Quiz 7

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.