We develop a method—based on linear algebra of interacting processes—to better understand the impact of energy resource productivity on societal prosperity. The first step toward a mathematical framework is to make more explicit what is meant by “prosperity”. An explicit definition is important because choosing a mathematical definition of prosperity will point to a specific means of estimation or modeling.

In this paper, we tentatively—and usefully, as will be shown—define prosperity as follows: a society is more prosperous if a larger fraction of the output from the economic system is free to put to use for discretionary uses. That is, a society is more wealthy when more of the produce of the economic system is not used simply to operate the basic economic system (i.e., inter-industry trade) but can be diverted to use by consumers as they see fit (i.e., final demand).

Many other reasonable measures of prosperity might exist. One might, for example, measure absolute material or energy output per capita (e.g., total kg or total MJ). We would counter, however, that what contributes to prosperity is not simply total economic output, but the fraction of that output that can be directed toward discretionary uses. As an illustration, imagine two societies where per-capita gross outputs from the chemicals sector are identical. If the first of these societies has extremely poor quality farmland and insect infestation, then the per-capita chemical industry output might be disproportionately dedicated to the agricultural sector to simply allow production of basic foodstuffs. Meanwhile, if the second society has a more bountiful set of farmland resources, the chemical sector outputs could be increasingly directed toward any manner of discretionary uses.

A prosperous society, therefore, is able to invest more of its produce (be it steel, glass, or high-skilled services) into discretionary activities. These might include advanced education, science, entertainment, temperature controlled living and working spaces, or discretionary travel. Obviously, such activities are possible only when productive sectors are efficient enough that some abundance is left over after the basic requirements of the subsistence economic processes are met.

How does this definition relate to energy supply? A simple example will illustrate the impacts of declining energy resource quality on prosperity. Figure 1 shows a “toy” two-sector economy with “energy” and “materials” sectors, with the magnitude of flows specified for an initial time period, called \(t_1\). The output functional units for each sector are 1 MJ of energy and 1 kg of materials, respectively. The energy sector self-consumes 10% of its output in processing and refining the energy source, and consumes 0.01 kg of material (e.g., steel) for every MJ of output. Similarly, the materials sector consumes 10 MJ of energy for every kg of material produced and requires 5% of its own output for operation.

Fig. 1 Illustrative two-sector model containing only an energy sector and a steel sector. These flows are for time period \(t_1\) before resource depletion occurs Full size image

Using the framework of Leontief et al., with notation from Miller and Blair (2009) we can write these flows as a transactions matrix A for \(t_1\) as:

(5)

We note that A is a “mixed units” matrix in which the first column represents inputs to the energy sector, and the second inputs to the materials sector. The rows, in contrast, represent so-called “inter-industry” outputs from each sector to each other sector (including itself). We can then construct the Leontief inverse:

(6)

Now, let us imagine that we desire 1 MJ of total energy output for final demand outside of this two sector economy (i.e., any activity whose outputs is not consumed by another productive sector). To support this final demand \(\mathbf f\) of energy of 1 unit (i.e., f T = [1,0]), not to be used by the steel sector, the total (gross) production required is:

(7)

Thus, the system must produce 1.26 MJ of total energy to allow final demand use of 1 MJ of energy. Similarly, for 1 kg of discretionary steel use the same procedure shows us that gross steel output must be 1.19 kg.

Let us suppose that in time period \(t_2\) energy resource depletion has lowered the average quality of oilfields being tapped in this simple economy. The oilfield must now be pumped harder and more wells must be drilled. For the sake of concreteness, let self-consumption of energy rise to 15% of gross output, and steel requirements double to 0.02 kg per MJ. The matrix in \(t_2\) then becomes:

(8)

and our scale vector of total demands is:

(9)

Note that both terms of s have changed. There are two ways to interpret the change that occurred. The first is that to support the use of one unit of energy for discretionary purposes, more total energy output is required. That is, gross energy output grows to 1.56 MJ. In this case there are two causes for this change: direct within-sector consumption increases due to increased pumping energy (i.e., \(A_{11}\) changes from 0.10 to 0.15), and increased indirect consumption due to increased steel requirements.

Another interpretation of the above changed s is that the fraction of energy sector gross production that is free to allocate to discretionary use drops. That is, for an energy sector of a given capacity for total gross output, less net output is available after inter-industry demands are met. We might define a ratio of the discretionary (final) energy use over the total energy use, f \(_1\)/s \(_1\), which declines from 0.79 MJ/MJ (or 1/1.26 ) in \(t_1\) to 0.64 MJ/MJ in \(t_2\).

This result points to a more general framework for estimating the impacts of resource depletion on prosperity. We will develop this framework below.

First, we must re-cast the definitions of matrix-based ERRs shown in previous work. We can say that:

$$\begin{aligned} \mathrm{NER} = \frac{\mathbf {f}_e}{\mathbf {s}_e-{\mathbf {f}}_e} \end{aligned}$$ (10)

where \(\mathbf f _e\) is the energy sector element of the demand vector (e subscript indicates the placeholder for energy sector vector element or energy sector column of array A). Or, otherwise expressed, final demand of energy divided by the energy consumed to supply that final demand. In contrast, the GER counts as output from the sector all consumed energy:

$$\begin{aligned} \mathrm{GER} = \frac{\mathbf {s}_e}{\mathbf {s}_e-\mathbf {f}_e} \end{aligned}$$ (11)

where subscript notation follows from above.

How might we measure discretionary use? Of course, the distinction between discretionary and non-discretionary (subsistence) consumption is always fuzzy. However, a useful corollary in IO models is given by intermediate (inter-industry) demand and final demand. If we make the assumption that our economic matrix represents primarily the “basic” or “subsistence” side of the economy, focused on supplying primary goods, then all other activity would be modeled as final demand. Final demand is what we want as consumers, and total demand (final plus intermediate demand) is what sectors must produce in total to supply our final demand. In the language of Miller and Blair, such a model has “exogenized” consumption that is discretionary. Any calls for discretionary goods would then be included in the final demand vector (Miller and Blair 2009). Alternatively, in IO terms we have made the system more “open” and less “closed” (Miller and Blair 2009).

A mathematical definition of prosperity can be defined using the fraction of production of a given resource that is “free” to be used in discretionary activities. Stated alternatively: for each unit of final demand that is desired for discretionary uses, how much additional requirement is there for production of the product simply to maintain the production system? For systems with high prosperity, a large fraction of a given resource (human labor, energy, materials) is free to allocate as we please. In systems that are closer to subsistence, a large fraction of the output of any product must be “plowed back” into powering industrial processes. A striking fact of a subsistence economy in comparison to the modern economy is that in a modern economy comparatively few of the hours we work, or very little of the material output we consume, are directly related to basic subsistence.

Thus, to generalize on the equation above, if we demand 1 unit of output as discretionary demand, how much total production must occur (intermediate + final demand)? This ratio represents the amount of each product that is effectively free to use in discretionary activities for a given amount of gross output. Because we define implicitly the system as open to all discretionary consumption, the desired discretionary output from the system is simply given by \(\mathbf f\) and the total requirements for each product are given by \(\mathbf s\) and, therefore, a measure of the efficiency of a sector i can be determined to be a ratio r:

$$\begin{aligned} \forall i \in 1\ldots n: {\mathbf {r}}_i = \frac{\mathbf {f}_i}{\mathbf {s}_i}, \end{aligned}$$ (12)

where n is the total number of productive sectors in the economy. We can imagine a threshold value defined as \(\theta _i\), which is a value of \(\mathbf r _i\) below which we cannot call a system prosperous. For example, we might require a threshold value \(\theta _i\) of 0.66 for all products, suggesting that a society cannot be considered prosperous if less than 66% of its productivity is free to consume in discretionary uses. As production of a given output becomes less efficient, this ratio \(\mathbf r\) of free material/energy/food/labor drops. We move toward a more subsistence economy, where more and more of our output is required to sustain the industrial system.

Note that inverting the production matrix \(\mathbf A\) results in the infinite series of interactions between all sectors. This means that energy productivity can be dragged down by drops in the efficiency of the labor sector, or changes in the efficiency of the materials production sector will affect all other sectors as well. It is not just rising or dropping energy extraction performance that affects productivity, but changes throughout the system.

To explore the dynamics of an economic system as energy resource productivity declines, we generate a template economy, and change the productivity of the energy sector to see how this effects the general level of prosperity. For ease of explanation, we work with a highly theoretical four-sector economy. This economy includes an energy sector, materials sector, food sector, and labor sector. We will show that this simple economy behaves in ways that are congruent with current theories about the impacts of energy resource productivity. This four-sector model has similarities with the five-sector capital, labor, energy, materials, and services (KLEMS) models used in some economic sub-disciples (Gullickson and Harper 1987).

A Template Order-of-Magnitude Economic System

To generate a working example below, we will work with a very simple four-sector model. Our aggregate sectors produce three generic physical products: energy, materials, and food with dimensionality of (MJ), (kg), and (kg), respectively. To include the effects of changes in labor intensity on energy sector outputs, we add a generic labor sector, which consumes the three physical products to supply hours of input labor (either physical or intellectual) to our modeled sectors (h). In each case, the intensity of inputs per unit of output in a sector is rounded to the nearest OM (power of 10).

One important common assumption across all sectors is the counting of labor hours. There are at least three ways of computing the number of hours required by each employee. Most narrowly, one might assume that each worker puts in \(\approx\)2000 h of labor per year. However, each worker needs to live all 8760 h of a year to perform \(\approx\)2000 yearly hours of labor. Looking more broadly, a worker may provide \(\approx\) 40 years of labor per 80 year lifetime (approximately ages 25–65) and thus lives two total life years for each year in the workforce. Thus, the ratio of total hours to productive hours is 8760*2/2000 = 8.76 h of life per hour of work.

We can also use macro-scale labor statistics from the United States to come to a similar figure. Working hours in 2014 were derived from BLS statistics (BLS 2015): 69.1% of the over-15 population was working on a given day, an average of 38.6 h per week (BLS 2015). The 16 years and older population in 2015 was 251.0 \(\times 10^6\) people (BLS 2015). Therefore, worker hours were 3.48 \(\times 10^{11}\), while total lived hours for the 320 million US population in 2015 were 2.80 \(\times 10^{12}\). Thus, \(\approx\)8.04 living hours are supported for each working hour. We use the broadest measure here, as one cannot have workers without also supporting youth and the elderly. We, therefore, assume 8 h of life required to be supported per productive worker hour.

Note that people serve two functions in this model: they are both consumers of final output and the suppliers of labor. We treat this complexity as follows: the basic subsistence requirements to provide labor to the economic process are meant to be included as part of the inter-industry A matrix, while any additional consumption to satisfy discretionary demands is treated as part of final demand. This distinction is largely theoretical in this work, but could be further explored in future studies with more granular datasets and matrices.

Another key methodological concern is the treatment of stocks and flows. In some cases, consumption to fuel a productive sector may occur up front (as in drilling of an oil well) while other consumption may resemble a “flow” that is required for each unit of additional production. We have (in this work) treated all consumption as a yearly flow, with any up-front capital investment averaged over the life of the project and, therefore, “smoothed” into a flow. Therefore, all steel assumed required by the oil industry is apportioned proportionally for each barrel as a “flow”. Future versions of the model could work with dynamic data to incorporate investment of upfront capital and labor.

Another important consideration the possible overlap between food and energy sectors. For one, the purpose of the food sector is simply to provide energy (sustenance) to the labor sector. Second, the energy sector consumes some food sector outputs (e.g., corn for ethanol production) in some regions in some times. In a sense, the energy sector serves to provide animating power (exergy) to the built environment and machinery, while the food sector fuels the labor sector. These complexities aside, we still consider it useful to distinguish between the food and energy sectors because they are so different and fundamentally important to the economy. This is particularly the case if one is looking at long-term trends in the shift between traditional and fossil-based energy systems.

Note that in all cases, self-consumption within a sector is modeled using the on-diagonal elements. For some cases where only net output at the end of any self consumption is available (e.g., for food) we leave the diagonal element blank.

Lastly, we do not consider human physical energy expenditure important for any sector, which implies a modern economy where much or all material processing, manufacturing, construction, and transport is machine powered.

Energy Sector Requirements

The energy sector consumes energy, materials, food and labor to produce energy outputs. For the energy sector, the requirements are defined as follows:

Energy consumption by the energy sector Direct energy consumption by the energy sector, per unit of energy output, is modeled using oil industry figures as indicative. From prior work (Brandt et al. 2015, 2012; Cleveland 2005; Guilford et al. 2011) the direct self-consumption of energy within the energy sector for producing crude oil from the Earth ranges from \(\le\)1 to 30% of the energy content of produced crude oil. The high end of this range is observed in thermal enhanced oil recovery and oil sands projects. The low end of the range would be observed during primary production of a high quality (i.e., Persian Gulf) oil field. In addition, refining of crude oil into finished fuel products requires \(\approx\)5 to 15% of the energy content of the crude oil, with higher values observed in processing heavy crudes. As an order-of-magnitude figure, we approximate the total energy self-consumption of both production and refining as 0.1 MJ/MJ.

Material consumption by the energy sector Material consumption per oil well was found by Brandt (2015) and Beath et al. (2012, 2014) to be of order 100s to 1000s tonnes of material per well. For example, Brandt’s default well has direct material consumption for wellbore, cement, surface facilities, and long-distance transport of 750 tonnes per well (Brandt 2015). In the US, average per-well lifetime productivity (estimated ultimate recovery per well) is of order 130,000 bbl, while global productivities are closer to 1 \(\times 10^6\) bbl per well. At a typical energy content of crude of 6100 MJ/bbl, these results equal 8 \(\times 10^8\) to 6\(\times 10^9\) MJ per well. Thus, if we assume material consumption of 1000 tonnes per well (1 \(\times 10^6\) kg) and conservative energy output of 1 \(\times 10^9\) MJ per well, consumption intensity is of order 10\(^{-3}\) kg of material per MJ of energy.

Food consumption by the energy sector Since food is consumed to provide labor services, direct food consumption per unit of energy produced is 0 kg/MJ.

Labor consumption by the energy sector ExxonMobil is used as oil industry reference. In 2014, ExxonMobil had 73.5 \(\times 10^3\) regular employees, and produced 4.097 \(\times 10^6\) bbl of oil equivalent per day (ExxonMobil 2015). Using the above 6100 MJ ber BOE, this amounts to 9.12 \(\times 10^{12}\) MJ per year. Of course, ExxonMobil operations will employ also non-ExxonMobil contractors or service company employees. Taking a high multiplier of 10 total employees per ExxonMobil regular employee, the per-worker productivity is \(1 \times 10^{-4}\) hr/MJ, only counting the worked hours. Correcting for the factor of 8 h lived per hour worked, we arrive at \(1 \times 10^{-3}\) lived hours per MJ.

Material Sector Requirements

For inputs to material production processes, we use steel as the example material commodity.

Energy consumption by the materials sector IEA data (IEA 2009) suggest that energy requirements for basic oxygen furnace are of order 12–14 MJ/kg, so we use a figure of 10 MJ/kg.

Material consumption by the materials sector No ready data are available on direct material consumption by the materials industry. A variety of works focus on the energy intensity of material production (Gutowski et al. 1986), and some which discuss loss and recycling during supply chains (Allwood et al. 1986), but little information is available on the material consumed for building material production systems. For simplicity we assume that 0.1 kg of material is required to produce 1 kg of material output.

Food consumption by the materials sector Since food is consumed to provide labor services, direct food consumption per unit of material produced is 0 kg/MJ.

Labor consumption by the materials sector Nippon Steel is used as an example. Nippon Steel has \(\approx\)84,000 employees, and raw steel output is 4.22 \(\times 10^{10}\) tonnes, so about 4 \(\times 10^{-6}\) h worked per kg of steel (NSSM 2016). Accounting for hours lived per hour worked, we round this to 1 \(\times 10^{-4}\) hrs lived per kg.

Food Sector Requirements

Energy consumption by the food sector Estimates of energy consumption by the US food sector vary widely. For example, a low-end estimate is given by US EIA Manufacturing Energy Consumption Survey for NAICS code 311 (“Food”) was 1.2 EJ/y in 2010. Assuming 1000 kg of food consumed per year by each of 309.3 million 2010 citizens (Pimentel et al. 2008), this amounts to 4 MJ/kg. That figure is almost certainly low due to lack of transport energy and likely missing other indirect energy. In contrast, Pimentel et al. (2008) suggest that direct and indirect consumption by the US food system was \(\approx\)75 MJ/kg food. We choose an intermediate order-of-magnitude between these at 10 MJ/kg.

Material consumption by the food sector Pimentel and Patzek (2005) and Patzek (2004) suggest that material requirements for equipment used for corn harvesting are about 55 kg per ha, while yields are about 7300 kg/ha-y. These data are unavailable for further study, so we use more recent work on machinery requirements for Illinois corn production (Stubbs 2013). These more recent figures suggest machinery requirements of 33–108 kg/ha, depending on farm size. For a machinery lifetime of 20 years, this amounts to 2–5 kg/ha-y. We assume the wider general agriculture industry is about 10 times more material intensive than corn production, so estimate 0.01 kg material consumed per kg of food.

Food consumption by the food sector In a modern food production system, much food is consumed internally within the sector to produce higher value products (e.g., grain used to produce meat and dairy). Because our overall energy consumption value is for the whole sector, as is our labor requirement, we do not count internal consumption in the food sector.

Labor consumption by the food sector US Bureau of labor force statistics puts 2014 employment in food production 972,000 people [Bureau of Labor Statistics (2016) codes 35-0000], while over 12 M people work in food serving and preparation. To avoid endogenizing the discretionary consumption of labor for prepared food in restaurants, we round up the more narrowly defined food production labor pool to 1 M. If these people all work full time (2000 h per year) to produce and serve the 1000 kg of food per person noted below, this amounts to 0.09 direct worked hours per kg of food. Accounting for hours worked per hour lived, we round this to 1 h/kg.

Labor Sector Requirements

Energy consumption by the labor sector The labor sector consumes energy directly in households. US consumption in households was 10 EJ, or 1 \(\times 10^{13}\) MJ. If we assume that \(\approx\) 1/3 of energy consumption was for non-discretionary uses, then basic household energy consumption per lived hour was 1 MJ/hr.

Material consumption by the labor sector House construction for two studied houses consumed \(\approx\)80–90 tonnes of material (Winistorfer et al. 2005). We, therefore, assume that a housing unit consumes 100 tonnes of material. Total housing starts were \(\approx\)1 million per year (NAHB 2016), leading to consumption of 1 \(\times 10^{11}\) kg, leading to consumption of 0.04 kg per lived hour. Adding in additional non-housing material, we arrive at 0.1 kg per lived hour.

Food consumption by the labor sector FAO [FAO (2003)Table 2.7] gives 2015 food consumption for industrial countries at 1064 kg per person per year, including indirectly consumed grain (e.g. grain consumed by animals). This amounts to 0.12 kg of food per lived hour, which we round to 0.1 kg/h lived.

Labor consumption by the labor sector Because the BLS statistics shown above include all labor sources, we do not include any factor of additional labor self-consumption to provide labor.

In comparison to some IO models where household demand is entirely exogenous and part of final demand, in this model we have partly endogenized household demand. Conceptually, we endogenize into the labor sector all household consumption required (at a basic level) to support the supply of labor, while the remaining discretionary household consumption is exogenous. This distinction, while only approximate and “fuzzy”, at least conceptually allows us to include labor provision within the system while recognizing that much of final demand is in fact consumed by laborers (though for discretionary uses).

After these calculations, the resulting “baseline” matrix for our simple order-of-magnitude four sector economy is given below in Eq. 13:

$$\begin{aligned} \begin{array}{ccccccc} &{} &{} \mathrm {En.} &{} \mathrm {Mat.} &{} \mathrm {Fd.} &{} \mathrm {Lab.} &{} \\ \mathrm {Energy} &{}&{} 0.1 &{} 10 &{} 10 &{} 1 &{}\\ \mathrm {Materials} &{}&{} 10^{-3} &{} 0.1 &{} 0.01 &{} 0.1 &{}\\ \mathrm {Food} &{}&{} 0 &{} 0 &{} 0 &{} 0.1 &{}\\ \mathrm {Labor} &{}&{} 10^{-3} &{} 10^{-4} &{} 0.1 &{} 0 &{}\\ \end{array} \end{aligned}$$ (13)

Fig. 2 Illustration of cross-sector flows in “baseline” example model Full size image

We also illustrate these flows in Fig. 2.

Exploring Impacts of Reduced Energy Sector Productivity on Our Model Economy

Now that our template four-sector model is complete, we explore ways to model reduced productivity of the energy sector. In reality, depletion of energy resources is a complex phenomenon, resulting in myriad physical impacts. We will model these complex impacts simply by perturbing the column of the A matrix that represents the inputs to the energy sector.

We do this by scaling the energy column of the matrix by a multiplier. In the Figures below, we explore the range of columnar multipliers from 0.5\(\times\) to 10\(\times\). This range is explored in 100 steps. For each value of the multiplier, the resulting GER is computed for each range of energy inputs as above.