Examples and exercises on marginal and average product functions

Fixed proportions

If there are two inputs and the production technology has fixed proportions, and input 2 is fixed at k, the total product function is
TP(z1) =az1 if z1 bk/a
bk if z1 > bk/a,
so that the marginal product function of input 1 is
MP(z1) =a if z1 bk/a
0 if z1 > bk/a
and the average product function of input 1 is
AP(z1) =a  if z1 bk/a
bk/z1 if z1 > bk/a.
These functions are illustrated in the following figure.

Specifically, if, for example F (z1, z2) = min{2z1,4z2} and the amount z2 of input 2 is fixed at 5, we have TP(z1) = min{2z1,20}, or

TP(z1) =2z1 if z1  10
20 if z1 > 10,
MP(z1) =2 if z1  10
0 if z1 > 10
and
AP(z1) =2 if z1  10
20/z1 if z1 > 10.

Perfect substitutes

If there are two inputs, input 2 is fixed at k, and the two inputs are perfect substitutes then the total product function takes the form
TP(z1) = az1 + bk
so that the marginal product function of input 1 is
MP(z1) = a,
the average product function of input 1 is
AP(z1) = a + bk/z1.
These functions are illustrated in the following figure.

Note that the marginal product is constant, independent of z1: no matter how much of input 1 the firm is using, the extra output it can obtain from one additional unit of the input is the same, equal to a.

Cobb-Douglas production function

If there are two inputs, input 2 is fixed at k, and the technology is described by a Cobb-Douglas production function then the total product function takes the form
TP(z1) = Az1ukv
so that the marginal product function of input 1 is
MP(z1) = Auz1u1kv
and the average product function of input 1 is
AP(z1) = Az1u1kv.
These functions are illustrated in the following figure.

Specifically, if A = 1 and u = v = 1/2, for example,

MP(z1) = (1/2)z11/2k1/2 = k1/2/(2z11/2)
and
AP(z1) = z11/2k1/2 = k1/2/z11/2.

Exercise

Find the marginal product and average product functions of input 1 for the production function
F (z1, z2) = z11/2 + z21/2
when z2 = k.

[Solution]


Copyright © 1997 by Martin J. Osborne