# Oligopoly A monopoly is when there is only one firm.

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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

27.01 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?

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

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