# Leontief Production Function

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This Demonstration studies the Leontief production function of the form in three dimensions. The main feature of the function is that it models the situation when production factors and are perfect complements to each other.

Contributed by: Timur Gareev (May 2018)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

A Leontief production function of the form

has all its optimal solutions lying on the line

.

Factors and are perfect complements in the model. To shift from one optimal solution to another, a producer has to change both factors in the established proportion . If we take an arbitrary point lying outside the optimal direction, a redundancy (inefficiency) exists. On the three-dimensional plot, the size of the inefficiency corresponds to the length of the blue or red arrow. The blue arrow shows how much of resource can be saved without reducing . The red arrow shows the same thing for .

The Leontief function has no partial derivatives in kink points lying on the optimal direction . Nonetheless, it has directional derivatives in points along the vector , which are equal to .

We also show blue and red angles with tangents and , respectively. The angles show that the model is the result of the intersection of two planes and , and changes to or move their respective planes.

Use the "top view" button to see a classical two-dimensional representation of the model.

Note also that in the context of consumer theory, the same model can be interpreted as a utility function.

## Permanent Citation