Question

3. We will examine the idea of oligopolies in this problem. The examples here will be based on the ideas of competition over quantity for identical goods (i.e. Cournot and Stackelberg) (a) (4 points) We begin with a 3 firm oligopoly and the assumption that each firm is using the minimal efficient scale, i.e. the lowest cost production method. Let market demand be P(Q) = 1200 5Q where Q 2 + 3. Assume each firm has a cost of C(qa) 180q. Show the how much each firm produces and the profit created in this situation, .e find the Nash-Cournot Equilibrium (b) (4 points) Suppose two of the firms from part (a) merge together with cost re- maining the same, i.e. Q q 2 where 2 is produced by what was firms 2 and 3. Show what happens to the price and quantity produced. Does merging in this case make sense, i.e. are profits higher here than the joint profits from part (a)? (c) (4 points) How does your answer change from part (b) if the two firms that merged together gained enough market power to allow them to act like a first mover. Show how quantities and profits change, i.e. find the Stackelberg Equilibrium (d) (4 points) Let demand remain the same with P(Q) 1200-5Q, but suppose when the two firms merged (Q q 2), they were able to lower their cost. The cost for each firm is: C1 (q1) = 180q1 and C2(g2) 90q2. Show the quantity produced and profit earned by each firm in this case based on the Cornot Equilibrium 4 Consider the following game

Answer 1

Market demand: P=1200-5Q where Q= q₁ +q₂ +q₃

a) C(q_i)=180q_i

For firm one,profit=Pq₁-C(q₁ )= q₁(1200-5(q₁ +q₂ +q₃ )) -180q₁

For maximizing profit,we differentiate it with respect to q1 and equate to 0

we get 1200-10q₁ -5q₂ - 5q₃ -180=0

we get (1200-5q₂ -5q₃)-180=10q₁

q₁ =102-1/2q₂ -1/2q₃

Similarly for other 2 firms we will get

q₂ =102-1/2q₁ -1/2q₂

and

q₃ =102-1/2q₁-1/2q₂

From all 3 we get q₁₌q₂ =q₃=Q/3

we get

Q/3=102-Q/6-Q/6

Q=102*3/2=153 and total profit for each firm is (1200-5Q)Q/3-180Q/3=13,005

b) For part 2

so profit (1200-5Q)q₁ -180q₁

differentiating we get 1200-10q₁ -5q₂ -180=0

q₁ = 102- 1/2q₂

Like before we get a similar function for firm 2(combination of firm 2 and 3) and so Q=Q/2

Q/2=102-Q/4

Q=102*4/3=136 and q₁₌q₂ =68

Profit=(1200-5Q)q₁ -180q₁ =23,120 which is less than 2*13,005=26,010 and so it doesn't make sense to merge firms

c) For stackelberg equiilibrium,first we find q₁ as a function of q₂(which will be decided by the first mover)

we get q₁ = 102- 1/2q₂ as before

Now for the leader,we will substitute this function and then maximize the profit

(1200-5(q₂ +102- 1/2q₂ ))q₂-180q₂

Differentiating and equating to 0 we get

q₂ =102

and q₁= 102-51=51

now q₂ =102 for first mover and q₁=51

Profit for firm 2=26,010 and profit for firm 1= 13,005 which is similar to profit for firm 1 and combination of profits of firm 2 and 3 in the first case

d) You can find this similar to case b) by adjusting the Cost function of the second firm from 180q₂ to 90q₂ and thus find the profit maximization of each seperately and then find q₁ and q₂.

Hope this helps,please comment if you would prefer me to solve the last one too.