Question

Question 1:

An incumbent in a market is faced with the threat of entry by another firm.  The market demand is linear:  

p=28-Q

and the cost function for each firm is C(qi)=4qi+F

In the first stage, the incumbent commits to its output.  In the second stage, the entrant decides whether to enter the market by paying the entry cost F and, if so, how much to produce.

1. Find the optimal output for the incumbent for the following values of F: 1, 4 and 9.
2. Provide a range for F within which production by the incumbent and entry deterrence are both profitable, if such a range exists.

(Hint:

• note that the choice for the incumbent of the monopoly output will be optimal ifthe [potential] entrant finds it unprofitable to come into the market when the incumbent chooses this output.
  • in this case, entry is blockaded

• if entry is not blockaded, then in order to decide on output, the incumbent must first determine what the entrant’s response would be to each level of output. There is some critical output, q such that if the incumbent chooses an output at or above this level, the entrant will stay out.

• if the monopoly decides to produce less that this amount, then which output will it produce? call this qa for “accommodate.”  Is this the Stackleberg output?

• it is critical to get the overall logic straight, in order to figure out which values we have to compare for the solution.

• you will find it helpful to set up the extensive-form game, and remember to solve the game “backwards”)

Answer 1

a).

Consider the given problem here the demand curve is given by, “P=28-Q”, => TR = P*Q = Q*(28-Q) = 28*Q – Q^2.

=> MR = d(TR)/dQ = 28 – 2*Q and the “MC” is given by, “MC=4”. So, at the optimum “MR” must be equal to “MC”.

=> MR = MC, => 28-2Q=4, => Q = 24/2, => Q=12, => P=28-Q=28-12=16, => P=16.

So, here the optimum profit maximizing “P” and “Q” are given by “Q=12” and “P=16”.

So, the profit is given by.

=> A1 = P*Q – C = 16*12 – F – 4*12 = 144 – F. So, if “F=1, 4, 9” then the profit is given by.

=> A1 = 143, 140 and 135 respectively.

b).

Now, let’s assume that there is an incumbent and an entrant, => the profit functions are given by.

=> A1 = (28-q1-q2)*q1 – F – 4*q1, => A1 = 28*q1 - q1^2 - q2*q1 – F – 4*q1. Now, the FOC require “dA1/dq1 = 0”.

=> 28 – 2*q1 - q2 – 4 = 0, => q1 = 12 – q2/2, be the reaction function of “incumbent” and similarly, the reaction function of “entrant” is given by.

=> q2 = 12 – q1/2. So, if we solve these two equation then the optimum solution is given by, “q1=q2=8”. So, the total production is given by “Q=q1+q2=16” and the market price is given by “P = 28-Q = 28-16 = 12”. So, here at the optimum both the firm will produce same amount of output and have the same type of cost function, => both will earn same profit.

=> Ai = P*q1 – C1 = 12*8 – F – 4*q1 = 12*8 – F – 4*8 = 64 – F, => Ai = 64-F. So, the profit will be positive for all “F < 64”. So, for “0 < = F < 64” then the “incumbent” as well as the “entrant” bot will earn the positive profit.