Identification of a 50-year scaling relating current global energy demands to historically cumulative economic production

Global economic production, or the GDP, has risen steadily relative to world primary energy demands, suggesting technological change is driving a gradual decoupling of society from its resource needs and associated pollution. Here show that in each of the 50 years following 1970 for which reliable data are available, one Exajoule of world energy was consumed to sustain each 5.50±0.21 trillion constant 2019 US dollars, not of yearly production or physical capital, but of running cumulative production summed over human history. The half-century for which this fixed ratio held covers two thirds of historical growth 5 in energy demands, so assuming its persistence, the implication is that society is not in fact decoupling from resource needs. Rather, it can be expected that future environmental impacts will be more strongly guided by past activities, or inertia, than is generally permitted within economic and climate modeling prescriptions that allow for policy to spur more rapid change.

guide for facilitating long-run dynamic predictions of future interactions between society, resource depletion and discovery, 25 and climate change.

Results
To avoid complications associated with the details of trade, this study is focused only on global quantities as described in the Materials and Methods below. Annual energy demands can be expressed as an instantaneous quantity E with units of power (e.g., Terawatts) or a yearly quantity E i wit units of energy (e.g., Exajoules). For example, E 2019 = 609 means that humanity 30 in 2019 was powered by 609 Exajoules or at a rate of 19.3 Terawatts. Annual economic production (Gross Domestic Product) or output is defined monetarily as the sum of tallied financial exchanges made to acquire final goods and services within a given year. After adjusting for inflation, we denote this quantity as Y i , expressed in units of constant 2019 USD, or as an instantaneous rate Y as 2019 USD per year.
Given that humanity's billions emerged from its past, the magnitude of civilization's annual energy demands might be 35 thought to be tied to a quantity that is not a rate -as for Y -but rather has accumulated through time and has units of currency. The first candidate we consider for such an accumulated quantity is economic capital K i , one of the primary factors of production. The second is a time integral of production, not just over one year -as is done in calculation of Y i -but over the entirety of history, what we term the world historically cumulative production W i = i j=1 Y j , or expressed in continuous form as The contribution of depreciation to W is addressed later.
Time series for Y i , K i , W i and E i are shown in Figure 1 covering a 50 year period between 1970 and 2019. Global energy consumption E increased by a factor of 2.8, production Y increased by a factor of 4.5 and economic capital K increased by a factor of 7.9. The ratio y = Y /E, sometimes termed the energy productivity, has trended steadily upward. Defining growth 45 rates in quantity X as R X = (1/X)dX/dt = d ln X/dt, a least-squares fit to the data gives R y = 1.00% per year. Meanwhile, the ratio k = K/E grew at rate R k = 1.96 % per year, nearly twice as fast as y, or a doubling time of 35 years. The economy appears to be becoming rapidly less energy intensive, suggesting that technological innovation is enabling more to be done with less (Sorrell, 2014).
It would seem natural to infer that the global economy is undergoing a long-term decoupling from resource constraints.

50
However, a comparison between W i and E i suggests otherwise. Relative to Y i and K i , cumulative production W i increased by a much smaller factor of 2.7, similar to that of E i . Expressed (for simplicity) as a continuous function, the ratio w = W/E has fluctuated to some degree, but the average tendency has been R w = −0.02 % per year, far less than either Y /E or K/E. 1.9% of W 2019 , sufficiently small as to plausibly approximate the origin. Thus, the relationship between W and E does not appear to be one only of correlation between two quantities, as for example has been noted for E and Y (Jarvis, 2018). Instead the two quantities have maintained a linear scaling over the half century period for which widely published data are available.
The average value is: in units of trillion 2019 USD of cumulative production per Exajoule of energy consumed each year.

Discussion
A quantity identified here as the historically cumulative global production W appears to be an economic expression of the rotational power of Lotka's Wheel, that is the capacity to drive the collective to-and-from of civilization's circulations through the relationship W = wE, where w is nearly a constant. Certainly, an objection might be raised that the past 50 years is too 65 short relative to the time span of humanity to draw meaningful conclusions about the relationship of cumulative production to energy demands. However, the period nonetheless covers roughly two-thirds of humanity's growth in consumptive demands, or 1.5 doublings in E, during which clearly a great deal changed in humanity's social and technological makeup.
With respect to an inflation-adjusted production relation, taking the first derivative of Eq. 1, and assuming W = wE for an expression for the production function that differs significantly from prior approaches that either ignore the role of energy altogether or express it as proportional to some non-integer exponent of E (Ayres et al., 2003;Keen et al., 2019) rather than a change in energy demands, that is its derivative with respect to time. The quantity W is highly smoothed because it is a summation, or integration, over history and the global economy. Although there is a strong apparent relationship of E to W , 75 variability in E cannot as easily be related to economic production on the scale of years. Nonetheless, calculated as a running decadal mean, the average ratio of production to change in energy consumption is in units of trillion 2019 USD per Exajoule consumed each year, a very similar value to Eq. 2 although more noisy being a differential. Implicitly, our collective societal assessment of the final inflation-adjusted value of goods and services Y appears 80 to correspond with "enlarging the wheel" or enabling it to "spin faster". This societal division into size and rotational velocity has been noted elsewhere, and in fact there is some suggestion that of a near equal division between the two independent modes of variability. A linear scaling has been noted between the magnitude of a city's population and how fast its inhabitants walk (Bettencourt et al., 2007). More globally, over the 50 year period considered, world population -as a measure of sizeincreased at an average rate of 1.46% per year. Meanwhile, per capita GDP -as a plausible metric for speed -increased at the 85 nearly equivalent rate of 1.55% per year. At some level the empirical nature of the result given by Eq. 2 stands on its own. Nonetheless, its simplicity may come across as counter-intuitive, especially considering that W is not directly tied to any current economic transaction, only to the past. By way of explanation, consider the circulations within our bodies, brains, and machines, and our activities such as housework, transport to and from work and the grocery store, and even conversation among family and friends, that all of these require 90 energy in some form. While each one of them may indirectly involve a financial transaction at some prior stage, for cleaning products, gasoline, or food, no purchase is in fact made at the point at which the energy is consumed.
A possible counterargument is that historically distant production cannot linger to contribute to current energy demands. Fig   trees grown for the enjoyment of Ancient Greeks would seemingly have nothing to do with the power consumption of internet servers today. In traditional economic accounting, current capital is formed through economic production Y after subtracting 95 both depreciation at rate δ and consumption C of goods and services, that is dK/dt = (Y − C) − δK. Expressing consumption as C = cY and adopting a simplified production function of form Y = βK where β is the production efficiency (or the inverse of the capital-to-output ratio), it follows that so that the capital exponential growth rate is R K = (β − δ ) where δ = δ + cβ and consumption itself can be viewed as a 100 form of depreciation of very short-lived capital. From data for Y i and K i , the value of β over the past 50 years has steadily declined at an average rate of 0.95% per year but the average value is approximately 0.24 or 24%. So, considering how capital grew at an average annual rate of 4.0% over the same period, the implication is that the annual rate δ at which capital has been devalued is approximately 24%−4% = 20%, that is a halving time of just 3.5 years. Well-known concerns may be raised about any comparison of rates of capital formation with capital valuation, and with how valuations of varied capital stocks should 105 be aggregated (Samuelson, 1966;Sraffa, 1975). Nonetheless, the persistence of past productivity clearly lasts for much longer.
We may no longer use the personal computers of the 1980s, but current devices are derived from that seminal transformation. The crux of the problem appears to be that the long-distant contributions of past civilizations to politics, science, athletics, architecture, and language cannot be monetized on an open market, yet without them the bulk of our modern infrastructure 110 for wealth-generation would be gone. As noted by Piketty, "All wealth creation depends on the social division of labor and on the intellectual capital accumulated over the entire course of human history," continuing "the total value of public and private capital, evaluated in terms of market prices for national accounting purposes, constitutes only a tiny part of what humanity actually values -namely, the part that the community had chosen (rightly or wrongly) to exploit through economic transactions in the marketplace" (Piketty, 2020). Capital valued in the market K is an order of magnitude smaller than historically cumu-115 lative production W , suggesting energy is required not just to sustain that which we believe available to be sold, but also the unspoken utility of that which has previously been produced. Civilization was not built in a day.

Ancient
There are important analogs in the biological and physical world that may provide a guide. The energy of a wheel's rotation is the product of its mass and the square of its radius and rotational frequency, all quantities that increase through a prior history of material and energetic increments. In a cloud, a snow crystal grows through the diffusion of vapor molecules. Current vapor The leaves of a deciduous tree enable photosynthesis that fuels fluid circulations through the exterior sapwood. Leaves die seasonally as the sapwood turns into heartwood that, while not actively connected to a larger rejuvenated leaf crown in the following year, structurally supports it (Shinozaki et al., 1964;Oohata and Shinozaki, 1979). Inevitably there are also loss 125 processes, such as friction for a wheel, moments of evaporation or breakup for a snow crystal, or disease and predation for a tree. But, provided the system is in its phase of growth, past consumption is the primary determinant of the system's continued energetic demands.

Conclusions
We have identified a nearly constant relationship between world historically cumulative inflation-adjusted economic production 130 and current energy demands that has held for the past half-century, a period during which resource consumptive demands nearly tripled. Whatever its explanation, its persistence would appear to place substantial bounds on humanity's future interactions with its environment. For one, it implies that present sustenance cannot be decoupled from past growth, implying a much greater role for inertia than has been broadly assumed, for example, in the integrated assessment models used to evaluate the coupling between humanity and climate (Nordhaus, 2017). Even if world GDP growth falls to zero from its recent levels close 135 to 3% per year, long-term decadal-scale resource demands and waste production would continue. More worryingly, the result suggests that it is only by way of collapse of the previous growth that led to the wealth we enjoy today, effectively by shrinking Lotka's wheel, will our resource demands and waste production decline. Eq. 1 offers no direct mathematical approach for such an event to occur, except perhaps through hyper-inflation, as this would lead to high values of the GDP deflator that in economic accounting yield values of the inflation-adjusted GDP much lower than the nominal GDP. Historically, hyper-inflation has been 140 associated with periods of societal contraction (Zhang et al., 2007) suggesting a possible link to decay.
On the topic of climate policy, a constant value of w implies that economic production can be decoupled from carbon dioxide emissions, but only provided a rapid switch to renewables or nuclear energy. All newly added energy production would need to be emissions free, which based on recent consumption growth rates works out to about 1 Gigawatt per day. Alternatively, or concurrently, some means would need to be devised for decoupling W from E by increasing the value of w. Given the value the years 1970 to 2019 is created from the average of the three datasets while using single sources where only one is available.
Economic production is tallied and averaged using The world historically cumulative production W i = i j=1 Y j requires for its calculation yearly estimates of Y j prior to 1970, for which we apply a cubic spline fit to the Maddison Database (Maddison, 2003) for years after 1 C.E. Adjustments are made to the Maddison dataset to account for the chosen inflation-adjusted year of the dataset and to convert from currency expressed in purchasing power parity dollars rather to market exchange units using as a basis the time period between 1970 and 1992 for 160 which concurrent MER and PPP statistics are available. The value for cumulative production in 1 C.E. W (1) is obtained by assuming that the population and W were growing equally fast at that time. Population data from 1.C.E and one century before and after show that global population was 170 million and growing at 0.059 % per year (United States Census Bureau, 2021).
While there are inevitable uncertainties in the reconstruction of W as with any other, the yearly values of W since 1970 that are emphasized here cover two-thirds of total growth and so the calculations are more strongly weighted by recent data that is 165 presumably most accurate. Thus, calculation of W , most particularly the conclusion that w is nearly a constant, can be shown to be relatively insensitive to uncertainty in the older statistics (Garrett et al., 2020).
Author contributions. T.J.G. and M.R.G conceived the study, T.J.G. analysed the results. All authors wrote and reviewed the manuscript.
Competing interests. The authors declare no competing interests.
Acknowledgements. This work was supported by the National Institute of Economic and Social Research and the Economic and Social