Profit maximization

Question

A firm faces the demand schedule P=184-4q and the TC function TC=q³-21q²+160q+40

 

What output will maximize profit?

 

What will maximum profit be?

Answer 1

If p = price, q = quantity, and TC = total cost, then:

 

P = Profit = Revenue - Cost

 

              = (price)(quantity) - cost

 

              = (184-4q)(q) - (q³-21q²+160q+40)

 

              = -q³ + 17q² +24q - 40

 

P' = -3q² + 34q + 24

 

   = (-3q - 2)(q - 12) = 0 when q = -2/3 or q = 12

 

Since q can't be negative, q = 12.

 

When 0<q<12, P'>0.  So P is increasing.

 

When q>12, P'<0.  So P is decreasing

 

Therefore, profit is maximized when q = 12.