EC 202 Lecture notes 14 Oligopoly I George Symeonidis
Oligopoly When only a small number of firms compete in the same market, each firm has some market power. Moreover, their interactions cannot be ignored. Each firm recognises that its decisions influence the decisions of other firms in the market and are in turn influenced by them. So a firm s decisions depend on what its rivals do or might do. There is no single model of oligopoly we will discuss several. But one method of analysis: Game theory.
Oligopoly: Assumptions Few sellers, or a small number of sellers with high market shares Price-making sellers Sellers behave strategically Many small, price-taking buyers Perfectly informed or not well informed buyers Homogeneous or differentiated product Conditions of entry can vary from completely blocked to perfectly free
A Simple Duopoly Game Two firms, 1 and 2. Each firm sells its own version of a product Each firm chooses the price of its product When choosing its price, a firm does not know the price the other firm has chosen There are only two pricing strategies, price high and price low All strategies and payoffs are known to both firms Firms interact only once
A Simple Duopoly Game Player 2 High Low Player 1 High 15, 15 0, 25 Low 25, 0 5, 5
A Simple Duopoly Game Player 2 High Low Player 1 High 15, 15 0, 25 Low 25, 0 5, 5
Duopoly The Nash equilibrium of this one-shot game is that both firms set a low price. We will see later that this result can change if the firms interact more than once. More generally and realistically, firms can choose between more than two prices (or quantities).
Cournot Duopoly Two firms, 1 and 2, selling the same product Each firm chooses the quantity that it sells Firms choose their respective quantities independently and simultaneously and interact only once Each firm has a cost function C i (q i ) No capacity constraints After the output decisions, the price adjusts according to the demand function p = p(q), where Q = q 1 + q 2 is the total output sold in the market Cost functions and demand are known to both firms
Cournot Duopoly We are looking for the Nash equilibrium of this game: a pair of quantities such that each firm is maximising its profit given the quantity of the other firm. The optimal output of firm 1 depends on firm 2 s output. Suppose firm 1 believes that firm 2 will produce q 20. What is the level of output that would maximise firm 1 s profit? More generally, what is the level of output that would maximise firm 1 s profit for any given level of output produced by firm 2? To find this, we must first obtain firm 1 s residual demand function.
Residual Demand Function for q 2 = q 20 P q 20 Market Demand (assumed to be linear) Firm 1 s residual demand 0 Q
Firm 1 s Best Response to q 2 = q 20 P q 20 MC1 The point where (residual) marginal revenue equals marginal cost gives the best response of firm 1 to q2 = q20. MR 1 0 q Q 1 (q 20 )
Firm 1 s Best Response Function For every possible output of firm 2, we can determine firm 1's best response. The sum of all these points is the best response function of firm 1. Firm 1 s best response function is also called firm 1 s reaction function: q 1 = R 1 (q 2 ). Firm 1 s reaction function gives us the level of output that maximises firm 1 s profit for any possible value of firm 2 s output.
Firm 1 s Reaction Function q 20 q 2 MC 1 Firm 1 s Reaction Function 0 R 1 (q 20 ) q 21 q 20 q 21 MC 1 0 R R 1 (q 20 ) 1 (q 21 ) q 1 0 R 1 (q 21 )
Slope of Firm 1 s Reaction Function The reaction function is downward sloping: the more firm 2 produces (or is expected to produce), the lower the chosen output of firm 1. In other words, the more aggressive firm 2 is, the less aggressive firm 1 is in response: we say that the firms actions, q 1 and q 2, are strategic substitutes.
Cournot Equilibrium We can derive firm 2 s reaction function in a similar manner. At a Nash Equilibrium, firm 1 s output must be a best response to firm 2 s output; and conversely, firm 2 s output must be a best response to firm 1 s output: q 1C = R 1 (q 2C ) and q 2C = R 2 (q 1C ). In other words, none of the firms has an incentive to change its output given the rival s output.
Cournot Equilibrium q 2 q 2 M R 1 The Cournot-Nash equilibrium is given by the intersection of the two reaction functions q 2 C R 2 0 q 1 C q 1 M q 1
Cournot : A Numerical Example Suppose the inverse demand is given by P = 100 Q P = 100 (q1 + q2) and that MC = AC = 10. What is firm 1's profit-maximising output when firm 2 produces, say, 50? Firm 1's residual demand is given by: P = 100 (q1 + 50) = 100 q1 50 = 50 q1. Therefore: MR1,50 = P + q1 (ΔP/Δq1) = 50 q1 + q1 (-1) = 50 2q1. Finally, MR1,50 = MC 50 2q1 = 10 q1,50 = 20.
Cournot : A Numerical Example More generally, when firm 2 produces any given level of output q2, what is firm 1's profit-maximising output? Firm 1's residual demand is given by: P = 100 (q1 + q2) = 100 q1 q2, where q2 is taken as given. Therefore MR1 = P + q1 (ΔP/Δq1) = 100 q1 q2 + q1 (-1) = 100 2q1 q2. Finally, MR1 = MC 100 2q1 q2 = 10 q1 = 45 q2/2. This is firm 1's reaction function. Similarly, one can compute firm 2's reaction function: MR2 = MC 100 q1 2q2 = 10 q2 = 45 q1/2.
Cournot : A Numerical Example At the Cournot equilibrium the two reaction functions intersect. In other words, both equations are satisfied. So to find the Cournot equilibrium we solve the system We get q1* = q2* = 30. q1 = 45 q2/2 q2 = 45 q1/2. Then from the demand curve we obtain P* = 40. Note that P* > MC. Finally, the profit of each firm i is: (P MC) qi = 900.
Cournot Equilibrium: Comparative Statics q 2 R 1 The effect of a reduction in firm 2 s marginal cost At the new Cournot-Nash equilibrium firm 1 produces less (and makes less profit) and firm 2 more (and makes more profit) R 2 R 2 0 q 1
Joint Profit Maximisation If the two firms could agree to choose q 1 and q 2 to maximise the sum of their profits, would they produce more or less than the Cournot output? In the Cournot model, expanding q 1 hurts firm 2 s profits and expanding q 2 hurts firm 1 s profits. When the firms act independently (as in the Cournot model), they do not take these negative externalities into account. But when they maximise joint profits, they do. So they produce less and the market price is higher than the Cournot price. In fact the total output produced is the monopoly output and the price is the monopoly price.
Joint Profit Maximisation q 2 R 1 Isoprofit curves For any output pair in this region both firms make a higher profit than the Cournot equilibrium profit R 2 0 q 1
Incentives to Deviate from Joint Profit Maximisation q 2 R 1 R 2 (q 1J ) q 2 J R 2 0 q 1J R 1 (q 2J ) q 1
Incentives to Deviate from Joint Profit Maximisation The graph also illustrates why firms cannot maximise joint profits when they interact only once. When the other member(s) of a cartel stick to the collusive ouput/price, each individual firm has an incentive to cheat : increase its quantity and hence increase its profit. Collusion is illegal in most countries, so firms cannot sign binding agreements. Does this mean cartels cannot exist? We will see later that collusion can be sustainable under repeated interaction.
Price Competition with Homogenous Product Two firms sell the same product The firms choose the price of their product independently and simultaneously, and they serve the demand that arises at this price The firms interact only once Both firms have the same constant average and marginal cost c No capacity constraints Cost functions and demand known to both firms
Bertrand (Price) Competition We are looking for a Nash equilibrium of this game, i.e. a pair of prices such that each firm is maximising its profit given the price of the other firm. Assumptions about demand: If p i > p j, firm i makes no sales and has zero profit, while firm j covers the entire market at price p j and makes profit of (p j c) D(p j ) If p i = p j, each firm obtains half of the overall demand at that price and makes profit of (p i c) D(p i )/2
Bertrand Competition: Equilibrium To find the Nash equilibrium, we consider cases: pi > pj > c: No, because firm i will want to reduce its price below pj and make a positive profit rather zero pi = pj > c: No, because either firm will want to reduce its price slightly and almost double its profit pi > pj = c: No, because firm j will want to increase its price slightly and make some profit rather zero pi < c and/or pj < c: No, because one or both firms will want to set price equal to c and stop making losses pi = pj = c: YES. This is a Nash equilibrium as none of the firms can increase its profit by deviating.
Bertrand Competition With a homogenous good and equal and constant marginal costs, price competition leads to marginal cost pricing just like perfect competition. Both firms make zero profit. So having control over price is not a guarantee to earn positive profits. This result is independent of the number of firms. How can the Bertrand paradox be resolved? 1. Capacity constraints 2. Product differentiation 3. Repeated interaction
Prices or Quantities? Quantity competition and price competition have very different implications. Quantity competition is less tough than price competition: prices and profits are higher. But do firms compete by setting quantities? Maybe in some markets There is a more general and useful interpretation of quantity competition: competition between firms with capacity constraints.
Prices or Quantities? Capacity-constrained firms have no incentive to reduce price a lot since they cannot produce a lot of extra output competition is less tough. In fact, Cournot outcomes can emerge from a twostage process where firms first choose their capacities and then, given these capacities, they choose prices. The Cournot model is appropriate when firms are capacity-constrained, the Bertrand model when they are not.
Commitment By committing to an action, a party limits its own freedom of action Paradoxically, in a strategic situation, commitment can help a player achieve higher payoffs relative to no commitment provided it is observed by its rivals On the other hand, observing the action of an opponent can make a player worse-off compared to not being able to observe
Stackelberg Duopoly In a Stackelberg duopoly game one of the two firms can credibly commit to moving first and the other can observe this commitment. Let s reconsider our quantity-setting duopoly game assuming that firm 1 commits to choosing its quantity first. Firm 2 will observe q 1 and then choose q 2 and firm 1 knows that.
Stackelberg Equilibrium Firm 1 knows that for each q 1 that it could choose, firm 2 will respond with the corresponding q 2 on its reaction function, i.e. firm 1 knows that choosing q 1 will trigger a response q 2 = R 2 (q 1 ). Firm 1 can therefore choose any point on firm 2 s reaction function as an equilibrium of the duopoly game. Firm 1 will choose the point on R 2 where its own profit is largest.
Stackelberg Equilibrium q 2 R 1 Cournot Equilibrium Stackelberg Equilibrium R 2 0 q 1
Stackelberg : A Numerical Example Inverse demand is given by P = 100 (q1 + q2). MC = AC = 10. First firm 1 (the leader ) chooses a level of output q1. Then firm 2 (the follower ) observes q1 and chooses its level of output q2. Recall that the reaction function of firm 2 is q2 = 45 q1/2. When firm 1 chooses q1 it anticipates the reaction function of firm 2. So it chooses q1 to maximise its profit subject to q2 = 45 q1/2.
Stackelberg : A Numerical Example Formally, the residual demand for firm 1 is given by: P = 100 (q1 + q2) = 100 (q1 + 45 q1/2) = 55 q1/2. So MR1 = P + q1 (ΔP/Δq1) = P + q1 (-1/2) = 55 q1. MR1 = MC 55 q1 = 10 q1* = 45. The reaction function of firm 2 then gives q2* = 22.5. P* = 32.5. Profit of firm 1 = (32.5 10) 45 = 1012.50. Profit of firm 2 = (32.5 10) 22.5 = 506.25.
Stackelberg versus Cournot Cournot price > Stackelberg price > Marginal cost. Why is P > MC? Because both firms have some market power (each acts as a monopolist on its residual demand curve). Profit of Stackelberg leader > Cournot profit. Profit of Stackelberg follower < Cournot profit (but positive). Why? The leader can commit to a level of output before the follower. Since the quantities are strategic substitutes, the higher the output of one firm, the lower that of the other. The leader exploits its firstmover advantage to grab the bulk of the market.
Quiz Question 1 Which of the following is correct for an equilibrium in the Cournot model of oligopoly? (a) Each firm sets own MR = MC (b) Each firm s choice of output depends on the other firms choices of output (c) Each firm makes positive profit at equilibrium (d) All of the above (e) Statements (b) and (c) are correct
Quiz Question 2 A firm s reaction function in a Cournot duopoly model shows: (a) The firm s profit maximising level of output as a function of the rival s marginal cost (b) The firm s profit maximising price as a function of the rival s price (c) The firm s demand as a function of the rival s output (d) The firm s profit maximising price as a function of its marginal cost (e) None of the above
Quiz Question 3 In the Stackelberg (leader-follower) quantity-setting model of duopoly: (a) Each firm produces less than the Cournot level and makes higher profit than the Cournot profit (b) Both firms produce less than the Cournot level and the leader makes more profit than the follower (c) The leader produces more and the follower less than the Cournot level and the leader makes more profit than the follower (d) The leader produces more and the follower less than the Cournot level and both firms make higher profit than the Cournot profit