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Oligopoly

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.

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Oligopoly

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  1. 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.

  2. 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?

  3. 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.

  4. 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.

  5. 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.

  6. 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

  7. 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? 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)*(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?

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

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