Classic Oligopoly Models: Bertrand and Cournot

Transcription

1 Classic Oligopoly Models: Bertrand and Cournot Class Note: There are supplemental readings, including Werden (008) Unilateral Competitive Effects of Horizontal Mergers I: Basic Concepts and Models, that complement this lecture 1 $ Potential efficiency gain from the merger to monopoly Williamson Tradeoff P M DWL of CS created because of merger to monopoly P C Q M mr mc mc M Ashenfelter et al (013) In June of 008 the U.S. Department of Justice approved a joint venture between Miller and Coors, the second and third largest firms in the industry. Despite substantially increasing concentration in an already concentrated industry, the merger was approved by the antitrust authority partially because it was expected to reduce shipping and distribution costs. p. Q C D Q Oligopoly (a few firms) & Game Theory Unlike monopoly, profit interdependency for oligopoly: a firm s profits depend on its own behavior and its rivals ; Use game theory Eg: Cellular telephone service in Toronto: Players: Rogers (Fido), Bell, Telus, Wind, Mobilicity Compete on price, services, and advertising To measure market power, add i or j subscript: Lerner Index j = p j mc j p j 3 Class, Page 1 of 11

2 Basic Elements & Assumptions Noncooperative Game Theory: Each player acts in its own self-interest and cannot credibly commit to do otherwise Allows collusion (cartels) Players rational: seek best payoffs ECO316H, ECO36H A strategic game: Players Rules: Timing of moves, available actions, info Assume each firm knows its profits & rivals profits for all outcomes Outcomes: Results of all possible actions combos Payoffs: Preferences over outcomes 4 Number of Time Periods Static game: One period and all players move simultaneously (long run) Extensive game (multi-period): Multiple period game where players move sequentially and maybe more than once Dynamic game (repeated game): Infinite period game where players repeatedly interact over time 5 Two Symmetric by Normal Form Games: Discrete Actions Cournot Firm : q = 9 Firm : q = 1 Firm 1: q 1 = 9 ($405, $405) ($337.50, $450) Firm 1: q 1 = 1 ($450, $337.50) ($360, $360) Each cell shows: (π 1, π ) Bertrand Firm : p = $0 Firm : p = $5 Firm 1: p 1 = $0 ($500, $500) ($65, $375) Firm 1: p 1 = $5 ($375, $65) ($56.50, $56.50) 6 Class, Page of 11

3 Best Response and NE Best response: A strategy s i is a best response for firm i to its rivals strategies s i if for all s i : π i s i, s i π i (s i, s i ) Given the rivals strategies, a firm could do no better than playing its best response Nash equilibrium (NE): A strategy profile s (s 1, s,, s n ) is a NE if for all s i and for all players i: π i (s i, s i ) π i s i, s i Given the rivals strategies, no one firm has an incentive to change its strategy 7 Finding NE (if exist, may be multiple) For discrete actions, find NE by checking every possible combination of strategies and asking if any firm would want to change its strategy If no change desired for any firm, then have a NE Otherwise, do not have a NE For continuous actions, find NE by looking for intersection of best response functions Each firm playing a best response given the other firms strategies: means none wish to change 8 Example: Find any and all NE Firm 1: p 1 = $9 Firm 1: p 1 = $10 Firm 1: p 1 = $11 Firm : p = $5 Firm : p = $6 Firm : p = $7 ($119, $14) ($16, $18) ($133, $1) ($10, $15) ($18, $0) ($136, $15) ($117, $16) ($16, $) ($135, $18) How about in Slide 6? 9 Class, Page 3 of 11

4 Initial Oligopoly Assumptions Consumers are price takers Market demand is downward sloping Each firm has a single choice variable Homogeneous products [note: will be relaxed] No capacity constraints Market equilibrium is static NE No entry, exit, product repositioning, etc. 10 In Class Exercise Teams complete a worksheet Market demand is P = Q Each firm sets its price to the nearest 10 cents E.g. $.60 but not $.65 Marginal costs are $1; there are no fixed costs Homogenous products Consumers buy from firm with lowest price: any higher-priced firm sells zero units If any firms are tied for lowest price, sales divided evenly Deadline: 5 minutes Publically reveal results 11 Illustrate the Bertrand Paradox Bertrand duopoly, homogenous goods Price is continuous: any price possible Costs: C 1 q 1 = cq 1 and C q = cq What is NE in Bertrand game: (p 1, p )? Is each firm playing its best response? Profits at NE? Both firms better at (p M, p M )? What is the best response if rival plays p M? If C 1 q 1 = F 1 + cq 1 and C q = F + cq? If C 1 q 1 = c 1 q 1 and C q = c q if c 1 < c? 1 Class, Page 4 of 11

5 Escape the Paradox But no paradox if goods differentiated, capacity constrained, interactions repeated E.g. Linear differentiated goods Bertrand model q 1 = a 1 b 11 p 1 + b 1 p q = a b p + b 1 p 1 Costs: C 1 q 1 = c 1 q 1 and C q = c q Why allow for asymmetric marginal costs? Before working with the linear model analytically and graphically, consider a general model 13 π j p 1,, p N = p j q j p 1,, p N C j q j π j p j = q j p 1,, p N + p j mc j q j p 1,, p N p j = set 0 p j mc j q j p j = q j p j mc j = 1 p j q j p j p j p j mc j p j = 1 ε j q j What is name of LHS? Do rival firms prices affect Firm j s price? In general, is ε j constant? If not, what could affect it? 14 Find Bertrand NE: (p 1,, p N ) Start with Firm i s profit function: π i p 1,, p N = TR i p 1,, p N TC i (p 1,, p N ) Take partial derivative of π i p 1,, p N wrt choice variable p i and set equal to 0 Solve for p i : Firm i s best response function Repeat for other firms (or use symmetry) Find intersection of all best response functions, which is the NE 15 Class, Page 5 of 11

6 Firm 1 s Best Response Function, Linear Bertrand Duopoly Model π 1 p 1, p = TR 1 p 1, p TC 1 p 1, p π 1 p 1, p = p 1 q 1 p 1, p c 1 q 1 p 1, p π 1 p 1, p = (p 1 c 1 ) a 1 b 11 p 1 + b 1 p π 1 = a p 1 b 11 p 1 + b 1 p b 11 (p 1 c 1 ) set = 0 1 a 1 11 p 1 + b 1 p + b 11 c 1 = 0 p 1 = a 1 + b 11 c 1 + b 1 p p What happens if Firm 1 lowers its marginal costs? p 1 = a 1 + b 11 c 1 + b 1 p 11 p 1 17 Firm s Best Response Function π p 1, p = TR p 1, p TC p 1, p π p 1, p = p q p 1, p c q p 1, p π p 1, p = (p c ) a b p + b 1 p 1 π p = a b p + b 1 p 1 b (p c ) = set 0 a p + b 1 p 1 + b c = 0 p = a + b c + b 1 p 1 18 Class, Page 6 of 11

7 p What happens if Firm 1 lowers its marginal costs? p 1 = a 1 + b 11 c 1 + b 1 p 11 p = a + b c + b 1 p 1 p Are any alternate price pairs NE? p 1 p 1 19 Classic Cournot Model Firms choose quantity (continuous, non-neg.) Use initial oligopoly assumptions N Including homogeneous goods: Q = j=1 q j π i q 1,, q N = P Q q i C i (q i ) First solve general case, then explore linear Cournot model: Market demand: P = a bq Costs: C 1 q 1 = c 1 q 1 and C q = c q 0 π j q 1,, q N = P(Q)q j C j q j π j = P Q + P(Q) q j Q q j mc set j = 0 P Q mc j = P(Q) Q q j P Q mc j P(Q) P Q mc j P(Q) 1 q j = Q P(Q) Q P(Q) Q = s j ε If symmetric (mc j = c for all j) then obtain: P Q c P(Q) = 1 Nε 1 Class, Page 7 of 11

8 Find Cournot NE: (q 1,, q N ) Firm i s profit function: π i q 1,, q N = P Q q i C i (q i ) Take partial derivative of π i q 1,, q N choice variable q i and set equal to 0 wrt Solve for q i : Firm i s best response function Repeat for other firms (or use symmetry) Find intersection of all best response functions, which is the NE Linear Cournot Duopoly Model Firm 1 s best response function: π 1 q 1, q = P(Q)q 1 C 1 q 1 π 1 = a bq c 1 q 1 π 1 = a bq 1 bq c 1 q 1 Firm s best response function: π q 1, q = P(Q)q C q π = a bq c q π = a bq 1 bq c q π 1 q 1 = a bq 1 bq c 1 bq 1 π q = a bq 1 bq c bq a q 1 bq c 1 = set 0 q 1 = a c 1 bq a q bq 1 c set = 0 q = a c bq 1 3 q What happens if Firm 1 lowers its marginal costs? q 1 = a c 1 bq Which firm has lower marginal costs in this diagram? q q = a c bq 1 q 1 q 1 4 Class, Page 8 of 11

9 Solve for NE Algebraically q 1 = a c 1 bq q = a c bq 1 q 1 = a c 1 b a c bq 1 q 1 = a c 1 + c + b q 1 4bq 1 = a c 1 + c + bq 1 q 1 = a c 1 + c 3b q = a c + c 1 3b 5 Symmetric Cournot Duopoly: A Special Case q q 1 = a c bq q 1 = a c 1 bq q a c bq = q = a c bq q = a c 3b q q = a c bq 1 q q 1 6 Linear Demand: Comparing Outcomes P = a bq Symmetric (c 1 = c = c) Cournot market output, Q = q 1 + q, is between that of a perfectly competitive market and a monopoly Market Structure Perfect Competition Cournot Duopoly Monopoly Market Output (Q) a c b a c 3 b 1 a c b 7 Class, Page 9 of 11

10 q q 1 = a c bq a c q q = a c bq 1 q a c b q 1 8 Welfare, Efficiency of Symmetric Cournot Duopoly P a P Q = a c 3 b q P a + c a c = π 3 1 = π = 9b a c a c PS = CS = 9b 9b 4 a c TS = DWL 9b c 1 = c = c a c a b Q DWL = 18b 9 Three Firms in Cournot Industry π 1 q 1, q, q 3 = P(Q)q 1 C 1 q 1 π 1 = a bq c 1 q 1 π 1 = a bq 1 bq bq 3 c 1 q 1 π 1 q 1 = a bq 1 bq bq 3 c 1 bq 1 a q 1 bq bq 3 c set 1 = 0 q 1 = a c 1 bq bq 3 However, if symmetric: q = a c bq bq q a c = 4b Q = 3 a c 4 b 30 Class, Page 10 of 11

11 Bertrand versus Cournot Bertrand: Firms choose price Output (capacity) more flexible than price Strategic variable is price Output adjusted to clear the market: meet demand at chosen price E.g. software, streaming movies (Netflix) Cournot: Firms choose quantity Price more flexible than output (capacity) Strategic variable is quantity Price adjusted to clear the market E.g. commodities (milk, wheat, chemicals, steel, etc.) 31 Slope of Best Response Strategic substitutes: Best response functions have negative slope Cournot, linear demand, constant mc Strategic complements: Best response functions have positive slope Bertrand, linear demand (differentiated goods), constant mc Implications for merger analysis 3 Looking Ahead Workshop 1, Tuesday, 11:10 1:00, this room Bring a laptop with Excel Download the merger simulation spreadsheets from our course site ahead of time Interactive parts and there are also slides posted Also, Assignment #1 given on Tuesday 33 Class, Page 11 of 11