Editing Production function (section)

===== Independent factors of production =====
[[Robert Solow]] and [[Joseph Stiglitz]] describe an approach to modelling energy as a factor of production which assumes the following:<ref name=":1">{{Cite journal |last=Stiglitz |first=Joseph E. |author-link1=Joseph Stiglitz |date=1974 |title=Growth with Exhaustible Natural Resources: The Competitive Economy |journal=The Review of Economic Studies |volume=41 |pages=139–152 |doi=10.2307/2296378 |jstor=2296378 |issn=0034-6527}}</ref>

* Labor, capital, energy input, and technical change (omitted below for brevity) are the only relevant factors of production,
* The factors of production are independent of one another such that the production function takes the general form <math>Q = f(L, K, E)</math>,
* Labor, capital, and energy input only depend on time such that <math>K = K(t), L = L(t), E = E(t)</math>.

This approach yields an energy-dependent production function given as <math>Q = A L^\beta K^\alpha E^\chi</math>.<ref name=":1" /><ref>{{Cite journal |last1=Kümmel |first1=Reiner |author-link1=Reiner Kümmel |last2=Ayres |first2=Robert U. |author-link2=Robert Ayres (scientist)|last3=Lindenberger |first3=Dietmar |date=2010-07-01 |title=Thermodynamic laws, economic methods and the productive power of energy |journal=Journal of Non-Equilibrium Thermodynamics |url=https://www.degruyter.com/document/doi/10.1515/jnetdy.2010.009/html |language=en |volume=35 |issue=2 |pages=145–179 |doi=10.1515/jnetdy.2010.009 |bibcode=2010JNET...35..145K |s2cid=73538957 |issn=1437-4358}}</ref> However, as discussed in more-recent work, this approach does not accurately model the mechanism by which energy affects production processes.<ref name=":0">{{Cite journal |last1=Keen |first1=Steve |author-link1=Steve Keen |last2=Ayres |first2=Robert U. |last3=Standish |first3=Russell |date=2019-03-01 |title=A Note on the Role of Energy in Production |url=https://www.sciencedirect.com/science/article/pii/S0921800917311746 |journal=Ecological Economics |language=en |volume=157 |pages=40–46 |doi=10.1016/j.ecolecon.2018.11.002 |bibcode=2019EcoEc.157...40K |s2cid=158863011 |issn=0921-8009}}</ref> Consider the following cases which support the revision of the assumptions made by this model:

* If workers at any stage of the production process rely on electricity to perform their jobs, a power outage would significantly reduce their maximum output, and a long-enough power outage would reduce their maximum output to zero. Therefore <math>L</math> should be modeled as depending directly on time-dependent energy input <math>E(t)</math>.

* If there were a power outage, machines would not be able to run, and therefore their maximum output would be reduced to zero. Therefore <math>K</math> should be modeled as depending directly on time-dependent energy input <math>E(t)</math>.

This model has also been shown to predict a 28% decrease in output for a 99% decrease in energy, which further supports the revision of this model's assumptions.<ref name=":0" /> Note that, while inappropriate for energy, an "independent" modelling approach may be appropriate for modelling other natural resources such as land.