult to untangle from peak oil impacts and the recession that started in 2007, which was at least partly due to increasing oil price [9,10]. Whatever the particular causative chain of events, a few recent trends appear: both oil and energy use have been declining in the United States, including a drop in total energy consumption from 99.3 quads in 2008 to 94.6 quads in 2009 [11,12]; global peak crude oil production-or something like it has occurred or is occurring (see Figure 1) [13,14] with many agreeing that world crude oil production peaked in 2005; the US's “Great Recession” officially lasted from December 2007 until June 2009 [15]; many financial entities are still in very rough shape after the financial crises that began in 2008, including many banks and Wall Street firms; the average inflation-corrected value of stocks has ceased increasing over the last decade [10], bonds have outperformed stocks over the last decade; and over the last two years most States and many municipalities have been forced to cut social and civil services to balance budgets. To what degree all of these effects are related to EROI is speculative, but worth speculating on.

The most explicit analysis of the EROI needed by society that we are aware of is Hall, Balogh and Murphy (2009) [1]., who made calculations on the energy required to refine, ship and transport fuels to their use destination, as well as to develop and maintain the infrastructure necessary to use them They used direct and indirect energy costs (EROI_stand) as recommended by Murphy and Hall in this special issue [16]. They concluded that the minimum EROI required for transportation fuels appeared to be in the vicinity of 3:1. That is to say, for every unit of energy consumed at the point of use, as in a car, at least three units of energy must be produced in order to (1) extract, refine and distribute the final fuel to the point of use in the form required by consumers, (2) manufacture the end-use machinery, and (3) build and maintain the infrastructure (i.e., roads and bridges) within which the fuel system operates. If the EROI was less than 3:1, then the fuel might be extracted but it could not be used to drive a transportation system. But this appears not to be the whole story.

No energy-producing entity (EPE, i.e., firm, National Oil Company, etc.) can produce a fuel over time (without subsidy) if it does not make a monetary profit, and it is not an EPE if it has a long-run EROI < 1:1. In other words, an EPE has the economic profit constraint of any other firm, so that it must sell its product (energy) for more than the monetary cost of the energy (direct and indirect) inputs required to produce it—plus it has to pay for the labor, profits and so on for the entire supply chain leading to the energy containing products it uses. These cost factors are normally accounted for in cash flow analyses of energy production businesses and processes, but are not always accounted for in EROI analyses. If we have a value for EROI that correlates to the same monetary costs of the full supply chain for energy production, then we should be able to estimate the cost of energy.

But the financial constraints are even stricter. For a firm to make a profit, it has to have some value of positive EROI because the energy flows associated with its costs (roughly 14–20 MJ per $2005 for the US oil and gas extraction industry, including direct and indirect costs [17,18]) are much less than the energy associated with a dollar's worth of its product. For example, if oil sells in 2005 for $61 per barrel (BBL) (containing 6,100 MJ), then each dollar gained by the oil company is associated with 100 MJ that has come out of the ground. If the EROI for the oil was 2:1, then the firm could not make a profit because for each 20 MJ invested in the business, at a cost of $1, only 40 MJ are output can be sold at a value of $0.40 [19]. Hence, at $61/BBL to make a profit a firm needs to have an EROI of at least 5:1, or alternatively if the price of oil were higher the firm could make a profit at a lower EROI. The conundrum is that as the price of oil goes up so does, historically, the price of everything else, at least eventually, including those things required to produce the oil. For example, cost for drilling US oil wells increased 270% from $150/ft in 2000 to $590/ft in 2007 (in $2007) [20] as the US first purchase price of oil increased 110% from $30/BBL to $63/BBL (in $2005) during the same time frame [21]. Over time the minimum EROI for a profit can be used as an investment guide for the company.

Our objective in this paper is to relate the EROI of energy produced by an EPE to the cost of energy and monetary return on investment (MROI) of that same firm, both theoretically and compared to historical empirical information for US energy sectors. This is not merely an academic exercise. As the EROI of our major fuels continues to decline [18,22] a major extension of this analysis is that economic profitability could stop long before EROI reaches 1:1. Our hypothesis for the analysis of this paper is that the biophysical characteristics of producing available energy, namely the EROI of the energy production process, dictate a limit on the price and profit margin for a firm to engage in energy production and exploitation. At least one other paper has addressed the important issue of relating EROI to price of energy, where the authors applied a statistical analysis of various fitted curves that are based upon a similar structure as we present [23]. Here, rather than optimize for a statistical correlation, we formulate an underlying basis for the relationship between price and EROI such that there is a physical basis for price and a framework for projecting future trends.

Our basic method was to develop a mathematical expression for the relation of the biophysical characteristic, EROI, of an exploitable energy resource to the economic conditions that makes the exploitation possible. We derive an equation that describes the general trends of certain parameters of interest, namely the EROI, the monetary return on investment (MROI), and the unit price of produced energy, p (e.g., $/BBL, $/MWh, etc.) sold in the market. At MROI = 1, the predicted price equals the producer price, or cost of production.

The definition for EROI is as shown in (1). EROI is the energy output (E_out) from an energy production system divided by the required energy inputs (E_in) to the system. Most EROI analysts calculate (1) without discounting future energy production versus energy produced and consumed today, and for simplicity, our analysis also assumes simple energy and cash flow accounting (i.e., we do not discount energy or money) [24].

Most analyses also imply that the relation for investments today are reflecting production today, whereas today's production is partly from yesterday's capital investments and today's capital investments are partly for tomorrow's production. The data from [18] used in this paper indicate that the ratio of indirect indirect E_in/direct E_in for US oil and gas has varied from less than 0.3 to over 2 in the years in which high oil prices induced large increases in exploration and drilling (see Figure 2).

We now deconstruct (1) into a form used to calculate results for this paper. However, for previous descriptions of the general framework for characterizing how to include different inputs and outputs in (1), see [25-29]. Note that both the numerator and denominator of (1) can be composed of multiple factors: M forms of energy outputs and N forms of energy inputs. For example, an analysis of a drilling operation producing oil, natural gas, and natural gas liquids must count the energy content of all three (e.g., M = 3) resources to calculate E_out. The same premise holds for calculating E_in. Relations for the energy outputs and inputs of an energy production system are shown in (2) and (3) as a function of energy intensity, e_i of production (or consumption), multiplied by the number of units of production (or consumption). Here, we assume e_i is expressed in units of energy divided by any other unit whether that by a physical quantity (e.g., tonnes), money (e.g., dollars), or otherwise. In (2), M is the number of output energy products, and m_i represents the unit production of the i^th energy product. In (3), N is the number of input products that have direct or indirect embodied energy, and n_i represents the unit consumption of the i^th input. Example units of energy production are barrels (BBLs) of oil, megawatt-hours (MWh) of electricity, thousand cubic feet (MCF) of natural gas, etc. An example calculation is E_out for oil production where the energy content of a barrel (BBL) of oil is approximately e= 6.1 GJ/BBL, and if m = 10 BBLs of oil are produced, then E_out = 61 GJ. For Equation (3), the i^th unit input can describe direct energy (e.g., a BBL of oil) or indirect energy (e.g., energy embodied in a ton of steel, hour of labor, etc.). See [25] and [30] for a full discussion of how to consider different energy inputs and outputs, including using energy quality factors, when calculating E_in and E_out.

Equation (3) should include both direct and indirect energy inputs and represents the common methodology for performing process-based and input-output based life cycle assessments (LCAs) [31]. Hence with proper data we can assess what part of the expenditure dollar went for direct energy and what part for the indirect energy that is responsible for the different energy/monetary ratios of inputs and products. For example, in an oil production system, the direct E_in calculated in (3) can be a summation of electricity (or better the fuel consumed during electricity generation) for running trailers, pumps, compressors, and computer equipment as well as diesel fuel consumed for operating trucks, pumps, and the drilling rig. However, it is insufficient to include only the direct energy inputs to capture all of the energy necessary for the full operation of the energy production system. EROI researchers additionally include measures for indirect energy inputs to consider energy inputs from operations outside of the energy producing operation itself. For example, oil derricks have towers made from steel, and one company may install and operate the drill, but another made the steel tower. Because the energy inputs required to make the steel are not performed on the site of the oil well, they are considered indirect energy and can be included in the analysis by knowing the energy required per unit or dollar of production (e.g., energy intensity e) and following (3). For example, in 2004 the average mass energy intensity of steel was e_st = 20,000 MJ/tonne [32]. Thus, to include the indirect energy inputs from steel in (3), n is number of tonnes of steel and e_i = e_st = 20,000 MJ/tonne. When physical units are not available analysts must use dollars of steel (e.g., n in units of $) and monetary energy intensity (e.g., e in units of MJ/$) for such dollars spent.

It is possible to estimate indirect E_in using nominal data from input-output (I-O) analyses of the entire economy. Examples of such analyses are those by Bullard, Herendeen, and Hannon in the 1970s [33], Costanza and Herendeen in the 1980s [34,35], and the somewhat less comprehensive or detailed but more recent Economic Input-Output LCA analyses by Carnegie Mellon [36]. These I-O analyses blend national-level economic and energy consumption data to analyze the impacts of complete economic sectors rather than individual technologies or processes. In using I-O analyses, the most aggregated value of e_i that characterizes energy consumption and economic cost is the economy-wide energy consumed for every one dollar of investment by the energy sector of interest (e.g., oil and gas extraction sector). If monetary e_i (e.g., in units of MJ/$) is not available for the sector or project of interest via an I-O analysis, then the overall monetary energy intensity, e, of the economy (e.g., state, country, world) can be used as the best proxy. However, investments of energy industries have higher than average energy intensity. Equation (4) represents energy inputs as a function of money invested and energy intensity of the investment, e_investment, in units of energy per money (e.g., MJ/$).

Alternatively, one can calculate e_investment using Equation (3) to calculate all energy inputs and dividing them by all money spent to purchase those inputs. For reconstructing the value of e_investment without I-O analyses, we can use (5).

To relate EROI to the e_investment, we substitute (2) and (4) into (1) to obtain (6), a working definition of EROI.

Thus, the higher the e_investment for energy business operations, the lower the EROI. Note that in the case of oil production, as the oil resources left to exploit get deeper, heavier or from more inhospitable areas, it is important to understand not only how much more direct energy (e.g., diesel fuel and electricity) is required for drilling deeper and pumping up the oil but also how much indirect energy (e.g., infrastructure, engineering, and planning) is required. If an oil resource primarily requires direct energy, this raises e_investment because fuels have high ratios of energy/$ by definition (e.g., if a fuel was sold for an energy/$ ratio below that of the average of the economy, then the firm selling the energy would be a net energy consumer and not a net energy producer). Therefore, as e_investment increases, this produces a further feedback on decreasing the EROI of the resource. Additionally, the steel, aluminum, and other heavy manufacturing materials that are required for new drilling and construction of power plants are also characterized by e_investment higher than economy average (but lower than for fuels), again creating a feedback for lowering the EROI per Equation (6).

We next create (7), where m_i are in physical units of produced energy and e_i are in units of as an analog to (6) to enable a relation between simple monetary return on investment, MROI and EROI.

From (7), we solve for the total money invested as: (8) $ investment = ∑ i = 1 M m i p i MROI

Substituting (8) and (5) into (6) and we obtain an equation that shows EROI as a function of important economic factors: (9a) EROI = ∑ i − 1 M m i e i ∑ i = 1 M m i p i ⋅ MROI e investment (9b) = ∑ i = 1 M m i e i ∑ i = 1 M m i p i ⋅ ∑ i = 1 N n i p i ∑ i = 1 N n i e i ⋅ MROI

We use Equation (9b) to explicitly solve EROI as a function of the individual components of e_investment. Assuming for simplicity that there is only one type of energy production (e.g., M = 1), we easily solve (9) for one output variable of interest as a function given values for all other variables. For example, useful relations are (10) and (11). Equation (10) specifies the requisite sales price, p, of a unit of energy production (e.g., $/BBL of oil) as a function of EROI and MROI, and (11) specifies the monetary return on investment as a function of EROI. In the following results section, we use (10) and (11) to demonstrate the current methodology.

The benefit of the relations described by (4–11) is that we have derived an equation with both EROI and MROI explicitly stated together. Previous research either defines EROI without relation to monetary profits, or derives EROI from economic data of a specific year, but still without a closed form function relating EROI and MROI. To properly use (10) and (11), one must make sure that the parameters all correspond to the same time frame and system boundaries or point in the supply chain of an energy production technology, business, or system. For example, if analyzing the price of oil from a specific field, then the inputs to (10) and (11) must be the expected MROI, EROI, and e_investment of production for that field only. If one is interested in the average price of oil from the entire United States oil and gas sector, then the inputs to (10) and (11) must relate to the entire sector. Thus, the structure of ((4)–(11)) should allow researchers to do both top-down economic sector analyses as well as bottom-up technology-specific analyses to analyze the entire energy supply chain. By reconstructing the top-down results from bottom-up techniques, better future energy projections may be possible. However, in practice, bottom-up process LCAs are more easily performed using E_in as defined in (3) because values for process-specific e_investment are generally not available.

In ((9)–(11)) the various factors are not independent of each other. Ideally, data and calculations for EROI can be made independent of economic inputs, and this is most plausible when considering direct energy inputs only in (3). In considering indirect energy inputs, however, (e.g., that energy required for producing steel used in oil well casing), often times only monetary data are available (e.g., money spent for purchasing steel), requiring a blend of available economic and energy intensity data (e.g., an aggregate value of e in units such as MJ/$) to estimate energy inputs. Additionally, when considering sector level analysis, economic data are generally all that are available. Thus, it is important to understand that EROI is not an independent function of e_investment as it appears to be considered in ((9)–(11)). For example, oil as refined diesel is a major input into drilling for oil (e.g., as fuel for drilling rigs). Thus, if the biophysical descriptor (i.e., EROI) decreases because of the need to consume more diesel in drilling to deeper oil resources, other input products (e.g. steel) can become more expensive in both money and energy if they depend upon oil for production and shipping. That is to say, as the price of oil gets higher, it can have a feedback making it more expensive to produce more oil. Additionally, EROI is inversely proportional to the energy intensity of investment in energy production while at the same time being proportional to the energy output per unit of production (e.g., BBLs of oil production at 6,100 MJ/BBL). By using (9), we can account for a situation in which the EPE pays a price for an energy resource input that is different than the price for which the EPE sells the same energy resource as an output. By breaking e_investment into a weighted sum of many investments as in (5) and (9), we can gain insight into the coupling of inputs from each sector or fuel (direct or indirect) upon EROI, and ultimately the price of energy required to make a given financial return. In practice such assessments often are very difficult because the energy companies (especially national oil companies) keep much of this information to themselves.

Also,Equations(10) and (11) show that as energy gets more expensive, partially characterized by decreasing energy intensity (e.g., energy per dollar) of investment in energy production, e_investment, then energy price increases at constant EROI. The counter-intuitive result from (10) is that as the energy intensity of investing in energy production increases, the price of energy necessary to make a constant profit decreases. The reason is that higher energy intensity purchases represent cheap energy inputs and the ability to make higher monetary returns for a given EROI.

To gain insight into our methods, we use Equation (10) to estimate results under representative historic economic conditions. We first use the example of US oil production and later repeat the analysis for natural gas and coal production. Our results indicate that Equations (9–(11)) act as broad but valid representations of the relations between EROI, MROI, and the stated technoeconomic factors.

Our equations derived in this paper appear to predict rather well the basic relations among profits, prices, and EROI This formulation, however, is not meant necessarily to predict short term prices, but rather characterize broad relations and the ways in which we believe that EROI drives large scale financial phenomenon and long-term energy investments. It is important to note that Equations (9–11) represent equilibrium conditions with no constraints on any required inputs. In reality there can be shortages in global oil supply or quickly increasing demand of infrastructure for oil and gas development (e.g., drilling rigs) that raise prices much faster (in time) than indicated by the theory of this paper. Thus, the theory of this paper can be viewed as describing a lower bound on price as it relates to EROI.

The data used in Guilford et al. (2011) [18] and Cleveland (2005) [26] do represent some dynamics in supply and demand with regard to oil production. Because the underlying data from the US Census of Mineral Industries is reported only every five years, there are few conclusions we can make regarding the rate at which the underlying EROI changes on annual or monthly time scales. By the method and demonstrations developed in this paper we have confirmed our major hypothesis that the biophysical characteristic of EROI is a major factor that can dictate the profit margin and price necessary for a firm to engage in energy production. The relations in Equations (9)–(11) illustrate that lower EROI energy systems have less potential profitability for their businesses.

Over the long run, any energy producing entity must produce both a monetary and energetic profit. In the terminology of this paper, this statement means that MROI > 1:1 and EROI > 1:1. The question remains how much greater than 1:1 EROI must be. Considering that the past calculations of US EROI of oil and gas estimate it to never have been less than 7 [18,22,26], we can infer that there is some value of EROI between 10 and 1 that oil becomes prohibitively expensive. As seen in Equation (10), as EROI decreases, price increases. By developing theoretical minimum EROI values for fuels and electricity, as in one of the current author's previous work [1], we can translate those critical EROI values into a price range. Conversely, we can look to translate critical price thresholds as feedback to inform derivation of minimum EROI values.

If a business is characterized by EROI < 1:1, then by definition it is an energy consuming business no matter what the profit of the company. Thus, the monetary investments of an energy business must consume less energy than its products provide. In terms of our nomenclature, this means that e_investment < e_product for an energy production business or sector. Considering the example in Section 3.1, the oil and gas extraction sector invested at e_investment =18.6 MJ/$2005 in 2007 to produce a product with energy intensity of e_oil = (6,100 MJ/BBL)/(63 $2005/BBL) = 97 MJ/$2005. Thus, based upon pure economic information, we can say that the oil and gas extraction sector multiplied the energy available to the economy by a factor of 5 (e.g. 97/18.6 = 5) times. Equivalently in 2007, for the natural gas case study presented in Section 3.2 the energy available to the economy was increased by a factor of 10.

Historically, EROI has been many multiples higher than MROI, but our derived relation itself does not necessarily point to the limit of profitability. Theoretically, firms can charge higher prices in an attempt to command their desired profitability, but there is a price at which consumers are unwilling to pay or that they will cut back on consumption. Additionally, the marginal, or lowest EROI, energy supply often dictates the overall market price (e.g., oil) such that producers with high EROI supplies and resources sell at a large profit. We do show that because EROI is a ratio, as it drops lower and lower, the necessary price (at constant profit) increases quickly in a nonlinear manner. That is to say at a constant profit (e.g., MROI =1.5) and e_investment =19 MJ/$2005, an EROI decrease from 5 to 2 (a 60% drop) has a much more dramatic absolute increase in price from $96/BBL to $240/BBL (a 150% increase), than a drop from 25 to 10 (a 60% drop) with an increase in price from $19/BBL to $48/BBL (also a 150% increase). Because EROI is a ratio, changes around low values are larger in the absolute sense than changes around high values. And because most consumers think linearly with budgets and incomes that do not quickly adjust to large absolute changes in oil price, small changes in EROI can quickly translate to budgetary difficulties for families, companies, and governments. This phenomenon of decreasing net energy might explain a lot of our present economic difficulties [18,22]. Thus, low EROI directly translates to high price, and because EROI has a physical basis for its derivation, it is an important method for double checking and forecasting future energy prices and profitability of energy businesses.

In future work, the relations derived in this paper set the stage for proper EROI and price comparisons of individual fossil and renewable energy businesses as well as the electricity sector as a whole. For example, by including the EROI of individual energy technologies, including the energy inputs for investments in electricity storage, transmission, and distribution systems, we can use physical-based modeling to assist in forecasting a future energy transition to renewables. Additionally, the presented relations provide a framework for incorporating EROI into larger economic systems models that can explore the feedbacks between the EROI and prices of different energy supplies.