Subadditivity

Subadditivity is a term from mathematics . The opposite concept is that of superadditivity .

In industrial economics, as an application of the mathematical term, this describes a state in which a good can be produced more cheaply by a single company than by several companies together.

definition

If a cost function is subadditive in , these units can always be produced by exactly one company at lower costs than by two or more companies, regardless of how the production volume is divided between these companies. ${\displaystyle x}$ ${\displaystyle x}$

A cost function is described as strictly subadditive for all if for any output quantity with and applies: ${\displaystyle K(x)}$ ${\displaystyle x}$ ${\displaystyle x_{i}}$ ${\displaystyle 0<x_{i}<x(i=1,2,\dots ,n)}$ ${\displaystyle \sum _{i=1}^{n}x_{i}=x}$

${\displaystyle K(x)<\sum _{i=1}^{n}K(x_{i})}$.

Importance for competition

If the cost function of an industry is in the sub-additive area across the entire quantity demanded, then this industry is also called a natural monopoly .

In perfect competition , if the price so the marginal cost corresponds subadditive cost structures will lead to a deficit, because the average costs and fixed costs include and in the relevant area above marginal cost (the fixed costs not included) are. This is a justification for regulating markets .

Explanations

One product case

If a single product is produced (one-product case), it is cheaper if a single supplier produces the entire quantity than if several suppliers produce the same quantity together. Formally this is expressed by: where the cost of production is the amount that suppliers would produce; these partial amounts add up to the total amount . ${\displaystyle x}$ ${\displaystyle x_{1},\dots ,x_{n}}$ ${\displaystyle K(x)<K(x_{1})+K(x_{2})+\dots +K(x_{n})}$ ${\displaystyle K}$ ${\displaystyle x_{1}+x_{2}+\dots +x_{n}=x}$ ${\displaystyle n}$ ${\displaystyle x}$

The reasons for the underlying increasing economies of scale for such homogeneous goods lie in economies of scale , for example in stochastic size savings and learning curve effects .

Multi-product case

In the multi-product case, there is subadditivity if a company can produce two products together at a lower overall cost than if two companies were to produce the same quantity of only one good . This condition is met if the average costs are falling in the relevant range and are above the marginal costs. ${\displaystyle K(x,y)<K(x,0)+K(0,y)}$

In the production of different (heterogeneous) goods in this way, economic effects (economies of scope) and cost complementarity come into play. In both cases, subadditivity also favors the presence of density advantages . However, economies of scale and economies of scope are neither a necessary nor a sufficient condition for subadditivity.

Individual evidence

1. ↑ ^{a b} Jörg Borrmann, Jörg Finsinger: Market and Regulation , Vahlen-Verlag, Munich 1999, ISBN 3-8006-2471-0 , p. 122.
2. ↑ Ulrich Blum: Applied Institutional Economics . Gabler Verlag; Edition: 2005 (January 1, 2005). ISBN 978-3409142731 . P. 35.

Web links

• Subadditivity - definition in the Gabler Wirtschaftslexikon