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.
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.
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.
y = z11/2z21/2and
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.
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.
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:
and
y if w1 < w2 z1*(y,w1,w2) = 0 or y if w1 = w2 0 if w1 > w2
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.
[Solution]