An industry demand curve faced by firms in a duopoly is P = 69 - Q, where Q = Q1 + Q2. MC for each firm is 0. How many units should each firm produce? How much money will each firm make?

You first need to establish the benefits function B. For each firm it is equal to the amount produced (q1 for firm 1 and q2 for firm 2) multiplied by the price P, minus cost C. It is

B1 = P. q1 - C1 = (69 - q1 - q2) q1 - C1

B2 = P. q2 - C2 = (69 - q1 - q2) q2 - C2

As firma Will maximize benefits we need the derivative in q1 and q2 for firms 1 and 2 respectively. This will give us

69 - 2q1 - q2 = 0

69 - q1 - 2q2 = 0

Note that the derivative of cost is null as marginal cost is null.

Thus,

q2 = 69 - 2q1

Replacing on the second equation:

69 - q1 - 138 + 4q1 = 0

-69 + 3q1 = 0

q1 = 69/3=23

Replacing in the q2 equation:

q2=69 - 46 = 23

To find the money they make replace in benefits function. First we find piece P=69-23-23=23. Thus:

B1=23*23-C1

B2=23*23-C2

As we don't have a value for C1 and C2 we can't compute a number for benefits. If you have these values you will have the benefits.