so that the marginal product function of input 1 is
TP(z1) = az1 if z1 bk/a bk if z1 > bk/a,
and the average product function of input 1 is
MP(z1) = a if z1 bk/a 0 if z1 > bk/a
These functions are illustrated in the following figure.
AP(z1) = a if z1 bk/a bk/z1 if z1 > bk/a.
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,
and
MP(z1) = 2 if z1 10 0 if z1 > 10
AP(z1) = 2 if z1 10 20/z1 if z1 > 10.
TP(z1) = az1 + bkso 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.
TP(z1) = Az1ukvso that the marginal product function of input 1 is
MP(z1) = Auz1u1kvand 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.
F (z1, z2) = z11/2 + z21/2when z2 = k.
[Solution]