Leontief production function

Leontief production function: capital (K), labor (L), output (Y)

The Leontief production function , named after Wassily Leontief , is a type (Type B) of the microeconomic production function . It is called linear limitational , since the production factors are in a fixed relationship to one another and in a fixed relationship to the output of a company or plant. The output volume reaches a limit if a production factor is not available in sufficient quantities.

In formal notation applies to the function

${\displaystyle f\colon \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{1},n\geq 1,\quad f(v)=a_{0}\min \left\{{\frac {v_{1}}{a_{1}}},{\frac {v_{2}}{a_{2}}},\dots ,{\frac {v_{n}}{a_{n}}}\right\}^{r}}$.

The Leontief production function is a CES production function with zero elasticity of substitution . It is homogeneous in degree and there are always constant returns to scale . ${\displaystyle r}$

example

Every “recipe” production in the kitchen or in the laboratory is an example of the Leontief production function. Do you need z. For example, for a cake made according to a recipe, eggs, grams of flour and liters of milk can be used to make cakes with available eggs, grams of flour and liters of milk . In this case the eggs are limitational; one could have made cakes with eggs . ${\displaystyle a_{1}=2}$ ${\displaystyle a_{2}=100}$ ${\displaystyle a_{3}=0{,}1}$ ${\displaystyle v_{1}=4}$ ${\displaystyle v_{2}=300}$ ${\displaystyle v_{3}=0{,}3}$ ${\displaystyle f(v)=\min \left\{{\tfrac {4}{2}},{\tfrac {300}{100}},{\tfrac {0{,}3}{0{,}1}}\right\}=\min \left\{2,3,3\right\}=2}$ ${\displaystyle v_{1}=6}$ ${\displaystyle f(v)=\min \left\{{\tfrac {6}{2}},{\tfrac {300}{100}},{\tfrac {0{,}3}{0{,}1}}\right\}=3}$

The production function forms the basis of the input-output analysis .

See also

• Cobb-Douglas production function
• production
• Gutenberg production function