Quiz 6

Q: If TC= 4q^{2}+8q+100 and P=88, find the maximum profit for a
perfectly competitive firm and determine the shutdown price.

A: Profit maximization is where MR=MC. The competitive assumption
is that P=MR, so MR=88. MC=dTC/dq=8q+8.

Thus the optimal quantity is where 88=8q+8 or q=10. Profits would be
88*10-4(10)^{2}-8(10)-100=300.

The shut down price is where AVC=MC. VC= 4q^{2}+8q, so AVC =
4q+8. Setting MC=AVC gives 8q+8=4q+8 or q=0. The value of AVC when
q=0 is 8. (Technically we do not have an AVC when q=0 as it would have
required us to divide VC by 0. But for and price above 8 the firm would
operate and for any price below 8 the firm would prefer to shutdown.)