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Cournot versus Stackelberg

x 1 x 2. Cournot versus Stackelberg. Cournot duopoly (simultaneous quantity competition) Stackelberg duopoly (sequential quantity competition). x 1. x 2. Homogeneous duopoly (linear case). Two firms (i=1,2) produce a homogenous goods. Outputs: x 1 and x 2 , X= x 1 +x 2

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Cournot versus Stackelberg

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  1. x1 x2 Cournot versus Stackelberg • Cournot duopoly (simultaneous quantity competition) • Stackelberg duopoly (sequential quantity competition) x1 x2

  2. Homogeneous duopoly(linear case) • Two firms (i=1,2) produce a homogenous goods. • Outputs: x1 and x2, X= x1+x2 • Marginal costs: MC1 and MC2 • Inverse demand function: • Profit function of firm 1:

  3. Cournot-Nash equilibrium • Profit functions: • Reaction functions: • Nash equilibrium:

  4. Note: alone leads to a price of . Reaction curve in the linear case x 2 x 1

  5. Computing the Cournot equilibrium(accommodation) • Profit function of firm 1 • Reaction function of firm 1 • Nash equilibrium

  6. x 2 Cournot-Nash equilibrium C x 1 Depicting the Cournot equilibrium

  7. Exercise (Cournot) • Find the equilibrium in a Cournot competition. Suppose that the demand function is given by p(X) = 24 - X and the costs per unit by c1 = 3, c2 = 2.

  8. Common interests • c1, c2obtaining government subsidies and negotiating with labor unions • a , b advertising by the chemistry industrie

  9. Exercise (taxes in a duopoly) Two firms in a duopoly offer petrol. The demand function is given by p(X)=5-0.5X.Unit costs are c1=0.2 and c2=0.5. a) Find the Cournot equilibrium and calculate the price. b) Now suppose that the government imposes a quantity tax t (eco tax). Who ends up paying it?

  10. Two approaches to cost leadership • Direct approach (reduction of own marginal costs)- change of ratio between fixed and variable costs- investments in research and development (R&D) • Indirect approach (“raising rivals’ costs”)- sabotage- minimum wages, enviromental legislation

  11. Direct approach, graphically x 2 marginal cost reduction of firm 1 equilibria: increase in production of firm 1 x 1

  12. Direct approach, analytically • Direct approach (reduction of your own marginal costs): <0 <0 >0 =0direct strategic effect effect

  13. x 2 equilibria: increase in production of firm 1 increase of marginal costs of firm 2 x 1 Indirect approach, graphically

  14. Indirect approach, analytically • Indirect approach (raising rival’s cost): =0 <0 <0 =0direct strategic effect effect

  15. Exercise (Strategic trade policy) • Two firms, one domestic (d), the other foreign (f), engage in Cournot competition on a market in a third country. Assume • The domestic government subsidizes its firm’s exports using a unit subsidy s. • Which subsidy s maximizes domestic welfare

  16. Blockaded entry, graphically x 2 firm 1 as a monopolist C M x 1

  17. Blockaded entry • Entry is blockaded for each firm: • Entry is blockaded for firm 2:

  18. Blockaded entry (overview) c 2 no supply firm 1 as a monopolist firm 2 as a monopolist duopoly c 1

  19. The Herfindahl index • The Herfindahl index measures the concentration in an industry. • Exercise:Which market is the most concentrated: • 2 firms with equal market shares, • 3 firms with shares of 0.8, 0.1 and 0.1 or • 3 firms with shares of 0.6, 0.2 and 0.2 ?

  20. n firms in Cournot competition • Total industry output: • Firm i’s profit function: • Firm i’s marginal revenue:

  21. Lerner index of monopoly power • First order condition: • Lerner index for one firm: • Lerner index for the industry:

  22. Stackelberg equilibrium • Profit function • Follower’s reaction function • Leader’s optimal quantity • Nash equilibrium:

  23. Finding the profit-maximizing point on the follower's reaction curve x 2 Accommodation Blockade or deterrence x 1

  24. Computing the Stackelberg equilibrium (accommodation) • Reaction function of firm 2: • Profit function of firm 1: • Nash equilibriumwith and

  25. Depicting the Stackelberg outcome (both firms produce) x 2 quantities in a Stackelberg equilibrium C S x 1

  26. Exercise (Equilibria) • Which is an equilibrium in the Stackelberg model? • Are there any additional Nash equilibria ?

  27. Cournot versus Stackelberg II • Profit function of firm 1 • First order condition for firm 1direct effect follower effect Cournot: 0

  28. Exercise (Stackelberg) • Find the equilibrium in a Stackelberg competition. Suppose that the demand function is given by p(X) = 24 - X and the costs per unit by c1 = 3, c2 = 2. • Possible or not? : ?

  29. Blockaded entry p blockaded entry for firm 2 x 1

  30. Reaction functions in the case of blockaded entry

  31. Deterring firm 2’s entry p x 1

  32. Reaction functions in the case of deterred entry

  33. Blockaded and deterred entry • Blockaded entry (firm 2): • Deterred entry (firm 2):

  34. Blockade and deterrence c 2 no supply blockade firm 1 as a monopolist deterrence firm 2 as a monopolist duopoly c 1

  35. Exercise I (entry and deterrence) 1) Suppose a monopolist faces a demand of the form p(X)=10-0.5X. The firm’s unit costs are 2. a) Find the profit-maximizing quantity and price. Is entry blockaded for a potential entrant with unit costs of 8? b) Assume now that the potential entrant’s unit costs decrease to 4. Is entry still blockaded? c) Find the limit output level and price. Should the incumbent deter?

  36. Exercise II (entry and deterrence) 2) In addition, the firms are supposed to face quasi-fixed costs of 1. Hence, the cost functions are given by a) What is the monopolist’s profit-maximizing quantity and price? b) Is entry blockaded for the potential entrant? c) Find the incumbent’s profit-maximizing output level. (Hint: Compare the profits.)

  37. The quantity cartel • The firms seek to maximize joint profits • Optimization conditions

  38. The cartel agreement • The optimization condition is given by • Each firm will be tempted to increase its profits by unilaterally expanding its output. • In order to maintain a cartel, the firms need a way to detect and punish cheating, otherwise the temptation to cheat may break the cartel.

  39. Cartel quantities x 2 quantities in a symmetric cartel C S CA x 1

  40. Summary • The cartel is unstable if the firms meet only once. Thus, the cartel agreement is not an equilibrium. • Supposing that the number of repetitions is unknown, the cartel agreement can supported in equilibrium.

  41. Exercise (cartel quantities) • Consider a cartel in which each firm has identical and constant marginal costs. If the cartel maximizes total industry profits, what does this imply about the division of output between the firms? • Solution:Nothing. Since all firms have the same marginal cost, it doesn’t matter which of them produces the output. Hal R. Varian, Intermediate Microeconomics

  42. price a monopoly (M) and cartel (CA) p M Cournot (C) p C Stackelberg (S) p S perfect competition (PC) p PC = c quantity X M X C X S X PC The outcomes of our models

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