Quiz 7

Q: Market demand is given by P=15-Q. There are two firms, each with TC=q^{2}/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 MC_{1}=MC_{2}.
Otherwise the cartel could increase profits simply by moving production from the
high cost location to the low cost location. Market Q=q_{1}+q_{2}=MC_{1}+MC_{2}=
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*3-3^{2}/2=22.5.

2) Suppose that it is firm 1 that defects and that firm 2 honors the cartel
agreement and produces q_{2}=3. Firm 1's problem is to maximize its
profits which are (15-q_{1}-q_{2})q_{1}-q_{1}^{2}/2.
Taking the derivative and setting it equal to 0 (which is the same as setting MR=MC)
we find that 15-2q_{1}-q_{2}-q_{1}=0. This means
that the best firm 1 can do given the choice of firm 2 is q_{1}=(15-q_{2})/3.
Since firm 2 is making q_{2}=3 firm 1 should make q_{1}=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*4-4^{2}/2=24.