Quantitative and Physical History

Efficiency Discounted Exponential Growth (EDEG) Approach


First published on February 23, 2020. Last updated on April 18, 2021.

Efficiency Discounted-Exponential Growth (EDEG) Approach

The efficiency-discounted exponential growth (EDEG) approach is relatively new, although functions similar in output have existed for over 100 years. A few rough simulations were conducted earlier (Ciotola 2009, 2010), but this approach is being further refined and structured. EDEG produces a model based on two mathematically simple components, but allows the addition of other, more sophisticated components.

Origins of This Approach

This author’s original efforts at developing the EDEG approach were heavily influenced by M. King Hubbert, a geologist who used such curves to model domestic petroleum production (including peak oil) as well as labor model.[7] Hubbert in turn was influenced by D. F. Hewett.

The author originally used the normal distribution approach (Ciotola, 2001) discussed by Hubbert, but this approach was not sufficiently broad for the authors need to model history, since few historical dynasties utilized petroleum in major quantities, nor did Hubbert’s attempts involve a driving tendency for history. The author began some related models for French and Spanish dynasties (2009) and more fully developed EDEG for petroleum modeling (Ciotola 2010).

Exponential Growth

In principle, a system that is capable of self-replication can experience pure exponential growth. Examples include bacteria, fire and certain nuclear reactions. Humans can self-replicate, so human societies can also experience pure exponential growth, in the form of y = et, where the value of e is approximately 2.72. An example of pure exponential growth is graphed below.

Quantity rises slowly, then much more rapidly as time passes

Pure exponential growth

Human social movements can also experience exponential growth.  Many philosophies have been subject to exponential growth, because the founder could teach others to teach yet others. Investment pyramid schemes can also experience exponential growth.

Certain types of growth not only grow, but cause changes in their environment to enable further growth.

Some futurists make dire projections of explosive population growth and resource consumption by extending a pure exponential function into the future.[2]In reality, exponential growth is never seen for long periods of time, due to limiting factors.

It is noticed that systems in both nature and human society often grow exponentially at times (Ciotola 2001; Annila 2010). Exponential growth essentially means that a system’s present growth is proportional to its present magnitude. For example, if a doubling of population (say of mice) is involved, then 10 mice will become 20 mice, while 20 mice will become 40 mice. A formula for exponential growth is:

\(y = ek^t\)

where \(y\) is the output, \(k\) is the growth rate, and \(t\) represents time.


Exponential decay of efficiency is produced by a Carnot engine operating across an exhaustible thermodynamic potential. It is a fundamental form of decay. The following is an equation for an exponential decay function:

\(y = e^{-k^t}\),

where \(k\) is the decay factor. This appears nearly identical to the exponential growth function, except that the exponent is negative.

Efficiency-Discounted Exponential Growth

Efficiency-Discounted Exponential Growth (EDEG) involves the consumption of a non-replenished resource over time by a system of reproducing agents. It can be useful for modeling many phenomena, including the mineral production of a mining region.

It is possible to create differential equations that produce EDEG. It is also possible to empirically synthesize functions that closely replicate EDEG. An analytical solution to EDEG differential equations from fundamental principles has not yet been created. However, the  differential equations can be iteratively calculated via a spreadsheet or computer program to produce useful results.

When developing an EDEG model, an effort should be made to express parameters in terms of a potential and changing efficiency. If the quantity of the most critical conserved resource is known, then the model for total consumption over time should match that quantity.

An EDEG function can be approximated by multiplying a pure exponential growth function by an efficiency function.

\(P = k_1 e^{k_2 t} (1 – \frac{Q}{Q_0})\),

where P is power (or production), \(k_1\) is a constant of proportionality, typically the initial prediction (or power), \(k_2\) is a growth factor, \(Q\) is the amount of a nonrenewable critical resource thus far consumed, and \(Q_0\) is the initial quantity of the nonrenewable critical resource.

\(k_1 e^{k_2 t}\)

represents the exponential growth component.

\((1 – \frac{Q}{Q_0})\)

represents the efficiency component.

Rise and fall model versus plotted actual data

Colorado San Juans gold production versus model

Notes and References

[1]Meadows, et al. makes this point in Limits to Growth.

[2]For an opposing, but still dire view, see D. H. Meadows et al, Limits to Growth, Universe Books, New York (1972).

[3]The work of Forrester on system dynamics and the Club of Rome project Limits to Growthby Meadows et al involve attempts to better understand these limiting factors. The work of M. K. Hubbert and Howard Scott on peak oil and technocratic governance are other examples.

[4]M. Butler, Animal Cell Culture and Technology.  Oxford: IRL Press at Oxford University Press, 1996.

[5]Ciotola, M. 1997. San Juan Mining Region Case Study: Application of Maxwell-Boltzmann Distribution Function. Journal of Physical History and Economics 1. Also see M. K. Hubbert.

[6]Heilbroner, R. L., The Worldly Philosophers 5thEd.  (Touchstone (Simon and Schuster), 1980.

[7]Such curves were proposed by W. Hewitt as early as 1926 (source unavailable).

[8]See Gibson, C., Spain in America. Harper and Row, 1966.

Additional References

Annila, A. and S. Salthe. 2010. Physical foundations of evolutionary theory. J. Non-Equilib. Thermodyn. 35:301–321.

Ciotola, M. 2001. Factors Affecting Calculation of L, edited by S. Kingsley and R. Bhathal. Conference Proceedings, International Society for Optical Engineering (SPIE) Vol. 4273.

Ciotola, M. 2003. Physical History and Economics. San Francisco: Pavilion Press.

Ciotola, M. 2009. Physical History and Economics, 2nd Edition. San Francisco: Pavilion of Research & Commerce.

Ciotola, M. 2010. Modeling US Petroleum Production Using Standard and Discounted Exponential Growth Approaches.

Hewett, D. F. 1929. Cycles in Metal Production, Technical Publication 183. New York: The American Institute of Mining and Metallurgical Engineers.

M. King Hubbert. 1956. Nuclear Energy And The Fossil Fuels. Houston, TX: Shell Development Company, Publication 95.

Hubbert, M. K. 1980. “Techniques of Prediction as Applied to the Production of Oil and Gas.” Presented to a symposium of the U.S. Department of Commerce, Washington, D.C., June 18-20.

Mazour, A. G., and J. M. Peoples. 1975. Men and Nations, A World History, 3rd Ed. New York: Harcourt, Brace, Jovanovich.

Schroeder, D. V. 2000. Introduction to Thermal Physics. San Francisco: Addison Wesley Longman.

Smith, D. A. 1982. Song of the Drill and Hammer: The Colorado San Juans, 1860–1914. Colorado School of Mines Press.


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