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

Hence the total cost function isz_{1}*(y,w_{1},w_{2}) =yandz_{2}*(y,w_{1},w_{2}) =y.

TC(TC is shown as a function ofy,w_{1},w_{2}) =w_{1}y+w_{2}y= (w_{1}+w_{2})y.

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

Hence total cost function isz_{1}*(y,w_{1},w_{2}) = 2yandz_{2}*(y,w_{1},w_{2}) =y.

TC(For fixed values ofy,w_{1},w_{2}) =w_{1}·2y+w_{2}y= (2w_{1}+w_{2})y.

Hence total cost function isz_{1}*(y,w_{1},w_{2}) =y/aandz_{2}*(y,w_{1},w_{2}) =y/b.

TC(For fixed values ofy,w_{1},w_{2}) =w_{1}·(y/a) +w_{2}(y/b) =y(w_{1}/a+w_{2}/b).

andy=z_{1}^{1/2}z_{2}^{1/2}

The second condition is equivalent tow_{1}/w_{2}= MRTS.

Hence substituting forw_{1}/w_{2}=z_{2}/z_{1}.

Isolatingy= ((w_{1}/w_{2})z_{1})^{1/2}·z_{1}^{1/2}= (w_{1}/w_{2}))^{1/2}z_{1}.

z_{1}=y(w_{2}/w_{1})^{1/2}andz_{2}=y(w_{1}/w_{2})^{1/2}.

Thus the total cost function is

TC(y,w_{1},w_{2}) =w_{1}y(w_{2}/w_{1})^{1/2}+w_{2}y(w_{1}/w_{2})^{1/2}= 2y(w_{1}w_{2})^{1/2}.

We see that once again TC is a linear function of output *y*, given the input prices *z*_{1} and *z*_{2}.

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 *w*_{1} < *w*_{2} then the
firm uses only input 1: the optimal input bundle in this case is (*y*,0). Finally, if *w*_{1} = *w*_{2} 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

yif w_{1}<w_{2}z_{1}*(y,w_{1},w_{2}) =[0, y]if w_{1}=w_{2}0 if w_{1}>w_{2}

and

0 if w_{1}<w_{2}z_{2}*(y,w_{1},w_{2}) =[0, y]if w_{1}=w_{2}yif w_{1}>w_{2}

Thus the total cost function is

w_{1}yif w_{1}<w_{2}TC( y,w_{1},w_{2}) =wyif w_{1}=w_{2}=ww_{2}yif w_{1}>w_{2}

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 *w*_{1} < *w*_{2} then the optimal input bundle is (*y*,0), and if *w*_{1} = *w*_{2} then the both (0,*y*) and (*y*,0) are optimal, and no other bundles are optimal. Thus the
conditional input demands are:

and

yif w_{1}<w_{2}z_{1}*(y,w_{1},w_{2}) =0 or yif w_{1}=w_{2}0 if w_{1}>w_{2}

0 if w_{1}<w_{2}z_{2}*(y,w_{1},w_{2}) =0 or yif w_{1}=w_{2}yif w_{1}>w_{2}

Thus the total cost function is

w_{1}yif w_{1}<w_{2}TC( y,w_{1},w_{2}) =wyif w_{1}=w_{2}=ww_{2}yif w_{1}>w_{2}

Once again, for given values of *w*_{1} and *w*_{2} the cost function is linear in output *y*.

[Solution]

Copyright © 1997 by Martin J. Osborne