Q: Market demand is given by P=15-Q. There are two firms, each with TC=q2/2.
1) If the firms form a cartel what would be the profits a firm?
2) If one firm honors the cartel agreement while the other firm defects, what would be the profits to the defecting firm?
A: For both firms MC=q.
1) The cartel would jointly decide how much output to make and where to produce. To maximize profits it must be that MC1=MC2. Otherwise the cartel could increase profits simply by moving production from the high cost location to the low cost location. Market Q=q1+q2=MC1+MC2= 2 MC. Thus MC for the cartel is Q/2. TR=(15-Q)Q so MR= 15-2Q. Setting MR=MC the cartel would produce Q/2=15-2Q or Q=6 units. The MC of the 6th units is 6/2=3. Thus each firm will set its production such that its MC=3. Hence each firm will make 3 units. The price will be 15-Q=15-6=9. The profits to a firm are 9*3-32/2=22.5.
2) Suppose that it is firm 1 that defects and that firm 2 honors the cartel agreement and produces q2=3. Firm 1's problem is to maximize its profits which are (15-q1-q2)q1-q12/2. Taking the derivative and setting it equal to 0 (which is the same as setting MR=MC) we find that 15-2q1-q2-q1=0. This means that the best firm 1 can do given the choice of firm 2 is q1=(15-q2)/3. Since firm 2 is making q2=3 firm 1 should make q1=4. The total market quantity would be 4+3=7 and the price would be 15-7=8. Firm 1's profits from defecting would be 8*4-42/2=24.