Oligopoly

1
Oligopoly

• A monopoly is when there is only one firm.
• An oligopoly is when there is a limited number of
  firms where each firms decisions influence the
  profits of the other firms.
• We can model the competition between the firms
  price and quantity, simultaneously sequentially.
• The model where firms that choose price
  simultaneously is Bertrand (week 5 tutorial).
• The model when firms choose quantity
  simultaneously (week 6 tutorial) is Cournot.

2
Example (from tutorial)

• We had price p13-Q. (we were choosing quantity).
  
• For a monopolist,
• r(q) qp(q) where p(q)13-q. Marginal revenue
  was 13-2q.
• We had constant marginal cost of 1. Thus, c(q)q.
• Profitq(13-q)-qq(12-q)
• What is the choice of q? What does this imply
  about p?
• Are slight mistakes very costly?

3
Quantity competition (Cournot 1838)

• ?1p(q1q2)q1-c(q1)
• ?2 p(q1q2)q2-c(q2)
• Firm 1 chooses quantity q1 while firm 2 chooses
  quantity q2.
• Say these are chosen simultaneously. An
  equilibrium is where
• Firm 1s choice of q1 is optimal given q2.
• Firm 2s choice of q2 is optimal given q1.
• If D(p)13-p and c(q)q, what the equilibrium
  quantities and prices.
• Take FOCs and solve simultaneous equations.
• Can also use intersection of reaction curves.

4
FOCs of Cournot

• ?1(13-(q1q2))q1-q1(12-(q1q2))q1
• Take derivative w/ respect to q1.
• Show that you get q16-q2/2.
• This is also called a reaction curve (q1s
  reaction to q2).
• ?2 (13-(q1q2))q2-q2 (12-(q1q2))q2
• Take derivative w/ respect to q2.
• Symmetry should help you guess the other
  equation.
• Solution is where these two reaction curves
  intersect. It is also the soln to the two
  equations.
• Plugging the first equation into the second,
  yields an equation w/ just q2.

5
Quantity competition (Stackelberg 1934)

• ?1p(q1q2)q1-c(q1)
• ?2 p(q1q2)q2-c(q2)
• Firm 1 chooses quantity q1. AFTERWARDS, firm 2
  chooses quantity q2.
• An equilibrium now is where
• Firm 2s choice of q2 is optimal given q1.
• Firm 1s choice of q1 is optimal given q2(q1).
• That is, firm 1 takes into account the reaction
  of firm 2 to his decision.

6
Stackelberg solution

• If D(p)13-p and c(q)q, what the equilibrium
  quantities and prices.
• Must first solve for firm 2s decision given q1.
• Maxq2 (13-q1-q2)-1q2
• Must then use this solution to solve for firm 1s
  decision given q2(q1) (this is a function!)
• Maxq1 13-q1-q2(q1)-1q1

7
27.01

• Which point does firm 2 prefer?
• If firm 1 fixes the quantity, what are firm 2s
  choices?
• For a given q1, what is firm 2s preferred
  choice?

Reaction curve for Firm 2.
8
27.02
Stackelberg Equilibrium
9
Collusion

• If firms get together to set prices or limit
  quantities what would they choose.
• D(p)13-p and c(q)q.
• Quantity Maxq1,q2 (13-q1-q2-1)(q1q2).
• Note by substituting p13-(q1q2), we get a
  problem w/ price choice Maxp (p-1)(13-p)
• Say that the fair collusion point is fixing a
  quantity and splitting it.
• This is the monopoly price and quantity! Show all
  4 possibilities (Cournot, Bertrand, Collusion,
  Stackelberg) on the q1, q2 graph?

10
27.05
Possible Cartel points (note they are Pareto
optimal). Why?
Cartel