Q: If TC= 4q2+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= 4q2+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.)