Leontief Production Function

​
technological coefficients
a
2
b
3
combination of factors
5
5
​
top view
initial view
optimal
This Demonstration studies the Leontief production function of the form
Q=min
x
a
,
y
b
in three dimensions. The main feature of the function is that it models the situation when production factors
x
and
y
are perfect complements to each other.

Details

A Leontief production function of the form
Q=min
x
a
,
y
b
has all its optimal solutions lying on the line
y=
b
a
x
.
Factors
x
and
y
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
b
a
. If we take an arbitrary point
(x,y)
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
x
can be saved without reducing
Q
. The red arrow shows the same thing for
y
.
The Leontief function has no partial derivatives in kink points lying on the optimal direction
b
a
. Nonetheless, it has directional derivatives in points along the vector
1,
b
a
, which are equal to
1
2
a
+
2
b
.
We also show blue and red angles with tangents
1
a
and
1
b
, respectively. The angles show that the model is the result of the intersection of two planes
Q=
1
a
x
and
Q=
1
b
y
, and changes to
a
or
b
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.

External Links

Cobb-Douglas Production Functions

Permanent Citation

Timur Gareev
​
​"Leontief Production Function"​
​http://demonstrations.wolfram.com/LeontiefProductionFunction/​
​Wolfram Demonstrations Project​
​Published: May 1, 2018