Editing Production function (section)

===Specifying the production function===

A production function can be expressed in a functional form as the right side of
:<math>Q = f(X_1,X_2,X_3,\dotsc,X_n)</math>
where <math>Q</math> is the quantity of output and
<math>X_1,X_2,X_3,\dotsc,X_n</math> are the quantities of factor inputs (such as capital, labour, land or raw materials). For <math>X_1=X_2=...=X_n=0 </math> it must be <math>Q=0</math> since we cannot produce anything without inputs.

If <math>Q</math> is a scalar, then this form does not encompass joint production, which is a production process that has multiple co-products. On the other hand, if <math>f</math> maps from <math>\mathbb{R}^{n}</math> to <math>\mathbb{R}^{k}</math> then it is a joint production function expressing the determination of <math>k</math> different types of output based on the joint usage of the specified quantities of the <math>n</math> inputs.

One formulation is as a linear function:
:<math>Q=a_1 X_1+a_2 X_2+a_3 X_3+\dotsb+a_n X_n</math>
where <math>a_1, \dots, a_n</math> are parameters that are determined empirically. Linear functions imply that inputs are perfect substitutes in production.
Another is as a [[Cobb–Douglas]] production function:
:<math>Q = a_0 X_1^{a_1} X_2^{a_2} \cdots X_n^{a_n} </math>
where <math>a_0</math> is the so-called [[total factor productivity]].
The [[Leontief production function]] applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, the output will not change. This production function is given by

:<math>Q = \min (a_1X_1, a_2X_2, \dotsc, a_n X_n).</math>

Other forms include the [[Constant Elasticity of Substitution|constant elasticity of substitution]] production function (CES), which is a generalized form of the Cobb–Douglas function, and the quadratic production function. The best form of the equation to use and the values of the parameters (<math>a_0, \dots, a_n</math>) vary from company to company and industry to industry. In the short run, production function at least one of the <math>X</math>'s (inputs) is fixed. In the long run, all factor inputs are variable at the discretion of management.

Moysan and Senouci (2016) provide an analytical formula for all 2-input, neoclassical production functions.<ref>see {{cite journal |first1=G. |last1=Moysan and |first2=M. |last2=Senouci |year=2016 |title=A note on 2-input neoclassical production functions |journal=Journal of Mathematical Economics |volume=67 |pages=80–86 |doi=10.1016/j.jmateco.2016.09.011 |s2cid=3581910 |url=https://hal.archives-ouvertes.fr/hal-01383290/document }}</ref>