Oligopoly. Economics Microeconomic Theory II: Strategic Behavior. Shih En Lu. Simon Fraser University (with thanks to Anke Kessler)

Transcription

1 Oligopoly Economics Microeconomic Theory II: Strategic Behavior Shih En Lu Simon Fraser University (with thanks to Anke Kessler) ECON 302 (SFU) Oligopoly 1 / 15

2 Topics 1 Quick review of game theory basics 2 Cournot competition 3 Bertrand competition ECON 302 (SFU) Oligopoly 2 / 15

3 Most Important Things to Learn 1 Solidify understanding of normal-form games: definitions, finding NE, relation between ISD and NE 2 Find best-response functions in Cournot game 3 Determine the Cournot outcome in an oligopolistic market 4 Determine the Bertrand outcome(s) in an oligopolistic market 5 Be able to explain how to rule out other outcomes in the Bertrand game ECON 302 (SFU) Oligopoly 3 / 15

4 Some Review Questions What s the difference between a strategy, an outcome and a Nash equilibrium? Does it ever make sense to compare different players payoffs? When you re doing ISD, do comparisons between player 2 s payoffs help determine whether you should delete a row? Why can a NE, whether or not it is in pure strategies, only involve strategies that survive ISD? If a game is dominance solvable, what can you say about its NE? If you need to find the NE of a game that s bigger than 2x2, what could you do first to reduce your workload? ECON 302 (SFU) Oligopoly 4 / 15

5 Oligopoly Armed with game theory, let s return to the study of market power. What happens when the number of sellers is small, but greater than one? Intuitively, we expect the firms to have some market power, but not as much as a monopoly, so we expect an intermediate outcome. First: Assume firms simultaneously pick quantities (Cournot model) Then: Assume firms simultaneously pick prices (Bertrand model) Later this semester: Firms make decisions sequentially ECON 302 (SFU) Oligopoly 5 / 15

6 Cournot Competition: Basic Model Let s start with the simplest case: two firms with the same constant marginal cost c produce a homogeneous good with linear market demand P = a bq. We look for a Nash equilibrium. Actions: Firm 1 picks q 1 0, firm 2 picks q 2 0. Infinitely many actions! Let s fix firm 2 s quantity q 2, and figure out firm 1 s best response. Firm 1 maximizes: (P c)q 1 = (a b(q 1 + q 2 ) c)q 1 = bq1 2 + (a bq 2 c)q 1 (This is strictly concave in q 1 for any q 2. Is firm 1 s best response ever a mixed strategy?) ECON 302 (SFU) Oligopoly 6 / 15

7 Cournot Competition: Basic Model (II) First-order condition: 2bq 1 + a bq 2 c = 0 a c 2b q 2 2 = q 1 Similarly, firm 2 s best response to firm 1 picking q 1 is: a c 2b q 1 2 = q 2 In a NE, firms best respond to each other, so both these best-response functions (or reaction functions) must hold. Thus we solve the system of equations, which gives: q 1 = q 2 = a c 3b ECON 302 (SFU) Oligopoly 7 / 15

8 Cournot Competition: Basic Model (III) The total market quantity is thus Q = q 1 + q 2 = 2 a c 3 b > 1 a c 2 b = q m. The price is therefore lower under a Cournot duopoly than under a monopoly. Deadweight loss is also lower, but this conclusion changes if fixed costs are high enough. You can check that the total profit is 2 9 (a c) 2 b. (a c) 2 b, which is less than the monopoly profit of 1 4 The Cournot duopoly fails to maximize profit: firms are not internalizing the negative effect of their production on the other firm s price. Firms could increase profit by each producing 1 2 q m (cooperating), but each would have an incentive to produce more (defect). ECON 302 (SFU) Oligopoly 8 / 15

9 Exercise Same setup (constant MC = c, demand P = a bq), but now n firms. Solve for the Cournot quantities. What happens as n? ECON 302 (SFU) Oligopoly 9 / 15

10 Additional Topics (Time Permitting) Graphical representation Merging firms with increasing marginal cost (you will see this on the problem set) ECON 302 (SFU) Oligopoly 10 / 15

11 Cournot Summary Firms simultaneously pick quantity. Look for Nash equilibria: where the best response functions intersect. With linear demand (and typically), a firm wants to produce less when others produce more (quantities are strategic substitutes). Total quantity and price are between monopoly and perfect competition cases. Sellers fail to maximize joint profit. Outcome converges to perfect competition outcome when the number of firms. ECON 302 (SFU) Oligopoly 11 / 15

12 Bertrand Competition The Cournot model delivered sensible results. But do we observe firms setting quantity or price? Bertrand model: firms simultaneously choose price. Let Q d (.) be the demand curve. Firm with lowest price p L sells Q d (p L ) units, while firm(s) with higher price(s) sell nothing. (Homogeneous good) The following is often, but not always, assumed: if n firms share the lowest price p L, then each sells 1 n Q d (p L ) units. ECON 302 (SFU) Oligopoly 12 / 15

13 Bertrand Competition: Simplest Case Two firms with the same constant marginal cost c (and no fixed cost) face downward-sloping demand that crosses c at a positive quantity. We look for pure-strategy Nash equilibria. (There sometimes exist NE that are not in pure strategies. They are hard to find, and we will ignore them.) For simplicity, we assume that any real number can be a price (not just multiples of 0.01). ECON 302 (SFU) Oligopoly 13 / 15

14 Bertrand Competition: Simplest Case (II) In a NE, can a firm pick a price below c? In a NE, can both firms pick prices strictly above c? In a NE, can one firm choose c and the other firm choose a price strictly above c? What case is left? Is that a NE? Graphical representation ECON 302 (SFU) Oligopoly 14 / 15

15 Bertrand Competition: Discussion Does the NE seem plausible? Some explanations: 1 Product differentiation 2 Capacity constraints 3 Collusion (later this semester) Even though in the examined case, the Bertrand model seems easier to solve than the Cournot model, it gets very messy very fast when some simplifying assumptions are omitted! See problem set for examples. ECON 302 (SFU) Oligopoly 15 / 15