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 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.

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.

The 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^β K^α

To calculate the 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