Minimum cost combination

Under a minimum cost combination (engl. Lowest-cost combination or least-cost combination ) refers in production theory or Mikroökonomie an optimal combination of two factors of production . The problem of determining this combination is also called cost-minimizing input selection .

In the case of a minimum cost combination, in accordance with the principle of economy, either a given quantity can be produced at minimal cost or, given a given cost budget, the produced quantity can be maximized. A minimum cost combination is, so to speak, the implementation of the economic principle in the production area of ​​a company: to achieve a given goal with the least amount of effort (minimum version) or to achieve as much as possible with a given effort (maximum version).

To simplify matters, only two input factors are often considered, such as labor and capital . A corresponding question could be: How much labor and capital should each be used to achieve a certain output at minimal cost?

Substitutional factor input conditions

The tangent point of the isocost line and the isoquant determine the minimum cost combination.

Substitutionality allows a certain output to be created through various efficient combinations of production factors. This is generally believed because many factors are to some extent substitutable. The form or existence of the minimum cost combinations thus depends strongly on the form of the underlying production function . For simplification, it is further assumed that the yield isoquant is convex. If this were linear or concave, the optimal combination would be an edge solution.

In this case, there is a minimum cost combination (necessary condition) if the marginal productivities of two factors are related to each other like their prices: the so-called marginal rate of technical substitution thus corresponds to the ratio of the factor prices.

Numerical example

In the two-goods case, the production function is given by (type: Cobb-Douglas function ). The prices of the input factors are and . The ratio of the derivatives of the production function according to its input factors should correspond to the ratio of the respective prices of the input factors: ${\displaystyle f(r_{1},r_{2})={\sqrt {r_{1}}}{\sqrt {r_{2}}}}$ ${\displaystyle r_{1}=1}$ ${\displaystyle r_{2}=3}$

${\displaystyle {\frac {\partial \,f(r_{1},r_{2})/\partial \,r_{1}}{\partial \,f(r_{1},r_{2})/\partial \,r_{2}}}{\stackrel {!}{=}}{\frac {p_{1}}{p_{2}}}}$.

This results in:

${\displaystyle {\begin{aligned}{\frac {{\frac {1}{2{\sqrt {r_{1}}}}}{\sqrt {r_{2}}}}{{\frac {1}{2{\sqrt {r_{2}}}}}{\sqrt {r_{1}}}}}&{\stackrel {!}{=}}{\frac {1}{3}}\\{\frac {2{\sqrt {r_{2}}}{\sqrt {r_{2}}}}{2{\sqrt {r_{1}}}{\sqrt {r_{1}}}}}&={\frac {1}{3}}\\{\frac {2r_{2}}{2r_{1}}}&={\frac {1}{3}}\Leftrightarrow {\frac {1}{3}}={\frac {r_{2}}{r_{1}}}\Leftrightarrow r_{2}={\frac {1}{3}}r_{1}\Leftrightarrow 3r_{2}=r_{1}\\\end{aligned}}}$

For a corresponding budget restriction (isocost line, total costs are e.g. 182), the optimal quantities of and can now be calculated: ${\displaystyle r_{1}^{*}}$ ${\displaystyle r_{2}^{*}}$

${\displaystyle {\begin{aligned}182=r_{1}+3r_{2}&\Rightarrow 182=r_{1}+3\left({\frac {1}{3}}r_{1}\right)=2r_{1}\Leftrightarrow {\underline {r_{1}^{*}=91}}\\&\Rightarrow 182=(3r_{2})+3r_{2}=6r_{2}\Leftrightarrow {\underline {r_{2}^{*}={\frac {91}{3}}}}\end{aligned}}}$.

This optimization problem can also be solved with the help of Lagrange multipliers .

Limitation production function

Different isoquants for the case of a limitational production function.

In the case of a single limitational function, all output quantities can only be realized with a single combination of factors. This is then also the minimum cost combination. Since the isoquant only consists of one point here, the question of determining a minimum cost combination does not actually exist, since there is only this one efficient production. In this case, the combination would be determined by the technical production conditions. Only efficient combination processes can also be cost-effective.

If two or more limitation functions are available, which all deliver the same result, they can be combined and substituted. Are for example the two linear limitational functions and available so it is possible with either of the two processes to manufacture each product, which represents a factor substitution in fact. ${\displaystyle x_{1}=2r_{1}+5r_{2}}$ ${\displaystyle x_{1}=5r_{1}+2r_{2}}$

Expansion path

Isocost isoquant diagram with four minimum cost combinations. A connecting line between these points represents the expansion path.

In the isoquant diagram, a minimum cost combination can be seen as a tangent point of the isoquant and the isocost line . The line connecting the minimum cost combinations for different production levels is called the expansion path .

In graphic terms, this means that starting with low costs, the costs are slowly increased. As a result, the isocost line shifts further and further outwards and touches one isoquant after the other. The tangent points are marked and connected. The result is an expansion path (shown in red in the figure).

literature

• Günter Fandel: Production. I. Springer-Verlag, Berlin 2007. Chapter IV The cost-theoretical selection problem: The minimum cost combination. ISBN 978-3-540-73140-5 , p. 233 ff.

Individual evidence

1. ↑ Bofinger, Peter. Fundamentals of Economics: An Introduction to the Science of Markets. Pearson Deutschland GmbH, 2011. p. 102.
2. ↑ Pindyck, Robert S. microeconomics. Pearson Deutschland GmbH, 2009. p. 316.
3. ↑ Minimum cost combination - Article in the Gabler Wirtschaftslexikon.
4. ↑ Minimum cost combination - Article at mikrooekonomie.de .
5. ↑ Joachim Schwalbach: Production Theory. Vahlen, 2014, p. 23.
6. ^ Corsten: Production Management . 6th edition, pp. 91f.
7. ↑ Wollenberg, Klaus. Economics: Introduction and basics with solutions / by Rainer Fischbach and Klaus Wollenberg. Vol. 1. Oldenbourg Verlag, 2007. p. 234.
8. ↑ Joachim Schwalbach: Production Theory. Vahlen, 2014, p. 21.
9. ↑ Expansion path - Article at www.mikrooekonomie.de .

Web links

• Minimum cost combination - Article in the Gabler Wirtschaftslexikon
• Minimum cost combination - Article at mikrooekonomie.de
• Minimum cost combination - brief description