Editing Production function (section)

====Natural resources====
{{See also | Nicholas Georgescu-Roegen #Criticising neoclassical economics (weak versus strong sustainability)}}
[[Natural resource]]s are usually absent in production functions. When [[Robert Solow]] and [[Joseph Stiglitz]] attempted to develop a more realistic production function by including natural resources, they did it in a manner economist [[Nicholas Georgescu-Roegen]] criticized as a "conjuring trick": Solow and Stiglitz had failed to take into account the [[laws of thermodynamics]], since their variant allowed man-made capital to be a complete substitute for natural resources. Neither Solow nor Stiglitz reacted to Georgescu-Roegen's criticism, despite an invitation to do so in the September 1997 issue of the journal ''[[Ecological Economics (journal)|Ecological Economics]]''.<ref name="Daly">{{cite journal |title= Forum on Georgescu-Roegen versus Solow/Stiglitz |journal= Ecological Economics |volume=22 |issue=3 |year=1997 |pages= 261–306 |doi= 10.1016/S0921-8009(97)00080-3 |author= Daly, H }}</ref><ref name="Daly2">{{cite book |last=Daly |first=Herman E. |author-link=Herman Daly |date=2007 |chapter=How long can neoclassical economists ignore the contributions of Georgescu-Roegen? |editor-link=Herman Daly |title=Ecological Economics and Sustainable Development. Selected Essays of Herman Daly |url=http://library.uniteddiversity.coop/Measuring_Progress_and_Eco_Footprinting/Ecological_Economics_and_Sustainable_Development-Selected_Essays_of_Herman_Daly.pdf |location=Cheltenham |publisher=Edward Elgar |isbn=978-1-84720-101-0 |via=United Diversity}}</ref>{{rp|127–136}}<ref name="Cohen">{{cite journal |last1=Cohen |first1=A. J. |last2=Harcourt |first2=G. C. |year=2003 |title=Retrospectives: Whatever Happened to the Cambridge Capital Theory Controversies? |journal=[[Journal of Economic Perspectives]] |volume=17 |issue=1 |pages=199–214 |doi= 10.1257/089533003321165010|doi-access=free }}</ref><ref>{{cite book |author-link=Robert Ayres (scientist) |first1=Robert U. |last1=Ayres |first2=Benjamin |last2=Warr |title=The Economic Growth Engine: How Useful Work Creates Material Prosperity |year=2009 |publisher=Edward Elgar |isbn=978-1-84844-182-8 }}</ref>

Georgescu-Roegen can be understood as criticizing Solow and Stiglitz's approach to mathematically modelling factors of production. We will use the example of energy to illustrate the strengths and weaknesses of the two approaches in question.

===== 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.

===== Inter-dependent factors of production =====
The "independent" energy-dependent production function can be revised by considering energy-dependent labor and capital input functions <math>L = L(E(t))</math>, <math>K = K(E(t))</math>. This approach yields an energy-dependent production function given generally as <math>Q = f(L(E), K(E))</math>. Details related to the derivation of a specific functional form of this production function as well as empirical support for this form of the production function are discussed in more-recently published work.<ref name=":0" /> Note that similar arguments could be used to develop more-realistic production functions which consider other depletable natural resources beyond energy:

* If a geographical region runs out of the natural resources required to produce a given machine or maintain existing machines and is unable to import more or recycle, the machines in that region will eventually fall into disrepair and the machines' maximum output would be reduced to near-zero. This should be modeled as significantly affecting the total output. Therefore, therefore <math>K</math> should be modeled as depending directly on time-dependent natural resource input <math>N(t)</math>.