Oligopoly
• A monopoly is when there is only one firm.
• An oligopoly is when there is a limited number of
firms where each firm’s 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.
Example (from tutorial)
• We had price p=13-Q. (we were choosing
quantity).
• For a monopolist,
– r(q)= q*p(q) where p(q)=13-q. Marginal revenue was
13-2q.
– We had constant marginal cost of 1. Thus, c(q)=q.
– Profit=q*(13-q)-q=q*(12-q)
– What is the choice of q? What does this imply about p?
• Are slight mistakes very costly?
Quantity competition (Cournot 1838)
• Л1=p(q1+q2)q1-c(q1)
• Л2= p(q1+q2)q2-c(q2)
• Firm 1 chooses quantity q1 while firm 2 chooses
quantity q2.
• Say these are chosen simultaneously. An
equilibrium is where
– Firm 1’s choice of q1 is optimal given q2.
– Firm 2’s 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.
FOCs of Cournot
• Л1=(13-(q1+q2))q1-q1=(12-(q1+q2))q1
– Take derivative w/ respect to q1.
– Show that you get q1=6-q2/2.
– This is also called a reaction curve (q1’s reaction to q2).
• Л2= (13-(q1+q2))q2-q2= (12-(q1+q2))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.
Quantity competition
(Stackelberg 1934)
• Л1=p(q1+q2)q1-c(q1)
• Л2= p(q1+q2)q2-c(q2)
• Firm 1 chooses quantity q1. AFTERWARDS,
firm 2 chooses quantity q2.
• An equilibrium now is where
– Firm 2’s choice of q2 is optimal given q1.
– Firm 1’s choice of q1 is optimal given q2(q1).
– That is, firm 1 takes into account the reaction of firm 2
to his decision.
Stackelberg solution
• If D(p)=13-p and c(q)=q, what the equilibrium
quantities and prices.
• Must first solve for firm 2’s decision given q1.
– Maxq2 [(13-q1-q2)-1]q2
• Must then use this solution to solve for firm
1’s decision given q2(q1) (this is a
function!)
– Maxq1 [13-q1-q2(q1)-1]q1
• Which point does firm 2 prefer?
• If firm 1 fixes the quantity, what are
firm 2’s choices?
• For a given q1, what is firm 2’s
preferred choice?
27.01
Reaction curve for Firm 2.
27.02
Stackelberg Equilibrium
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)*(q1+q2).
• Note by substituting p=13-(q1+q2), 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?
Possible Cartel points (note they are Pareto
optimal). Why?
27.05
Cartel