Editing Production function (section)

===Criticisms of the production function theory===

There are two major criticisms{{which|date=September 2014}} of the standard form of the production function.<ref>On the history of production functions, see {{cite journal |first=S. K. |last=Mishra |year=2007 |title=A Brief History of Production Functions |journal=Working Paper |ssrn=1020577 }}</ref>

====On the concept of capital====
During the 1950s, '60s, and '70s there was a lively debate about the theoretical soundness of production functions (see the [[Capital controversy]]). Although the criticism was directed primarily at aggregate production functions, microeconomic production functions were also put under scrutiny. The debate began in 1953 when [[Joan Robinson]] criticized the way the factor input [[Capital (economics)|capital]] was measured and how the notion of factor proportions had distracted economists. She wrote:

"The production function has been a powerful instrument of miseducation. The student of economic theory is taught to write <math>Q = f (L, K)</math> where <math>L</math> is a quantity of labor, <math>K</math> a quantity of capital and <math>Q</math> a rate of output of commodities. [They] are instructed to assume all workers alike, and to measure <math>L</math> in man-hours of labor; [they] are told something about the index-number problem in choosing a unit of output; and then [they] are hurried on to the next question, in the hope that [they] will forget to ask in what units K is measured. Before [they] ever do ask, [they] have become a professor, and so sloppy habits of thought are handed on from one generation to the next".<ref>{{cite journal |title=The Production Function and the Theory of Capital |first=Joan |last=Robinson |year=1953 |journal=[[Review of Economic Studies]] |volume=21 |issue= 2|pages= 81–106 |doi= 10.2307/2296002|jstor=2296002 }}</ref>

According to the argument, it is impossible to conceive of capital in such a way that its quantity is independent of the rates of [[interest]] and [[wages]]. The problem is that this independence is a precondition of constructing an isoquant. Further, the slope of the isoquant helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.

====On the empirical relevance====
As a result of the criticism on their weak theoretical grounds, it has been claimed that empirical results firmly support the use of neoclassical ''well behaved'' aggregate production functions. Nevertheless, [[Anwar Shaikh (economist)|Anwar Shaikh]]  has demonstrated that they also have no empirical relevance, as long as the alleged good fit comes from an accounting identity, not from any underlying laws of production/distribution.<ref name="Shaikh">{{cite journal |last=Shaikh |first=A. |year=1974 |title=Laws of Production and Laws of Algebra: The Humbug Production Function |journal=[[Review of Economics and Statistics]] |volume=56 |issue=1 |pages=115–120 |doi= 10.2307/1927538|jstor=1927538 }}</ref>

====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>.