Examples and exercises on the cost function for a firm with one variable input

Example: fixed proportions

If the firm's production function has fixed proportions then its total product function is
TP(z1) =az1 if z1 bk/a
bk if z1 > bk/a
Thus it is not possible for the firm to produce more than bk units of output---if you like, its cost of doing so is infinite---and if y bk then the amount of input 1 needed to produce y units of output is the value of z1 that satisfies
y = az1,
so that
z1 = y/a.
Thus the firm's variable cost of production is
w1y/a if y bk
 if y > bk.
Thus its total cost of production is
STCk(y) =w1y/a + w2k if y bk
 if y > bk.
This function is shown in the following figure.

Example: perfect substitutes

If the firm's production function has perfect substitutes then its total product function is
TP(z1) = az1 + bk.
That is, if it has z1 units of input 1 then the amount of output it produces is
y = az1 + bk.
Thus if it wants to produce at most bk units of output it need to use no amount of input 1; if it wants to produce more than bk units of output then it needs (y  bk)/a units of z1. That is, the amount of input 1 that it needs is
z1 =0 if y bk
(y  bk)/a if y > bk.
Thus the firm's variable cost of producing y units of output is
VC(y) =0 if y bk
w1(y  bk)/a if y > bk.
and its total cost is
STCk(y) =w2k if y bk
w1(y  bk)/a + w2k if y > bk.
This function is shown in the following figure.

Example: a Cobb-Douglas production function

Suppose that firm's production function is
F (z1, z2) = (z1z2)1/2.
(This function is an example of a Cobb-Douglas production function.) Then the firm's total product function is
TP(z1) = z11/2k1/2.
Thus the amount of input 1 needed to produce y units of output is the value of z1 that satisfies
y = z11/2k1/2.
Isolating z1, we have
z1 = y2/k.
Thus the firm's variable cost of producing y units of output is
VC(y) = w1y2/k
and its total cost is
STCk(y) =w1y2/k + w2k.
This function is shown in the following figure.

Exercise

A firm's production function is
F (z1, z2) = min{2z1,4z2}.
The amount z2 of input 2 is fixed at 20, and the prices of the inputs are w1 = 2 and w2 = 10. Find the firm's (short run) cost functions: VC, FC, STC, SMC, AVC, AFC, and SAC.

[Solution]

Exercise

A firm's production function is
F (z1, z2) = z11/4z21/2.
The amount z2 of input 2 is fixed at 4, and the prices of the inputs are w1 = 2 and w2 = 4. Find the firm's (short run) cost functions: VC, FC, STC, SMC, AVC, AFC, and SAC.

[Solution]


Copyright © 1997 by Martin J. Osborne