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'''Constant returns to scale''' occur when a company increases its [[Input]] by x% and its output also increases by x%. If a company with constant returns to scale doubles its facilities, workers, and materials, the amount of products it makes will also double.

A process displays '''constant returns to scale''' when increasing all [[input]]s by a factor of ''s'' leads to an increase in output by the same factor.  In [[economics]], this is in terms of the [[production function]] of the process &mdash; for instance, that of a [[business]], [[company]], or economy as a whole.  Mathematically, if the function is <math>F(\vec{x})</math>, where <math>\vec{x} \in \bold{R}^{n}</math> is a vector of inputs, constant returns to scale are characterized by <math>F(s\vec{x}) = sF(\vec{x})</math>, where <math> s > 0 </math>.

 

A function displaying constant returns to scale is [[homogeneous]] of degree one, and by Euler's Theorem can be written as:

 

<math> F(\vec{x}) = x_1\frac{\partial F}{\partial x_1} + x_2\frac{\partial F}{\partial x_2} + ... + x_n\frac{\partial F}{\partial x_n}</math>

 

Since for a [[competition|competitive]] market, the payment to each factor of production (<math>x_i</math>) is <math>\frac{\partial F}{\partial x_i}</math>, this implies that that the total payments to all factors of production exhaust output (''F''), and the market has zero [[profit]].

 

==Examples of functions displaying constant returns to scale==

 

 

*:The variables have the interpretation of Y as output, L as labour, K as capital, and A is multifactor productivity.  In general, empirical studies have shown that the [[United States]] economy has approximately <math>\alpha = 0.33</math>.

 

[[category:economics]]

[[category:economics]]