The Pricing Game
In chapter 3, we claimed that $80 is the common price that yields B. B. Lean and Rainbow's End the highest possible joint profit in the pricing game. Here, as promised, we provide the supporting calculations.

The demand function for Rainbow's End is Q_RE = 2800 − 100P_RE + 80P_BB, where P_RE and P_BB are the prices charged by Rainbow's End and B. B. Lean respectively. Since the cost per unit for each firm is $20, Rainbow's End's profit is

Q_RE(P_RE − 20) = (2800 − 100P_RE + 80P_BB) (P_RE − 20)

and similarly for B. B. Lean. So if the firms collude and charge a common price P, the profit of each is

(2800 − 20P) (P − 20) = 20 (140 − P) (P − 20).

Profits are zero at either P = 20 (because the profit margin is zero) or 140 (because demand is zero). At this point, you could just use Excel solver to find that joint profits are maximized at the midway point, P = (140 + 20)/2 = 80. If you would like to use calculus, read on:

To find the maximum, we take the first derivative of the profit function and set it equal to zero:

Profit = (2800 − 20P) (P − 20) = −20P² + 3200P − 56000

d(Profit)/dp = −40P + 3200 = 0 at P = 80

We confirm that P = 80 is a maximum by checking that the second derivative is negative:

d²(Profit)/dp² = −40,

which it is. Therefore, joint profits are maximized when they both charge $80.