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Course outline I Introduction Game theory Price setting – monopoly – oligopoly Homogeneous Quantity setting goods – monopoly – oligopoly Process innovation 1 Quantity and cost competition Bertrand versus Cournot Simultaneous quantity competition (Cournot) Sequential quantity competition (Stackelberg) Quantity Cartel Concentration and competition 2 Price or quantity competition? Cournot (1838) x1 1 x2 2 Bertrand (1883) p1 1 p2 2 3 Capacity + Bertrand = Cournot Bertrand (1883) criticized Cournot’s model (1838) on the grounds that firms compete by setting prices and not by setting quantities. Kreps and Scheinkman (1983) defended Cournot’s model. They developed a two-stage game with capacities k1 p1 1 k2 p2 2 and proved that capacities in a Nash equilibrium are determined by Cournot’s model. 4 Cournot versus Stackelberg Cournot duopoly (simultaneous quantity competition) x1 1 x2 2 Stackelberg duopoly (sequential quantity competition) 1 x1 x2 2 5 Homogeneous duopoly (linear case) Two firms (i=1,2) produce a homogenous good. Outputs: x1 and x2, X= x1+x2 Marginal costs: c1 and c2 Inverse demand function: pX a bX a bx1 x2 Profit function of firm 1: 1 x1, x2 p X x1 c1x1 a bx1 x2 c1 x1 6 Cournot-Nash equilibrium Profit functions: 1 ( x1 , x2 ), 2 ( x1 , x2 ) Reaction functions: x ( x2 ) arg max x1 1 ( x1 , x2 ) R 1 x ( x1 ) arg max x2 2 ( x1 , x2 ) R 2 C C Nash equilibrium: ( x , x ) 1 2 x1R ( x2 ) x1 C C x2 ( x1 ) x2 R C C 7 Computing the Cournot equilibrium (accommodation) Profit function of firm 1 1 ( x1 , x2 ) p( X ) x1 c1x1 (a b( x1 x2 ) c1 ) x1 Reaction function of firm 1 a c1 x2 a c2 x1 x ( x2 ) R 1 analogous : x ( x1 ) R 2 2b 2 2b 2 Nash equilibrium C a 2c1 c2 C a 2c2 c1 x1 , x2 3b 3b ( a c1 c2 ) p C 3 ( a 2c1 c2 ) 2 (a 2c2 c1 ) 2 1 C analogous : 2 C 9b 9b 8 Depicting the Cournot equilibrium x2 x1R ( x2 ) M Cournot-Nash x 2 C equilibrium C x2 R x2 ( x1 ) x1C x1M x1 9 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 and c2 = 2. 20 23 S. : x C 1 and x C 2 3 3 10 Common interests c 1, c 2 Obtaining government subsidies and negotiating with labor unions. a , b Advertising by the agricultural industry (e.g. CMA). 11 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? dp 2 S. : a) p 1.9 b) dt 3 12 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 13 Direct approach, analytically 1 c1 , c2 1 c1 , c2 , x1 c1 , c2 , x2 c1 , c2 C C C Direct approach (reduction of your own marginal costs): 1 1 1 x1 1 x2 C C C 0 c1 c1 x1 c1 x2 c1 0 0 0 0 0 direct strategic effect effect 14 Direct approach, graphically x2 x1R ( x2 ) equilibria: increase in production of firm 1 R x2 ( x1 ) x1 15 Exercise (direct approach) Who has a higher incentive to reduce own costs, a monopolist or a firm in Cournot- Duopoly? 16 Indirect approach, analytically c1 , c2 1 c1 , c2 , x c1 , c2 , x c1 , c2 C 1 C 1 C 2 Indirect approach (raising rival’s cost): d1 C 1 1 dx2 1 dx1 C C 0 c2 x2 dc2 x1 dc2 dc2 =0 <0 <0 =0 0 direct strategic effect effect 17 Indirect approach, graphically x2 equilibria: increase in production of firm 1 x1R ( x2 ) R x2 ( x1 ) x1 18 Reaction curve in the linear case x2 a c2 x1 a c2 if x1 M x2 ( x1 ) 2b R 2 b x2 0 otherwise R x2 ( x1 ) x1L x1 a c2 Note: x L 1 alone leads to a price of c2. b 19 Blockaded entry, graphically x2 x1R ( x2 ) C firm 1 R as a monopolist M x2 ( x1 ) x2 M x2 0 L M x1 x x 1 1 20 Blockaded entry Entry is blockaded for each firm: c1 a and c2 a Entry is blockaded for firm 2: c1 a and a c2 a c1 x x L 1 M 1 i.e. b 2b 1 c2 p (c1 ) (a c1 ) M 2 21 Blockaded entry (overview) c2 no firm 1 as a supply a monopolist 1 a 2 firm 2 as a duopoly monopolist 1 2 a a c1 22 Cournot – Executive summary A duopoly will occur only, if entry is blockaded for other firms. Firms have common and competing interests with respect to demand and cost functions. There are two approaches to cost leadership. The direct approach is to lower your own marginal cost. The indirect approach is known as “raising rivals’ costs“. 23 Stackelberg equilibrium Profit functions 1 ( x1 , x2 ), 2 ( x1 , x2 ) Follower’s reaction function (2nd stage) x2 ( x1 ) arg max x2 2 ( x1 , x2 ) R Leader’s optimal quantity (1st stage) x1S arg max x1 1 x1 , x2 ( x1 ) R Nash equilibrium: ( x1S , x2R ) 24 Finding the profit-maximizing point on the follower’s reaction curve x2 Accommodation M x2 Blockade or R x 2 deterrence R x2 ( x1 ) x1 x L a c2 x1 1 b 25 Computing the Stackelberg equilibrium (accommodation) Reaction function of firm 2: a c2 x1 x ( x1 ) R 2 2b 2 Profit function of firm 1: a c2 x1 1 ( x1 , x ( x1 )) a b x1 R 2 c1 x1 2b 2 Nash equilibrium S a 2c1 c2 R x1 , x2 2b a 2c1 3c2 a 2c1 c2 with x2 S and p S 4b 4 26 Depicting the Stackelberg outcome (both firms produce) x2 x1R ( x2 ) quantities in a Stackelberg equilibrium M x2 C C x 2 x S 2 S R x2 ( x1 ) x1C S x 1 x1 27 Exercise (equilibria) Which is an equilibrium in the Stackelberg model? xS 1 R , x2 x1S , xS 1 ,x R 2 , xC 1 ,x C 2 ? Are there any additional Nash equilibria ? 28 Cournot versus Stackelberg Profit function of firm 1 1 ( x1 , x2 ) p( X ) x1 C1 ( x1 ) First order condition for firm 1 dR1 dp dX dp dx1 dx2 R p( X ) x1 p( X ) x1 dx1 dX dx1 dX dx1 dx1 R dp dp dx2 ! p( X ) x1 x1 MC1 ( x1 ) dX dX dx1 direct effect follower or strategic effect, Cournot: 0, Stackelberg: >0 29 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. S. : x1S 10, x2 R Possible or not: ? C 1 S 1 30 Exercise (three firms) Three firms compete in a homogenous good market with X(p)=100-p. The costs are zero. At stage 1, firm 1 sets its quantity; at stage 2, firms 2 and 3 simultaneously decide on their quantities. Calculate the price on the market! 50 S. : p 3 31 Blockaded entry p blockaded entry for firm 2 a p( x1 ) c2 p1M M c1 x1L x1M L x1 x 1 32 Reaction functions in the case of blockaded entry x2 x1R ( x2 ) R x2 ( x1 ) L M x1 x 1 x 1 33 Profit function of firm 1 in the case of blockaded entry of firm 2 1 1 M 1 x1 ,0 1 x1 , x2 x1 R x1L x1M x1 34 Deterring firm 2’s entry p a p( x1 ) p1M c2 L 1 c1 M x1 x1L x1 35 d1 x1 , x2 x1 R Deterrence pays, dx1 0 L x1 1 1 M 1 x1 ,0 1 x1 , x2 x1 R x1M x1L x1 36 d1 x1 , x2 x1 R Deterrence does not pay dx1 0 L x1 1 1 M 1 x1 ,0 1 x1 , x2 x1 R x1M x1S x1L x1 37 Blockaded and deterred entry I Entry is blockaded for each firm: c1 a and c2 a Blockaded entry (firm 2): c 2 p M ( c1 ) or x1L x1M and c1 a a c1 c2 and c1 a 2 38 Blockaded and deterred entry II Deterred entry (firm 2): a c1 – Entry is not blockaded if c2 p1M 2 – Deterrence pays if d1 x1 , x2 x1 R 1 1 1 3 0 bx1 a c2 c1 a c2 c1 L dx1 L x1 2 2 2 2 1 2 c2 a c1 3 3 Deterrence if 1 2 a c1 a c1 c2 3 3 2 39 Blockade and deterrence c2 no supply a blockade firm 1 as a monopolist 1 a 2 1 firm 2 as a a 3 duopoly monopolist a 1 2 a c1 40 Exercise (entry and deterrence) Suppose a monopolist faces a demand of the form p(X)=4-0.25X. The firm’s unit costs are 2. a) Find the profit-maximizing quantity and price. Is entry blockaded for a potential entrant (firm 2) with unit costs of 3.5? b) How about unit costs of c2=1? c) Find firm 1’s limit output level for c2=1. Should the incumbent deter entry? S. : a) blockaded entry, b) entry is not blockaded c) 1 0.5 12 1 , accomodation S L 41 Deterrence and sunk costs I We now introduce quasifix costs of 3: p(X)=4-0.25X Leader' s cost function 3 2 x1 , x1 0 C1 x1 0, x1 0 Follower' s cost function 3 x2 , x2 0 C2 x2 0, x2 0 42 Deterrence and sunk costs II b) Entry blockaded ? ? ? x 4, M 1 p 4 3 M 1 4,0 1 0 1 0,0 x1 4 M p M 4 3 Comparison p x 2 is not sufficient M 1 c x2 x1 6 x1 x2 x1 4 R 1 R M 2 2 4,4 4 0,25 4 4 4 3 1 4 1 0 Entry not blockaded 43 Deterrence and sunk costs III c) Should firm 1 deter? x1Lq limit quantity w ith quasifixed costs, x1Lq x1L (why?) 0 2 x1Lq , x2 x1Lq p X x2 x1Lq C2 x2 x1Lq R R R 4 1 x1Lq x2 x1Lq x2 x1Lq 3 x2 x1Lq 4 R R R 4 1 x1Lq 6 1 x1Lq 6 1 x1Lq 3 6 1 x1Lq 4 2 2 2 2 8 x1Lq 6 1 x1Lq 3 6 1 x1Lq 5 1 2 2 Lq 2 15 x 3 4 Lq 1 x 5 4 Lq 1 1 16 x 1 3 6 1 x1Lq 2 Lq 2 6 x 3 2 Lq 1 1 16 x 1 x1Lq1 12 4 3, x1Lq : x1Lq 2 12 4 3 12 x L 1 44 Deterrence and sunk costs IV 1 x1Lq ,0 4 1 12 4 3 0 12 4 3 0 3 24 8 3 4 0,71 1 x1S , x2 x1S (see exercise " entry and deterrence " ) 1 S R 2 deterrence pays 45 Deterrence and sunk costs V x2 6 x2 x1 R x1S x1M x1Lq 12 x1 46 Deterrence and sunk costs VI x1S 7,2 x1M 4 x1S 2 0,5 1,44 3 4 9 CF Accommodation Deterrence Accommodation Blockade 47 Strategic trade policy xd 1 s xf 2 Two firms, one domestic (d), the other foreign (f), compete on a market in a third country. The domestic government subsidizes its firm’s exports using a unit subsidy s. The subsidy grants the domestic firm an advantage that is higher than the subsidy itself (Brander / Spencer (1981, 1983)). 48 Exercise (Strategic trade policy) In the setting just described, assume c : c1 c2 and p(X)=a-bX. Since the two firms sell to a third country, the rent of the consumers is without relevance and domestic welfare given by W s C c s, c sxd c s, c d C Which subsidy s maximizes domestic welfare? ac S. : s * 4 49 Solution (Strategic trade policy) - interpretation Direct effect of subsidy for domestic welfare is zero. Strategic effect: d x C f 0 xf x f s <0 <0 R ,s x d Cournot-Nash- s equilibria x R xd c s, c xd c, c !! C S d xM (firm d Stackelberg leader) xR f M xd x 50 Strategic trade policy - problems The recommendation depends on whether there is price or quantity competition. „One can always do better than free trade, but the optimal tariffs or subsidies seem to be small, the potential gains tiny, and there is plenty of room for policy errors that may lead to eventual losses rather than gains.“ Trade Policy and Market Structure; Helpman/Krugman, S. 186 51 Stackelberg – Executive Summary Time leadership is worthwhile: in a Stackelberg equilibrium the leader realizes a profit that is higher – than the follower’s and – his own in a Cournot equilibrium. Costs of entry (even in the form of identical quasifix costs) make the follower’s deterrence easier. Strategic trade policy may conceivably pay. 52 Example: The OPEC Cartel The best known cartel is the OPEC, which was formed in 1960 by Saudi Arabia, Venezuela, Kuwait, Iraq and Iran. Each member nation must agree to an individual output quota, except for Saudi Arabia, which adjusts its production as necessary to maintained high prices. In 1982, OPEC set an overall output limit of 18 million barrels per day (before 31 million). Production quota at 28 million barrels per day effective July 1, 2005. 53 The quantity cartel The firms seek to maximize joint profits 1 ( x1 , x2 ) 2 ( x1 , x2 ) p( X )( x1 x2 ) C1 ( x1 ) C2 ( x2 ) Optimization conditions ( 1 2 ) dp ! p ( X ) ( x1 x2 ) MC1 ( x1 ) 0 x1 dX ( 1 2 ) dp ! p ( X ) ( x1 x2 ) MC2 ( x2 ) 0 x2 dX Compare monopoly with two factories. 54 The cartel agreement The optimization condition is given by d1 dp ! dp p( X ) x1 MC1 ( x1 ) x2 0 dx1 dX dX 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. 55 Cartel quantities x2 quantities in a symmetric cartel x1R ( x2 ) M x2 C C x2 S 1 M x2 x2 S R x2 ( x1 ) 2 K 1 x x x x M C S M x1 2 1 1 1 1 56 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? Intermediate Microeconomics; Hal R. Varian 57 Cartel – Executive Summary If all firms keep the cartel agreement, they can increase their profits compared to Cournot competition. Nevertheless cartels are unstable from a static point of view. However, cartel agreements may be stable from the point of view of repeated games. 58 The outcomes of our models price a pM monopoly (M) and cartel (K) pC Cournot (C) pS Stackelberg (S) perfect competition (PC) p PC = c X M X C X S X PC quantity 59 Antitrust laws and enforcement, Germany laws – Gesetz gegen unlauteren Wettbewerb (1896) – Gesetz gegen Wettbewerbsbeschränkungen (GWB), (1957) enforcement – Bundeskartellamt 60 Ck concentration ratio xi Setup: n firms, si and s1 s2 sk sn X k k Definition: Ck si , for n identical firms : Ck ,k n i 1 n monopoly: n k 1 k 1 n perfect comp.: lim k 0 for identical firms n n Exercise: Calculate C2 for 2 firms with equal market shares, 3 firms with shares of 0.1, 0.1 and 0.8 or 3 firms with shares of 0.2, 0.6 and 0.2 ? 61 GWB, §19 (3) One firm is called „market dominating“ if C1>1/3. A group of firms is called „market dominating“ if Ck 1 / 2, k 3 or Ck 2 / 3, k 5. 62 The Herfindahl (Hirschman) index monopoly : H 1 Definition: 1 2 n identical firms : H n xi n n H si 2 perfect competitio n i 1 X i 1 (n ) : H 0 Exercise: Calculate H for 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 ? 63 n firms in Cournot competition Total industry output: X x1 x2 ... xn Firm i’s profit function: i ( x1 ,..., xn ) p( x1 ... xn ) xi Ci xi Firm i’s marginal revenue: dp dp d x1 xn dp MR ( xi ) p xi p xi p xi 1 dxi dX dxi dX xi X dp p 1 p 1 si X p dX X ,p 64 Lerner index of market power First order condition: dp dX ! MR( xi ) p( X ) xi MC ( xi ) dX dxi Lerner index for one firm: p p 1 si p MCi ! X,p si p p X,p Lerner index for the industry: n p MCi ! n si H si p si i 1 i 1 X,p X,p 65 Exercise (Replication) In a homogenous good market there are m identical costumers and n identical firms. Every costumer demands the quantity 1-p at price p. The cost function of firm j is given by C j x j 0,5x 2 . j a) Calculate the inverse market demand function! b) Calculate the reaction function of firm j and the total market output X C x1C x2 xn and pC in the symmetric Cournot- C C equilibrium! Hint: Use X j x1 ... x j 1 x j 1 .. xn c) Now the number of firms and costumers is multiplied by . Calculate again pC and MCj! Prove that for the gap between price and marginal costs converges to zero! S. : a) p X 1 m X m n b) X C n , pC 1 Theorie der Industrieökonomik; Bester 66 m 1 n m 1 n

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