Examples and exercises on total product functions

Fixed proportions

Suppose that there are two inputs and the production technology has fixed proportions, so that the production function takes the form
F (z1, z2) = min{az1,bz2}.
If z2 is fixed at k, then the total product function is
TP(z1) = min{az1,bk}
or equivalently
TP(z1) =az1 if z1 bk/a
bk if z1 > bk/a
This function is shown in the following figure.

Perfect substitutes

If there are two inputs, and these inputs are perfect substitutes then the production function takes the form
F (z1, z2) = az1 + bz2.
If z2 is fixed at k, the total product function is
TP(z1) = az1 + bk.
This function is shown in the following figure.

Cobb-Douglas production function

If there are two inputs, and production is described by a Cobb-Douglas production function then the production function takes the form
F (z1, z2) = Az1uz2v
If z2 is fixed at k, then the total product function is
TP(z1) = Az1ukv.
For u = 1/2 this function is shown in the following figure.

Exercise

Find the total product function for the production function
F (z1, z2) = z11/2 + z21/2
when z2 = k.

[Solution]

Copyright © 1997 by Martin J. Osborne