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Chapter 15 Imperfect Competition Nicholson and Snyder, Copyright ©2008 by Thomson South-Western. All rights reserved. Short-Run Decisions: Pricing & Output • When there are only a few firms in a market, predicting output and price can be difficult – how aggressively do firms compete? – how much information do firms have about rivals? – how often do firms interact? Short-Run Decisions: Pricing & Output • Bertrand model – two identical firms choosing prices simultaneously for identical products • end up with situation similar to perfect competition • Cartel model – firms act as a group • end up with the monopoly outcome Short-Run Decisions: Pricing & Output In the Bertrand model, output would be Q* and price would be P* Price Under the cartel model, output would be Q** and price would rise to P** P** MC=AC P* D MR Q** Q* Quantity Short-Run Decisions: Pricing & Output • Cournot model – firms set quantities rather than prices • end up with a result between the Bertrand and the cartel models Short-Run Decisions: Pricing & Output It is important to know where the industry ends up because total welfare depends on price and quantity Price Under Bertrand, there is no DWL P** MC=AC P* The cartel model implies a DWL D MR Q** Q* Quantity Bertrand Model • Two identical firms producing identical products at a constant MC = c • Firms choose prices p1 and p2 simultaneously – single period of competition • All sales go to the firm with the lowest price – sales are split evenly if p1 = p2 Nash Equilibrium of the Bertrand Model • The only pure-strategy Nash equilibrium is p1* = p2* = c – both firms are playing a best response to each other • neither firm has an incentive to deviate to some other strategy Nash Equilibrium of the Bertrand Model • If p1 and p2 > c, a firm could gain by undercutting the price of the other and capturing all the market • If p1 and p2 < c, profit would be negative Nash Equilibrium of the Bertrand Model • The same result will arise for any number of firms n 2 • The Nash equilibrium of the n-firm Bertrand game is p1* = p2* = … = pn*= c Bertrand Paradox • The Nash equilibrium of the Bertrand model is identical to the perfectly competitive outcome • It is paradoxical that competition between as few as two firms would be so tough Cournot Model • Each firm chooses its output qi of an identical product simultaneously • Total industry output Q = q1 + q2 +…+ qn determines the market price P(Q) – P(Q) is the inverse demand curve corresponding to the market demand curve Cournot Model • Each firm recognizes that its own decisions about qi affect price – P/qi 0 • However, each firm believes that its decisions do not affect those of any other firm – qj /qi = 0 for all j i Cournot Model • The FOC for profit maximization are i P Q P ' Q qi C 'i (qi ) 0 qi • The firm maximizes profit where MRi = MCi Cournot Model • Price exceeds marginal cost by P ' Q q i Cournot Model • Price will exceed marginal cost, but industry profits will be lower than in the cartel model – social welfare is greater in the Cournot model than in the cartel situation Cartel Model • In the cartel model, each firm chooses qi for each firm so as to maximize total industry profits n n n j 1 j 1 j 1 j P Q q j C j (q j ) Cartel Model • The FOC for a maximum gives n n j P Q P ' Q q j C 'i (qi ) 0 qi j 1 j 1 • This is the same result as Cournot, except that price exceeds marginal cost by n P ' Q q j P ' Q Q j 1 Natural Springs Duopoly • Assume that there are two owners of natural springs – firm’s cost of pumping and bottling qi liters is Ci(qi) = cqi – each firm has to decide how much water to supply to the market • The inverse demand function is P(Q) = a – Q Natural Springs Duopoly • In the Bertrand game the two firms set price equal to marginal cost P* = c total output = Q* = a – c *i = 0 total profit for all firms = * = 0 Natural Springs Duopoly • The solution for the Cournot model is similar 1 = P(Q)q1 – cq1 = (a – q1 – q2 – c)q1 2 = P(Q)q2 – cq2 = (a – q1 – q2 – c)q2 a q2 c q1 2 a q1 c q2 2 Natural Springs Duopoly • The Nash equilibrium will be q1* = q2* = (a – c)/3 total output = Q* = (2/3)(a – c) P* = (a + 2c)/3 1* = 2* = (1/9)(a – c)2 total profit for all firms = * = (2/9)(a – c)2 Natural Springs Duopoly • The objective function for a perfect cartel involves joint profits 1 + 2 = (a – q1 – q2 – c)q1 + (a – q1 – q2 – c)q2 • The FOCs for a maximum are 1 2 1 2 a 2q 1 2q 2 c 0 q 1 q 2 Natural Springs Duopoly • These FOCs do not pin down the market shares for the firms in a perfect cartel total output = Q* = (1/2)(a – c) P* = (1/2)(a + c) total cartel profit = * = (1/4)(a – c)2 Cournot Best-Response Diagrams • We can also show each firm’s bestresponse function graphically – the intersection of these best-response functions is the Nash equilibrium Cournot Best-Response Diagrams The intersection of the firms’ bestresponse functions is the Nash equilibrium q2 a-c BR1(q2) The Nash equilibrium is where q1* = q2* = (a – c)/3 a c 2 a c 3 BR2(q1) a c a c 3 2 a-c q1 Cournot Best-Response Diagrams A change in a firm’s marginal cost will shift its best-response function q2 BR1(q2) If firm 1’s marginal cost rises, its bestresponse-function will shift in and there will be a new Nash Equilibrium BR2(q1) q1 Varying the Number of Cournot Firms • The Cournot model can represent the whole range of outcomes by varying the number of firms – n = perfect competition – n = 1 perfect cartel / monopoly total output = Q* = (1/2)(a – c) P* = (1/2)(a + c) total cartel profit = * = (1/4)(a – c)2 Varying the Number of Cournot Firms • In equilibrium, identical firms will produce the same share of output qi = Q/n • The difference between price and marginal cost becomes P’(Q)Q/n – this wedge term gets smaller as the number of firms gets larger Prices or Quantities? • Moving from price competition to quantity competition changes the outcome dramatically – an advantage of the Cournot model is the realistic implication that the increases in the number of firms makes the market more competitive • but real-world firms tend to set prices rather than quantities Capacity Constraints • Firms must have unlimited capacity for the Bertrand model to generate the Bertrand paradox – more realistically, firms may not have an unlimited ability to meet all demand Capacity Constraints • Consider a two-stage game – firms build capacity in the first stage – firms choose prices p1 and p2 in the second stage – sales of firms cannot exceed the capacity chosen in the first stage Capacity Constraints • If the cost of building capacity is sufficiently high, the equilibrium of this game is the same as the Nash equilibrium of the Cournot model – firms choose the price at which quantity demanded equals total capacity Price Leadership Model D represents the market demand curve Price SC SC represents the supply decisions of all of the n-1 firms in the competitive fringe D Quantity 34 Price Leadership Model Price We can derive the demand curve facing the industry leader SC P1 P2 For a price of P1 or above, the leader will sell nothing For a price of P2 or below, the leader has the market to itself D Quantity 35 Price Leadership Model Between P2 and P1, the demand for the leader (D’) is constructed by subtracting what the fringe will supply from total market demand Price SC P1 PL D’ P2 MC’ MR’ QL D The leader would then set MR’ = MC’ and produce QL at a price of PL Quantity 36 Price Leadership Model Price SC Market price will then be PL P1 The competitive fringe will produce QC and total industry output will be QT (= QC + QL) PL D’ P2 MC’ MR’ QC QL QT D Quantity 37 Price Leadership Model • This model does not explain how the price leader is chosen or what happens if a member of the fringe decides to challenge the leader • The model does illustrate one tractable example of the conjectural variations model that may explain pricing behavior in some instances 38 Stackelberg Leadership Model • The assumption of a constant marginal cost makes the price leadership model inappropriate for Cournot’s natural spring problem – the competitive fringe would take the entire market by pricing at marginal cost (= 0) – there would be no room left in the market for the price leader 39 Stackelberg Leadership Model • There is the possibility of a different type of strategic leadership • Assume that firm 1 knows that firm 2 chooses q2 so that q2 = (120 – q1)/2 • Firm 1 can now calculate the conjectural variation q2/q1 = -1/2 40 Stackelberg Leadership Model • This means that firm 2 reduces its output by ½ unit for each unit increase in q1 • Firm 1’s profit-maximization problem can be rewritten as 1 = Pq1 = 120q1 – q12 – q1q2 1/q1 = 120 – 2q1 – q1(q2/q1) – q2 = 0 1/q1 = 120 – (3/2)q1 – q2 = 0 41 Tacit Collusion • Tacit collusion is not the same as an explicit cartel – can only be enforced through punishments internal to the market Product Differentiation • Firms often devote considerable resources to differentiating their products from those of their competitors – quality and style variations – warranties and guarantees – special service features – product advertising 43 Product Differentiation • The law of one price may not hold, because demanders may now have preferences about which suppliers to purchase the product from – there are now many closely related, but not identical, products to choose from • We must be careful about which products we assume are in the same market 44 Product Differentiation • The output of a set of firms constitute a product group if the substitutability in demand among the products (as measured by the cross-price elasticity) is very high relative to the substitutability between those firms’ outputs and other goods generally 45 Product Differentiation • We will assume that there are n firms competing in a particular product group – each firm can choose the amount it spends on attempting to differentiate its product from its competitors (zi) • The firm’s costs are now given by total costs = Ci (qi,zi) 46 Product Differentiation • Because there are n firms competing in the product group, we must allow for different market prices for each (p1,...,pn) • The demand facing the ith firm is pi = g(qi,pj,zi,zj) • Presumably, pi/qi 0, pi/pj 0, pi/zi 0, and pi/zj 0 47 Product Differentiation • The ith firm’s profits are given by i = piqi –Ci(qi,zi) • In the simple case where zj/qi, zj/zi, pj/qi, and pj/zi are all equal to zero, the first-order conditions for a maximum are i p i C i pi q i 0 q i q i q i i p i C i qi 0 z i z i z i 48 Product Differentiation • At the profit-maximizing level of output, marginal revenue is equal to marginal cost • Additional differentiation activities should be pursued up to the point at which the additional revenues they generate are equal to their marginal costs 49 Product Differentiation • The demand curve facing any one firm may shift often – it depends on the prices and product differentiation activities of its competitors • The firm must make some assumptions in order to make its decisions • The firm must realize that its own actions may influence its competitors’ actions 50 Spatial Differentiation • Suppose we are examining the case of ice cream stands located on a beach – assume that demanders are located uniformly along the beach • one at each unit of beach • each buyer purchases exactly one ice cream cone per period – ice cream cones are costless to produce but carrying them back to one’s place on the beach results in a cost of c per unit traveled 51 Spatial Differentiation L Ice cream stands are located at points A and B along a linear beach of length L A E B Suppose that a person is standing at point E 52 Spatial Differentiation • A person located at point E will be indifferent between stands A and B if pA + cx = pB + cy where pA and pB are the prices charged by each stand, x is the distance from E to A, and y is the distance from E to B 53 Spatial Differentiation L a x y b a+x+y+b=L A E B 54 Spatial Differentiation • The coordinate of point E is pB p A cy x c pB p A x Lab x c 1 pB p A x L a b 2 c 1 p A pB y L a b 2 c 55 Spatial Differentiation • Profits for the two firms are 1 pA pB p A2 A p A (a x ) (L a b ) p A 2 2c 1 pA pB pB2 B pB ( b y ) (L a b ) pB 2 2c 56 Spatial Differentiation • Each firm will choose its price so as to maximize profits A 1 pB p A (L a b ) 0 p A 2 2c c B 1 p A pB (L a b ) 0 pB 2 2c c 57 Spatial Differentiation • These can be solved to yield: ab pA c L 3 ab pB c L 3 • These prices depend on the precise locations of the stands and will differ from one another 58 Spatial Differentiation L a x y b Because A is somewhat more favorably located than B, pA will exceed pB A E B 59 Spatial Differentiation • If we allow the ice cream stands to change their locations at zero cost, each stand has an incentive to move to the center of the beach – any stand that opts for an off-center position is subject to its rival moving between it and the center and taking a larger share of the market • this encourages a similarity of products 60 Tacit Collusion • Repeating the stage game T times does not change the outcome – the only subgame perfect equilibrium is to repeat the stage-game Nash equilibrium in each of the T periods Tacit Collusion • If the stage game is repeated infinitely, the folk theorem applies – any feasible and individually rational payoff can be sustained each period as long as the discount factor () is close enough to 1 Tacit Collusion • Suppose two firms in a duopoly agree to tacitly collude to sustain the monopoly price by using a grim trigger strategy • Successful tacit collusion provides the profit stream V collude M M M 1 2 M ... 2 2 2 2 1 Tacit Collusion • If a firm deviates, it will earn all of the monopoly profit for itself in the current period – the deviation will trigger the grim strategy of marginal cost pricing for all future periods – the stream of profits from deviating is Vdeviate = M Tacit Collusion • For deviation not to be profitable, it must be that Vcollude Vdeviate M 1 M 2 1 1 2 Tacit Collusion • Suppose only 2 firms produce a medical device that is produced at constant average and marginal cost of $10 • The demand for the device is Q = 5,000 – 100P Tacit Collusion • If the Bertrand game is played in a single period, each firm will charge $10 and a total of 4,000 devices will be sold • At the monopoly price, each firm would earn a profit of $20,000 Tacit Collusion • Collusion at the monopoly price is sustainable if 1 20,000 40,000 1 1 2 Tacit Collusion • Now, suppose there are n firms – monopoly profit is $40,000, but each firm’s share is 40,000/n • n firms can successfully collude on the monopoly price if 40,000 1 40,000 n 1 1 1 n Investment, Entry, and Exit • Even when making long-run decisions, an oligopolist must consider how rivals will respond • Crucial to these decisions is how easy it is to reverse a decision once it has been made Investment, Entry, and Exit • Absent strategic considerations, a firm would value flexibility and reversibility • But commitment has value as well – firm can gain first-mover advantage Sunk Costs and Commitment • Sunk costs are expenditures on irreversible investments – these allow the firm to produce in the market but have no residual value if the firm leaves the market – could include expenditures on unique types of equipment or job-specific training of workers First-Mover Advantage in the Stackelberg Model • This model is similar to the duopoly version of the Cournot model except firms move sequentially – firm 1 (the leader) chooses q1 first – firm 2 (the follower) chooses q2 after seeing q1 First-Mover Advantage in the Stackelberg Model • We can solve the model by backward induction – begin with output of the follower (q2) • this results in a best-response function for Firm 2 [BR2(q1)] – substitute BR2(q1) into Firm 1’s profit function 1 = P(q1 + BR2(q1))q1 – C1(q1) First-Mover Advantage in the Stackelberg Model • The FOC is 1 P Q P ' Q q1 P ' Q BR '2 q1 q1 C '1 (q1 ) 0 q1 S – this is the same FOC as in the Cournot model except for the addition of the strategic effect of Firm 1’s output on Firm 2 (S) First-Mover Advantage in the Stackelberg Model • The strategic effect will lead Firm 1 to produce more than it would have in a Cournot model – this leads Firm 2 to lower output – if Firm 2 lowers output, the market price will rise, increasing Firm 1’s revenue from existing sales First-Mover Advantage in the Stackelberg Model • The strategic effect would not occur if – the leader’s output was unobservable to the follower – the leader could reverse its output choice in secret • The leader must be able to commit or else firms are back in the Cournot game Stackelberg Springs • Recall the natural springs duopoly discussed earlier – this time we will assume they choose output levels sequentially – Firm 1 is assumed to be the leader – Firm 2 is assumed to be the follower Stackelberg Springs • Solving for Firm 2’s output, we get its best-response function a q1 c q2 2 • Substituting Firm 2’s best-response function into Firm 1’s profit function, 1 a q1 c 1 a q 1 c q 1 a q 1 c q 1 2 2 Stackelberg Springs • Taking the FOC, 1 a 2q 1 c 0 q 1 2 • This means that q * 1 a c 2 1 a c 2 8 * 1 q * 2 a c 4 1 a c 2 16 * 2 Contrast with Price Leadership • In the Stackelberg game, the leader uses a “top dog” strategy – aggressively overproduces to force the follower to scale back production – the leader earns more (than it would in the Cournot game), while the follower earns less Contrast with Price Leadership • The leader could follow a “puppy dog” strategy – increases its price, producing less output than in a simultaneous-move game – acts less aggressively, leading its rival to compete less aggressively Contrast with Price Leadership • The crucial difference between these two games is that the slopes of the best-response functions differ – “top dog” strategy leads to a downwardsloping best-response function for Firm 2 – “puppy dog” strategy leads to an upwardsloping best-response function for Firm 2 Strategic Entry Deterrence • In some cases, first-mover advantages may be large enough to deter all entry by rivals – however, it may not always be in the firm’s best interest to create that large a capacity Deterring Entry of a Natural Spring • We will now add an entry stage to the Stackelberg Natural Springs example – Firm 2 must decide whether to enter the market after seeing Firm 1’s output level – entry for Firm 2 requires a sunk cost, K2 • Firm 1 incurred sunk cost before the start of the game – we will assume a = 120 and c = 0 Deterring Entry of a Natural Spring • We start by calculating Firm 1’s profit if it accommodates entry – this was done in earlier example q1acc = (a – c)/2 = 60 1acc = (a – c)2/8 = 1,800 Deterring Entry of a Natural Spring • Next, we compute Firm 1’s profit if it deters entry – Firm 1 needs to produce and amount high enough that Firm 2 cannot earn enough profit to cover sunk cost Deterring Entry of a Natural Spring • Firm 2’s best-response function is q2 = (120 – q1)/2 • Substituting into Firm 2’s profit function gives us 120 q 2 2 det 1 2 K 2 Deterring Entry of a Natural Spring • Setting Firm 2’s profit to zero yields q 1det 120 2 K 2 1det 2 K2 (120 2 K2 ) Deterring Entry of a Natural Spring • The final step is to compare 1acc with 1det • The level of K2 at which the firm would be indifferent is K2 = 77 – if K2 < 77, entry is cheap and Firm 1 would have to increase its output to 102 to deter entry Signaling • The ability to signal is another firstmover advantage – if a second mover has incomplete information about the market, it may try to watch the first-mover to learn about market conditions – the first mover may distort its actions to manipulate what the second mover learns Entry-Deterrence Model • Consider a game where two firms choose a price for their differentiated products – Firm 1 is a first mover – Firm 2 is a second mover Entry-Deterrence Model • Firm 1 has private information about its marginal costs – High costs with a probability of Pr(H) – Low costs with a probability of Pr(L) = 1 – Pr(H) • In period 1, Firm 1 serves the market alone – at the end of the period, Firm 2 observes p1 and considers entry Entry-Deterrence Model • If Firm 2 enters, it faces a sunk cost of K2 and learns the true nature of Firm 1’s costs • The firms then behave as duopolists in the second period – choosing prices for differentiated products Entry-Deterrence Model • If Firm 2 does not enter, it obtains a payoff of zero – Firm 1 serves the market alone • Assume there is no discounting between periods Entry-Deterrence Model • Let Dit = duopoly profit for firm i if Firm 1 is of type t (low-cost, high-cost) • Assume that D2L < K2 < D2H – Firm 2 earns more than its entry cost only if Firm 1 is high-cost Entry-Deterrence Model • If Firm 1 is low cost, it has only one relevant action – setting the monopoly price (p1L) • If Firm 1 is high cost, it has two possible actions – set the monopoly price associated with its type (p1H) – choose the same price as the low-cost type (p1L) Entry-Deterrence Model • Let M1t = Firm 1’s monopoly profit if it is of type t • Let R = the loss in Firm 1’s profit if it is high-cost, but chooses p1L Entry-Deterrence Model Separating Equilibrium • In a separating equilibrium, the different types of the first-mover must choose different actions • There is only one possibility for Firm 1 – the low-cost type chooses p1L – the high-cost type chooses p1H Separating Equilibrium • Firm 2 sees Firm 1’s actions – stays out is Firm 1 charges p1L – enters if Firm 1 charges p1H • Would a high-cost Firm 1 prefer to charge a price of p1L? – only if R < M1H – D1H Pooling Equilibrium • If R < M1H – D1H, the high type would like to pool with the low type if pooling deters entry – pooling deters entry if Firm 2’s prior belief that Firm 1 is the high type is low enough that Firm 2’s expected payoff from entering is less than zero Predatory Pricing • The incomplete-information model of entry deterrence may explain why a firm would engage in predatory pricing – charging an artificially low price to prevent potential rivals from entering or to force existing rivals to exit Barriers to Entry • In order for a market to be oligopolistic, there must be barriers to entry – sunk cost to enter – government intervention (patents, licensing) – search costs faced by consumers – product differentiation (brand loyalty) – entry deterrence by existing firms Long-Run Equilibrium • Suppose there are a large number of symmetric firms that are potential entrants into a market – make the decision simultaneously • Entry requires a sunk cost, K • Let n = number of firms that decide to enter Long-Run Equilibrium • Let g(n) = profit earned by a firm (not including sunk cost) – we would expect g’(n) < 0 Long-Run Equilibrium • The sub-game perfect equilibrium number of firms (n*) will satisfy two conditions – they earn enough to cover their entry costs • g(n*) K – an additional firm cannot cover its entry cost • g(n*+1) < K Long-Run Equilibrium • Is the long-run equilibrium efficient? • A benevolent social planner would choose n to maximize CS(n) + ng(n) – nK – CS(n) is equilibrium consumer surplus – ng(n) is equilibrium gross profits – nK is total expenditure on sunk entry costs Long-Run Equilibrium • The long-run equilibrium number of firms (n*) may be greater or less than the social optimum (n**) depending on two effects – the appropriability effect – the business-stealing effect Long-Run Equilibrium • The appropriability effect – the social planner takes account of increased consumer surplus from lower prices – firms do not • This implies that n** > n* Long-Run Equilibrium • The business-stealing effect – entry causes the profits of existing firms to fall – the marginal firm does not consider the drop in other firms’ profits when making its entry decision (the social planner would) • This implies that n* > n** Feedback Effect • The feedback effect is that the more profitable a market is for a given number of firms, the more firms will enter the market, making the market more competitive and less profitable than it would be if the number of firms was fixed Monopoly on Innovation • The dissipation effect – competition dissipates some of the profit from innovation and thus reduces the incentives to innovate • The replacement effect – firms gain less in incremental profit and thus have less incentive to innovate if the new product replaces an existing product Competition for Innovation • New firms are not always more innovative than existing ones – the dissipation effect may counteract the replacement effect • Dominant firms apply for “sleeping patents” to prevent entry – patents that are never implemented Important Points to Note: • One of the most basic oligopoly models, the Bertrand model, involves two identical firms that set prices simultaneously – the equilibrium resulted in the Bertrand paradox • even though the oligopoly is as concentrated as possible, the two firms act as perfect competitors Important Points to Note: • The Bertrand paradox is not the inevitable outcome in an oligopoly but can be escaped by changing assumptions – allowing for quantity competition, differentiated products, search costs, capacity constraints, or repeated play leading to collusion Important Points to Note: • As in the Prisoners’ Dilemma, firms could profit by coordinating on a less competitive outcome – this outcome will be unstable unless firms can explicitly collude by forming a legal cartel or tacitly collude in a repeated game Important Points to Note: • For tacit collusion to sustain supercompetitive profits, firms must be patient enough that the loss from a price war in future periods to punish undercutting exceeds the benefit from undercutting in the current period Important Points to Note: • Whereas a nonstrategic monopolist prefers flexibility to respond to changing market conditions, a strategic oligopolist may prefer to commit to a single choice – the firm can commit to the choice if it involves a sunk cost that cannot be recovered if the choice is later reversed Important Points to Note: • A first mover can gain an advantage by committing to a different action from what it would choose in the Nash equilibrium of a simultaneous game – to deter entry, the first mover should commit to reducing the entrant’s profits – if it does not deter entry, the first mover should commit to a strategy that leads its rival to compete less aggressively Important Points to Note: • Holding the number of firms in an oligopoly constant in the short run, an introduction of a factor that softens competition will raise firms’ profit – an offsetting effect in the long run is that entry will now be more attractive • reducing oligopoly profit Important Points to Note: • Innovation may be even more important than low prices for total welfare in the long run – determining which oligopoly structure is the most innovative is difficult because offsetting effects are involved • dissipation • replacement