Scale elasticity

In production theory , the elasticity of scale indicates the percentage by which the production quantity (output) increases if the input quantities of all production factors (inputs) are increased by one percent at the same time.

With a differentiable production function , the scale elasticity can be calculated from the partial production elasticities : with = scale elasticity, = number of inputs, = partial production elasticity of the -th input, = production volume and = input volume of -th input.
${\displaystyle \lambda =\sum _{i=1}^{n}\sigma _{i}=\sum _{i=1}^{n}{\frac {\partial y}{\partial x_{i}}}\cdot {\frac {x_{i}}{y}}}$
${\displaystyle \lambda }$ ${\displaystyle n}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle i}$ ${\displaystyle y}$ ${\displaystyle x_{i}}$ ${\displaystyle i}$

If the production function is linearly homogeneous , then the scale elasticity is 1. An increase in the input quantities of all production factors by one percent also leads to an increase in the production quantity by one percent.

Of positive economies of scale , increasing returns to scale or a scale elasticity greater than one is used when the production volume is increased by more than one percent when all factors are simultaneously increased by one percent.

From negative economies of scale , falling economies of scale or a scale elasticity smaller one is when the production volume is increased by less than one percent when all factors are simultaneously increased by one percent.

With the CES production function , the scale elasticity can be read off directly as the sum of the exponents ( exponents ) of the input quantities (e.g. work A and capital K). This applies in particular to the Cobb-Douglas production function , which is part of the CES production functions.

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  Fig. 1: Scale elasticity less than 1

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  Fig 2: Scale elasticity equal to 1

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  Fig. 3: Scale elasticity greater than 1