## Examples and exercises on the cost function for a firm with two variable inputs

### Example: a production function with fixed proportions

Consider the fixed proportions production function F (z1z2) = min{z1z2} (one worker and one machine produce one unit of output). An isoquant and possible isocost line are shown in the following figure.

To produce y units, the firm wants to use y units of each input, no matter what the input prices are. Thus the conditional input demands are

z1*(y,w1,w2) = y and z2*(y,w1,w2) = y.
Hence the total cost function is
TC(y,w1,w2) = w1y + w2y = (w1 + w2)y.
TC is shown as a function of y, for some fixed values of w1 and w2, in the following figure.

### Example: a production function with fixed proportions

Consider the fixed proportions production function F (z1z2) = min{z1/2,z2} (two workers and one machine produce one unit of output). An isoquant and possible isocost line are shown in the following figure.

To produce y units, the firm want to use 2y units of input 1 and y units of input 2, no matter what the input prices are. Thus the conditional input demands are

z1*(y,w1,w2) = 2y and z2*(y,w1,w2) = y.
Hence total cost function is
TC(y,w1,w2) = w1·2y + w2y = (2w1 + w2)y.
For fixed values of w1 and w2, this function is linear in y, like the TC function for the previous example.

### Example: a general production function with fixed proportions

Consider the general fixed proportions production function F (z1z2) = min{az1bz2}. To produce y units, the firm wants to use y/a units of input 1 and y/b units of input 2, no matter what the input prices are. Thus the conditional input demands are
z1*(y,w1,w2) = y/a and z2*(y,w1,w2) = y/b.
Hence total cost function is
TC(y,w1,w2) = w1·(y/a) + w2(y/b) = y(w1/a + w2/b).
For fixed values of w1 and w2, this function is linear in y, line the TC function for the previous example.

### Example: a Cobb-Douglas production function

Consider the production function F (z1z2) = z11/2z21/2. The isoquants of this function are smooth and convex to the origin, and for any input prices the firm optimally uses a positive amount of each input. Thus the conditional input demands satisfy the two conditions
y = z11/2z21/2
and
w1/w2 = MRTS.
The second condition is equivalent to
w1/w2 = z2/z1.
Hence substituting for z2 in the first condition, we find that the conditional input demands satisfy
y = ((w1/w2)z1)1/2·z11/2 = (w1/w2))1/2z1.
Isolating z1 and then substituting into the equation w1/w2 = z2/z1 to obtain z2, we conclude that the conditional input demands are
z1 = y(w2/w1)1/2 and z2 = y(w1/w2)1/2.

Thus the total cost function is

TC(y,w1,w2) = w1y(w2/w1)1/2 + w2y(w1/w2)1/2 = 2y(w1w2)1/2.

We see that once again TC is a linear function of output y, given the input prices z1 and z2.

### Example: a production function in which the inputs are perfect substitutes

Consider the production function F (z1, z2) = z1 + z2, in which the inputs are perfect substitutes. An isoquant and some isocost lines for the case in which w1 > w2 are shown in the following figure.

For such input prices, the optimal input bundle is (0,y): the firms uses only input 2. The reason is clear: the inputs may be substituted for one another one-for-one, so if the price of input 1 exceeds the price of input 2 then the firm uses only input 2. Similarly, if w1 < w2 then the firm uses only input 1: the optimal input bundle in this case is (y,0). Finally, if w1 = w2 then the isocost lines have the slope 1, the same as the isoquant. Thus the firm is indifferent about the input bundle it uses.

In summary, the conditional input demands are

 y if w1 < w2 z1*(y,w1,w2) = [0,y] if w1 = w2 0 if w1 > w2

and

 0 if w1 < w2 z2*(y,w1,w2) = [0,y] if w1 = w2 y if w1 > w2

Thus the total cost function is

 w1y if w1 < w2 TC(y,w1,w2) = wy if w1 = w2 = w w2y if w1 > w2

As in the previous examples, for any fixed values of the input prices the total cost function is linear in output y.

### Example: a production function with nonconvex isoquants

Suppose that the production function is F (z1z2) = (z12 + z22)1/2. In this case the isoquants are quarter-circles. An isoquant and some isocost lines for a case in which w1 > w2 are shown in the following figure.

We see that in this case the optimal input bundle is (0,y). If w1 < w2 then the optimal input bundle is (y,0), and if w1 = w2 then the both (0,y) and (y,0) are optimal, and no other bundles are optimal. Thus the conditional input demands are:

 y if w1 < w2 z1*(y,w1,w2) = 0 or y if w1 = w2 0 if w1 > w2
and
 0 if w1 < w2 z2*(y,w1,w2) = 0 or y if w1 = w2 y if w1 > w2

Thus the total cost function is

 w1y if w1 < w2 TC(y,w1,w2) = wy if w1 = w2 = w w2y if w1 > w2

Once again, for given values of w1 and w2 the cost function is linear in output y.

### Exercise

A firm uses two inputs to produce output; both inputs may be varied. Its production function is y = min{z1,z2/2}. (That is, there are fixed proportions; one unit of input 1 and two units of input 2 efficiently produce one unit of output.) The firm wishes to maximize its profit. Find the firm's cost function TC(y,w1,w2) (where y is output and w1 and w2 are the input prices).

[Solution]