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4820–1 Introduction Static oligopoly Geir B. Asheim Introduction & Static oligopoly Introduction 4820–1 Static oligopoly Geir B. Asheim Department of Economics, University of Oslo ECON4820 Spring 2010 Purpose 4820–1 Introduction Static oligopoly Geir B. Asheim Applies game theory to analyze strategic competition Present theoretic models that are consistent with stylized facts Introduction Outline Preliminary analysis Static oligopoly Associates predictions with equilibrium behavior (Must be supplement by empirical analysis; not covered here) Examples: Telenor Mobil, NetCom, others Rimi, Rema, others SAS, low price airlines What is strategic competition? 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Outline Preliminary analysis Static oligopoly American term: Industrial organization. The study of activities within an industry, mainly with respect to competition among the ﬁrms in a product market. Few ﬁrms, seeking to avoid or soften competition Why? The model of perfect competition is unrealistic, because in many industries, large proﬁts, with p > MC Who set the prices? The ﬁrms Can they inﬂuence prices? Yes, if they are few But: diﬃcult to ﬁnd a general model of imperfect competition Many models with varying applications So what does it then mean to explain stylized facts? Characteristica 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Outline Preliminary analysis Static oligopoly Competition in price or quantities Static oligopoly (4820–1) One-time competition or long-term relationship Dynamic oligopoly (4820–2&3) Homogeneous or diﬀerentiated products Product diﬀerentiation (4820–4&5) Competition from potential entrants Entry (4820–6&7) Symmetric or asymmetric information Information (4820–8&9) Other topics: Research & development (4820–10) Auctions (4820–11) Vertical integration (4820–12) Mergers (4820–13) Noncooperative games and strategic behavior 4820–1 Introduction Static oligopoly Geir B. Asheim Assume that the proﬁt of ﬁrm i (i = 1, 2) depends its own action ai and the action aj of the other ﬁrm j: Πi (ai , aj ) Introduction Outline Preliminary analysis Static oligopoly The pair (a1∗ , a2∗ ) is a Nash equilibrium if, for each i (i = 1, 2), Πi (ai∗ , aj∗ ) ≥ Πi (ai , aj∗ ) for all ai One interpretation: If the ﬁrms agrees on (a1∗ , a2∗ ), then the agreement is self-enforcing in the sense that each ﬁrm has not an incentive to deviate from the agreement, provided that it believes that the other ﬁrm will stick to its part. Best response functions: Strategic complements and substitutes 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Outline Preliminary analysis Static oligopoly First-order condition for a Nash equilibrium: For each ﬁrm i, Πii (ai∗ , aj∗ ) = 0 Second-order condition: For each ﬁrm i, Πiii (ai∗ , aj∗ ) ≤ 0 If Πiii (ai , aj ) < 0 for all (ai , aj ), then i’s best response function Ri (aj ) can be deﬁned by Πii (Ri (aj ), aj ) = 0. Ri (aj ) = Πiij (Ri (aj ), aj ) −Πiii (Ri (aj ), aj ) Strategic complements if Ri (aj ) > 0. Strategic substitutes if Ri (aj ) < 0. Static oligopoly 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Price competition: Bertrand model Static oligopoly Bertrand model Cournot model Price or quantity Quantity competition: Cournot model Price or quantity as strategic variable? Bertrand model with homogeneous products 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Structure: (1) Prices set (2) Demand determines quantities D(·): continuous and strictly decreasing demand function for all p with D(p) > 0; D(p) = 0 for p ≥ p̄. Firms 1 and 2 have constant unit cost c and set prices p1 and p2 simultaneously. Sales are given by: ⎧ ⎪ if pi < pj ⎨ D(pi ) 1 Di (pi , pj ) = D(pi ) if pi = pj 2 ⎪ ⎩ 0 if pi > pj Proﬁt is given by: Πi (pi , pj ) = (pi − c)Di (pi , pj ). Write Πm = maxp (p − c)D(p) and p m = arg maxp (p − c)D(p). We have that 0 ≤ Π1 + Π2 ≤ Πm . Why? Unique Nash equilibrium: p1 = p2 = c. Bertrand paradox Relaxing competition 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Firms set prices. But the conclusion of the Bertrand model — that two ﬁrms are suﬃcient for perfectly competitive outcomes — is unrealistic. Competition can be relaxed by Not no capacity contraints, but capacity constraints (Bertrand oligopoly model with capacity constraints; Cournot oligopoly model) Not homogeneous products, but product diﬀerentiation (Bertrand oligopoly model with heterogeneous products; other models) Not a static game, but repeated long-term interaction Cournot model (1) 4820–1 Introduction Static oligopoly Geir B. Asheim Structure: (1) Quantities set (2) Demand determines price Assume ﬁrms ﬁrst choose quantities q1 and q2 and that these are sold at price P(q1 + q2 ) in the market, where P(·) is determined by P(D(p)) = p, with P(0) = p̄. Each ﬁrm i’s proﬁt: Introduction Static oligopoly Bertrand model Cournot model Price or quantity Πi (qi , qj ) = P(qi + qj )qi − C (qi ) max P(qi + qj )qi − C (qi ) qi First-order condition: Πii = P(qi + qj ) − C (qi ) + P (qi + qj )qi = 0 For each qj , let Ri (qj ) denote i’s best response: Πii (Ri (qj ), qj ) = 0 Cournot model (2) 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Nash equilibrium (q1∗ , q2∗ ) satisﬁes, for each ﬁrm i: qi∗ = Ri (q2∗ ) or, equivalently, P(qi∗ + qj∗ ) − C (qi∗ ) + P (qi∗ + qj∗ )qi∗ = 0 This can be rewritten as Li = αi , i is the Lerner index for ﬁrm i, αi ≡ qQi is ﬁrm i’s where Li ≡ P−C P market share, and ≡ − PP Q is the elasticity of demand. Consider a special case: D(p) = 1 − p and Ci (qi ) = ci qi . Consider the generalization to the case of n ﬁrms. Whereas the Cournot model with n = 1 corresponds to monopoly, the Cournot model approaches perfect competition as n → ∞. Concentration indices 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity the m-ﬁrm concentration ratio (for m < n), which adds up the m highest shares in the industry: m Rm ≡ αi i=1 (ordering the ﬁrms so that α1 ≥ · · · ≥ αm ≥ · · · ≥ αn ) the Herﬁndahl index, which is equal to the sum of the squares of the market shares: m αi2 RH ≡ i=1 the entropy index, which is equal to the sum of the shares times their logarithms: m Re ≡ αi ln αi i=1 Total proﬁts under Cournot competition 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Assume that each ﬁrm i has constant unit costs ci . n n Π= Πi = (p − ci )qi i=1 i=1 n n = pLi αi Q = p αi αi Q i=1 i=1 n α2 = pQ = pQ RH i=1 If = 1 so that pQ = k, then Π = kRH implying that the Herﬁndahl index yields an exact measure of industry proﬁtability under these assumptions. Is price or quantity the strategic variable? 4820–1 Introduction Static oligopoly Geir B. Asheim We have seen that price competition yields diﬀerent predictions than quantity competition. Introduction Static oligopoly Bertrand model Cournot model Price or quantity So which model is “right”? To analyze this, consider a setting where quantities/capacities are determined before price. Capacity constrained price competition requires that we specify rationing rules. Rationing rules 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Assume that ﬁrm 1 has capacity constraint q̄1 , and that ﬁrm 1 sets a lower price (p1 < p2 ). What is the demand facing ﬁrm 2? The eﬃcient-rationing rule: A consumer with the highest willingness-to-pay is served ﬁrst. D(p2 ) − q̄1 if D(p2 ) > q̄1 D̃2 (p2 ) = 0 otherwise . The proportional-rationing rule: Any consumer willing to pay a good oﬀered at p has an equal change of buying at p. D(p1 ) − q̄1 . D̃2 (p2 ) = D(p2 ) D(p1 ) Why is eﬃcient rationing eﬃcient? More on eﬃcient rationing 4820–1 Introduction Static oligopoly Geir B. Asheim Let the ﬁrms have capacity constraints q̄1 and q̄2 , and assume no cost of production up to these constraints. If p1 > p2 , then q2 = min{q̄2 , D(p2 )} Introduction Static oligopoly Bertrand model Cournot model Price or quantity If p1 < p2 , then q2 = min{q̄2 , max{0, D(p2 ) − q̄1 }} If p1 = p2 , then 2) q̄2 , D(p 2 D(p2 ) 2 + max 0, − q̄1 D(p2 ) = min q̄2 , max 2 , D(p2 ) − q̄1 q2 = min Price competition with capacity constraints The Bertrand-Edgeworth model 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Let R10 (q2 ) and R20 (q1 ) be the ﬁrms’ Cournot best response functions when costs are zero. Assume eﬃcient rationing. Case 1: Small capacity (q̄1 ≤ R10 (q̄2 ) and q̄2 ≤ R10 (q̄1 )) Larger proﬁt by producing less that q¯1 ? No! Nash equilibrium: p1 = p2 = P(q̄1 + q̄2 ) Case 2: Intermediate capacity ((q̄1 > R10 (q̄2 ) or q̄2 > R10 (q̄1 )) & (q̄1 < D(0) or q̄2 < D(0))) Complicated; Nash equilibria involve mixed strategies. Case 3: Large capacity (q̄1 ≥ D(0) and q̄2 ≥ D(0)) Like ordinary Bertrand model with zero costs: p1 = p2 = 0. Capacity competition followed by price competition 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Structure: (1) Capacities set (2) Prices set (3) Demand determines quantities In the subgames starting at stage (3), all possible proﬁles of capacities and prices must be considered. Use the concept of subgame-perfect equilibrium. Assume eﬃcient rationing. For “small” capacities, prices are determined as in the Cournot model. Hence, if the cost of capacities leads to “small” capacities, then capacities will be determined as in the Cournot model. Kreps & Scheinkman (1983) show formally that ﬁrms will choose “small” capacities (by showing that at least one ﬁrm has a proﬁtable deviation if one ﬁrm does not choose a “small” capacity): With eﬃcient rationing, the game considered here leads to the Cournot outcome. So what is the strategic variable? 4820–1 Introduction Static oligopoly Geir B. Asheim Introduction Static oligopoly Bertrand model Cournot model Price or quantity Principle: The least ﬂexible variable is the strategic variable Usually, capacities are determined before production, which in turn is determined before price. Exceptions?