Question

Consider the following market game: An incumbent firm, called firm 3, is already in an industry.

Two potential entrants, called firms 1 and 2, can each enter the industry by paying the entry cost of 2. First, firm 1 decides whether to enter or not. Then, after observing firm 1's choice, firm 2 decides whether to enter or not. Every firm, including firm 3, observes the choices of firms 1 and 2.

After this, all of the firms in the industry (including firm 3) compete in a Cournot oligopoly, where they simultaneously and independently select quantities. The price is determined by the inverse demand curve p = 12-Q, where Q is the total quantity produced in the industry. Firm 3's total cost is  c₃(q₃) = q₃. The total cost of the entering firm i is given by c_i(q_i) = 6.3q_i where i can be 1 or 2. (Remember that firms 1 and 2 have to pay the fixed cost of 2 to enter and that neither firm 1 or firm 2 might enter.)

What is the total quantity produced in the market?

Expert Answer

Answer 1

Solution:

Firms demand curve, p=12-Q, where Q = q1+q2+q3

Given TC of firm 3, TC3= c3(q3) =q3

Ci(qi) =6.3qi, i= 1 or 2

P=12-(q1+q2+q3)

=12-q1-q2-q3

Firm 3, TR3 = p.q3

(12-q1-q2-q3)q3

=12q3-q1q3-q2q3-q3²

MR3= 12-q1-q2-2q3

TC3= C3(q3)=q3

MC= d/dq3= 1

We know MR=MC,

12-q1-q2-2q3=1

=-q1-q2-2q3=1-12

= q1+q2+2q3=11 ---------------------- eqn (1)

Now let i=1

Firm 1, C1(q1)=6.3 q1+ 2 ( where 2 is the fixed cost)

MC=6.3

TR1=p.q1

=12q1-q1²-q2q1-q3q1

MR1= 12-2q1-q2-q3

MR1=MC1

=12-2q1-q2-q3=6.3

=-2q1-q2-q3=6.3-12

=2q1+q2+q3=5.7-------------------------------eqn (2)

Firm 2, i=2

C2(q2)=6.3 q2+2, where 2 is the fixed cost

MC2= d/dq2=6.3

TR2= p.q2

=(12-q1-q2-q3)q2

=12q2-q1q2-q2²-q3

MR2= 12-q1-2q2-q3

MR2=MC2

=12-q1-2q2-q3=6.3

=-q1-2q2-q3=6.3-12

=q1+2q2+q3=5.7----------------------------------eqn (3)

Now, solving equation 1 and 2

q1+q2+2q3=11

2q1+q2+q3=5.7 (subtracting)

-q1       +q3=5.3-----------------------------------eqn (4)

Again solving eqn 1 and 3

q1+q2+2q3=11

q1+2q2+q3=5.7 (subtracting)

-q2+q3=5.3-----------------------------------eqn(5)

Now, solving equation 2 and 3

2q1+q2+q3=5.7

q1+2q2+q3=5.7 (subtracting)

q1-q2 =     0

q1=q2-----------------------------eqn (6)

putting eqn (6) in eqn(4)

-q1+q3=5.3------------------------------eqn(7)

Solving equation 7 and 5

q2+q3=5.3

-q2+q3=5.3

    2q3=10.6

q3=10.6/2 = 5.3

q3=5.3

q2=5.3-5.3=0

q2=q1=0

Therefore =, the total quantity produced in the market is q1=0,q2=0q3=5.3

Q= 0+0+5.3=5.3