====== Elasticity of Production ======

The elasticity of production, also called **output elasticity**, is the percentaje change in the production of a good by a firm, divided the percentage change in an input used for the production of that good, for example, labor or capital.

The elasticity of production shows the **responsiveness** of the output when there is a change in one input.

It is defined as de proportional change in the product, divided the proportional change in the quantity of an input.

**For example**, if a factory employs 10 people, and produces 100 chairs per day. If the number of people employed in the factory increases to 12, that is, a 20% increase, and the number of chairs produced per day increases to 110 (that is, a 10% increase), the elasticity of production is:

**ΔQ/Q / ΔL/L** = 10/100 / 2/10 = 0.1 / 0.2 = 0.5

If the production function contains only one input, the elasticity of production measures the degree of [[https://www.econowiki.com/doku.php?id=en:returns-to-scale|returns to scale]]. In this case: 

- if the elasticity of production is 1, the production has **constant returns to scale**, at that point. 

- if the elasticity of production is greater than one, the production has **increasing returns to scale** at that point. 

- if the elasticity of production is less than one, the production has **decreasing returns to scale** at that point.


===== Using a production function =====

If a production function, for example: Q=f(K,L), is used to calculate the input, and the function is diferentiable, the elasticity of production can be calculated using derivatives:

(∂Q/Q) / (∂L/L) 

This is the same as:

(∂Q/∂K) / (Q/K)

That is, the marginal product of capital, divided the average product of capital.

==== Example: Elasticity of Production of a Cobb Douglas Production Function ====

The [[:en:cobb-douglas-production-function|Cobb-Douglas production function]] is a function that is used a lot in economics. The form of a Cobb-Douglas production function is:

Q(L,K) = A L<sup>β</sup>  K<sup>α</sup>

To calculate the [[:en:cobb-douglas-output-elasticity|elasticity of production of the Cobb-Douglas production function]], with respect to K, we must find the proportional change in the production, divided the proportional change in K:

\begin{equation} \frac {\frac{\partial Q}{Q}} {\frac{\partial K}{K}} = \frac {\frac{\partial Q}{\partial K}} {\frac{Q}{K}} = \frac { α A L ^{β} K^{α-1} }{ \frac {A L^β K^α}{K} } \end{equation}

\begin{equation} = \frac { α A L ^{β} K^{α} K^{-1} }{ \frac {A L^β K^α}{K} } \end{equation}

\begin{equation} = \frac { \frac {α A L ^{β} K^{α}}{K} }{ \frac {A L^β K^α}{K} } \end{equation}

\begin{equation} = \frac { \frac {α Q}{K} }{ \frac {Q}{K} } \end{equation}

\begin{equation} = α \end{equation}

[[production-function-example]]