View source for Cobb-Douglas function

In a mathematical setting, the Cobb–Douglas function is defined as (see [[#References|[a2]]]):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301302.png" /> is a positive constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301303.png" /> is a positive fraction. The primary application of the Cobb–Douglas function has been in agriculture and industrial production. This is the reason that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301304.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301306.png" /> are usually named  "output" ,  "capital"  and  "used labour power to get the output Q" .

Setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301307.png" />, one may express <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301308.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c1301309.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

or

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013010.png" /></td> </tr></table>

For given values for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013012.png" />, the magnitude of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013013.png" /> will proportionately affect the level of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013014.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013015.png" /> may be considered as an efficient parameter, i.e. as an indicator of the state of technology.

The function defined by (a1) is homogeneous concerning the factor variables and linear by some logarithm application.

Ch.W. Cobb and P.H. Douglas published their article [[#References|[a14]]] in 1928, and one of its applications has been described in [[#References|[a15]]] (see also [[#References|[a6]]]). In [[#References|[a7]]], a detailed description is given of a mathematical model of the  "production and consumption variation under uncertainty in a one-sector economy" , resulting in the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013017.png" /> is the rate of capital depreciation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013018.png" /> is the aggregate rate of consumption. In (a3), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013019.png" /> is the production function, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013020.png" /> is instantaneously determined when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013023.png" /> are known.

For applications of statistical and mathematical models using the  "production function" , for constructing optimal trajectories in a medium- and long-term run, see [[#References|[a8]]] and [[#References|[a10]]]; these also contain numerical analyses concerning the production, costs and work productivity.

Mathematical methods and models of long-run growth (in particular, the Solow model) using an enlarged production function are given in [[#References|[a13]]].

Macro-economical models regarding investment spending and the rental cost capital, as well as the real rate of interest, using a  "production function"  are given in [[#References|[a4]]]. Problems regarding comparative relations of a few national economies have been studied with production functions of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013027.png" /> have been defined previously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013028.png" /> represents the natural resources of the nation, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130130/c13013029.png" /> is the technical progress; see, e.g., [[#References|[a3]]]. A computer programming solution tool for statistical and production functions can be found in [[#References|[a5]]].

The Cobb–Douglas function, regarded as a utility function, and preference functions with applications in micro-economics are given in [[#References|[a12]]]. The theory and applications of production functions with multiple inputs and of different shapes, not only as a factor products, are well explained in [[#References|[a9]]].

Mathematical models in insurance and risk theory, constructed using a  "production function"  with imperfect and asymmetric information, are very interesting; see [[#References|[a1]]], [[#References|[a11]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Boyer,   G. Dionne,   R. Khilstrom,   "Insurance and the value of publicly available information"  T. Band Fomby (ed.)  TaeKun Seo (ed.) , ''Studies in the Economics of Uncertainty''  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Chang,   "Fundamental methods of mathematical economics" , McGraw-Hill  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Dioury,   "Économie internationale" , Décarie Éditeur Montréal  (1998)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Dornbush,   S. Fisher,   G. Sparks,   "Macro economics" , McGraw-Hill Ryerson  (1982)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Fogiel,   "The statistics problem solver" , REA NYNY  (1983)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W.W. Haynes,   "Managerial economics: Analysis and cases" , The Dorsey Press  (1963)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Karlin,   H.M. Taylor,   "Some stochastic differential equation models" , '''2''' , Acad. Press  pp. Chapt 15: Diffusion Processes</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R.G. Lipsey,   P.O. Steiner,   "Economics" , Harper and Row  (1969)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  K.K. Seo,   B.J. Winger,   "Managerial economics" , Richard D. Irwin  (1979)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  F. Todor,   "Trajectoires optimales dans l'économie du Québec et du Canada"  ''Gazette Sci. Math. Québec'' , '''13''' :  3  (1990)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  F. Todor,   "Sur un modèle mathematique appliqué à l'étude du risque avec l'information imparfaite et asymetrique"  ''Anal. Univ. Ovidiu Constantza Romania Ser. Mat.''  (1993)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  H.R. Varian,   "Intermediate microeconomics: a modern approach" , Norton  (1987)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  H.R. Williams,   J.D. Huffnagle,   "Macroeconomic theory: Selected readings" , Meredith Corp.  (1969)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  Ch.W. Cobb,   P.H. Douglas,   "Theory of production"  ''Amer. Economic Review'' , '''March'''  (1928)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  E. Heady,   J. Dillon,   "Agricultural production functions" , Iowa State Univ. Press  (1961)</TD></TR></table>