Bertrand and Hotelling

2 Assume: Many Buyers Few Sellers  Each firm faces downward-sloping demand because each is a large producer compared to the total market size  There is no one dominant model of oligopoly… we will review several.

3 1. Bertrand Oligopoly (Homogeneous) Assume: Firms set price* Homogeneous product Simultaneous Noncooperative *Definition: In a Bertrand oligopoly, each firm sets its price, taking as given the price(s) set by other firm(s), so as to maximize profits.

4 Definition: Firms act simultaneously if each firm makes its strategic decision at the same time, without prior observation of the other firm's decision. Definition: Firms act noncooperatively if they set strategy independently, without colluding with the other firm in any way

5 How will each firm set price?  Homogeneity implies that consumers will buy from the low-price seller.  Further, each firm realizes that the demand that it faces depends both on its own price and on the price set by other firms  Specifically, any firm charging a higher price than its rivals will sell no output.  Any firm charging a lower price than its rivals will obtain the entire market demand.

6 Definition: The relationship between the price charged by firm i and the demand firm i faces is firm i's residual demand In other words, the residual demand of firm i is the market demand minus the amount of demand fulfilled by other firms in the market: Q 1 = Q - Q 2

7 Example: Residual Demand Curve, Price Setting Quantity Price Market Demand Residual Demand Curve (thickened line segments) 0

8  Assume firm always meets its residual demand (no capacity constraints)  Assume that marginal cost is constant at c per unit.  Hence, any price at least equal to c ensures non-negative profits.

9 Example: Reaction Functions, Price Setting and Homogeneous Products Price charged by firm 1 Price charged by firm 2 45° line p2*p2* Reaction function of firm 1 Reaction function of firm 2 p1*p1* 0

10 Thus, each firm's profit maximizing response to the other firm's price is to undercut (as long as P > MC) Definition: The firm's profit maximizing action as a function of the action by the rival firm is the firm's best response (or reaction) function Example: 2 firms Bertrand competitors Firm 1's best response function is P 1 =P 2 - e Firm 2's best response function is P 2 =P 1 - e

11 So… 1. Firms price at marginal cost 2. Firms make zero profits 3. The number of firms is irrelevant to the price level as long as more than one firm is present: two firms is enough to replicate the perfectly competitive outcome!

12 If we assume no capacity constraints and that all firms have the same constant average and marginal cost of c then… For each firm's response to be a best response to the other's each firm must undercut the other as long as P> MC Where does this stop? P = MC (!)

13 Bertrand Competition Homogenous good market / perfect substitutes Homogenous good market / perfect substitutes Demand q=15-p Demand q=15-p Constant marginal cost MC=c=3 Constant marginal cost MC=c=3 It always pays to undercut It always pays to undercut Only equilibrium where price equals marginal costs Only equilibrium where price equals marginal costs Equilibrium good for consumers Equilibrium good for consumers Collusion must be ruled out Collusion must be ruled out

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19 Sample result: Bertrand “I learnt that collusion can take place in a competitive market even without any actual meeting taking place between the two parties.” Two Firms Fixed Partners Two Firms Random Partners Five Firms Random Partners “Some people are undercutting bastards!!! Seriously though, it was interesting to see how the theory is shown in practise.”

20 Hotelling’s (1929) linear city Why do all vendors locate in the same spot? Why do all vendors locate in the same spot? For instance, on High Street many shoe shops right next to each other. Why do political parties (at least in the US) seem to have the same agenda? For instance, on High Street many shoe shops right next to each other. Why do political parties (at least in the US) seem to have the same agenda? This can be explained by firms trying to get the most customers. This can be explained by firms trying to get the most customers.

21 Hotelling (voting version) Party BParty A If Party A shifts to the right then it gains voters. Voters vote for the closest party. L R Party BParty A L R Each has incentive to locate in the middle.

22 Hotelling Model Party BParty A L R Average distance for voter is ¼ total. This isn’t “efficient”! Party BParty A L R Most “efficient” has average distance of 1/8 total.

23 Further considerations: Hotelling Firms choose location and then prices. Firms choose location and then prices. Consumers care about both distance and price. Consumers care about both distance and price. If firms choose close together, they will eliminate profits as in Bertrand competition. If firms choose close together, they will eliminate profits as in Bertrand competition. If firms choose further apart, they will be able to make some profit. If firms choose further apart, they will be able to make some profit. Thus, they choose further apart. Thus, they choose further apart.

24 Price competition with differentiated goods Prices p A and p B Prices p A and p B Zero marginal costs Zero marginal costs Transport cost t Transport cost t V value to consumer V value to consumer Consumers on interval [0,1] Consumers on interval [0,1] Firms A and B at positions 0 and 1 Firms A and B at positions 0 and 1 Consumer indifferent if Consumer indifferent if V-tx- p A = V-t(1-x)- p B Residual demand q A =(p B - p A +t)/2t for firm A Residual demand q A =(p B - p A +t)/2t for firm A Residual demand q B =(p A - p B +t)/2t for firm B Residual demand q B =(p A - p B +t)/2t for firm B

25 Price competition with differentiated goods Residual demand q A =(p B - p A +t)/2t for firm A Residual demand q A =(p B - p A +t)/2t for firm A Residual demand q B =(p A - p B +t)/2t for firm B Residual demand q B =(p A - p B +t)/2t for firm B Residual inverse demands Residual inverse demands p A =-2t q A +p B +t, p B =-2t q B +p A +t Marginal revenues must equal MC=0 Marginal revenues must equal MC=0 MR A =-4t q A +p B +t=0, MR B =-4t q B +p A +t=0 MR A =-2(p B - p A +t)+p B +t=0, MR B =-2(p A - p B +t)+p A +t=0 MR A =2p A -p B -t=0, MR B =2p B -p A -t=0 p A =2p B -t; 4p B -2t-p B -t=0; p B =p A =t Profits t/2 Profits t/2

26 Assume: Firms set price* Differentiated product Simultaneous Noncooperative As before, differentiation means that lowering price below your rivals' will not result in capturing the entire market, nor will raising price mean losing the entire market so that residual demand decreases smoothly

27 Example: Q 1 = P 1 + P 2 "Coke's demand" Q 2 = P 2 + P 1 "Pepsi's demand" MC 1 = MC 2 = 5 What is firm 1's residual demand when Firm 2's price is $10? $0? Q 1 10 = P = P 1 Q 1 0 = P = P 1

28 Example: Residual Demand, Price Setting, Differentiated Products Each firm maximizes profits based on its residual demand by setting MR (based on residual demand) = MC Coke’s price Coke’s quantity MR 0 Pepsi’s price = $0 for D 0 and $10 for D

29 Example: Residual Demand, Price Setting, Differentiated Products Each firm maximizes profits based on its residual demand by setting MR (based on residual demand) = MC Coke’s price Coke’s quantity Pepsi’s price = $0 for D 0 and $10 for D D0D0 D 10

30 Example: Residual Demand, Price Setting, Differentiated Products Each firm maximizes profits based on its residual demand by setting MR (based on residual demand) = MC Coke’s price Coke’s quantity MR 0 D0D0 Pepsi’s price = $0 for D 0 and $10 for D 10 0 MR D 10

31 Example: Residual Demand, Price Setting, Differentiated Products Each firm maximizes profits based on its residual demand by setting MR (based on residual demand) = MC Coke’s price Coke’s quantity 5 MR 0 D0D0 Pepsi’s price = $0 for D 0 and $10 for D 10 0 D 10 MR

32 Example: Residual Demand, Price Setting, Differentiated Products Each firm maximizes profits based on its residual demand by setting MR (based on residual demand) = MC Coke’s price Coke’s quantity MR 0 D0D0 Pepsi’s price = $0 for D 0 and $10 for D 10 0 D 10 MR

33 Example: MR 1 10 = 55 - Q 1 10 = 5  Q 1 10 = 50  P 1 10 = 30 Therefore, firm 1's best response to a price of $10 by firm 2 is a price of $30

34 Example: Solving for firm 1's reaction function for any arbitrary price by firm 2 P 1 = 50 - Q 1 /2 + P 2 /2 MR = 50 - Q 1 + P 2 /2 MR = MC => Q 1 = 45 + P 2 /2

35 And, using the demand curve, we have: P 1 = 50 + P 2 /2 - 45/2 - P 2 /4 …or… P 1 = P 2 /4…reaction function

36 Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products Pepsi’s price (P 2 ) Coke’s price (P 1 ) P 2 = P 1 /4 (Pepsi’s R.F.) 27.5

37 Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products Pepsi’s price (P 2 ) Coke’s price (P 1 ) P 1 = 110/3 P 1 = P 2 /4 (Coke’s R.F.) P 2 = P 1 /4 (Pepsi’s R.F.) 27.5

38 Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products Pepsi’s price (P 2 ) Coke’s price (P 1 ) P 1 = 110/3 P 2 = 110/3 P 1 = P 2 /4 (Coke’s R.F.) P 2 = P 1 /4 (Pepsi’s R.F.) Bertrand Equilibrium 27.5

39 Equilibrium: Equilibrium occurs when all firms simultaneously choose their best response to each others' actions. Graphically, this amounts to the point where the best response functions cross...

40 Example: Firm 1 and firm 2, continued P 1 = P 2 /4 P 2 = P 1 /4 Solving these two equations in two unknowns… P 1 * = P 2 * = 110/3 Plugging these prices into demand, we have: Q 1 * = Q 2 * = 190/3  1 * =  2 * =  =

41 Notice that 1. profits are positive in equilibrium since both prices are above marginal cost!  Even if we have no capacity constraints, and constant marginal cost, a firm cannot capture all demand by cutting price…  This blunts price-cutting incentives and means that the firms' own behavior does not mimic free entry

42  Only if I were to let the number of firms approach infinity would price approach marginal cost. 2. Prices need not be equal in equilibrium if firms not identical (e.g. Marginal costs differ implies that prices differ) 3. The reaction functions slope upward: "aggression => aggression"

43 Back to Cournot Inverse demand P=260-Q1-Q2 Inverse demand P=260-Q1-Q2 Marginal costs MC=20 Marginal costs MC=20 3 possible predictions 3 possible predictions Price=MC, Symmetry Q1=Q2 Price=MC, Symmetry Q1=Q Q1=20, Q1=120, P=20 Cournot duopoly: Cournot duopoly: MR1=260-2Q1-Q2=20, Symmetry Q1=Q Q1=20, Q1=80, P=100 Shared monopoly profits: Q=Q1+Q2 Shared monopoly profits: Q=Q1+Q2 MR=260-2Q=20, Q=120, Q1=Q2=60, P=140

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46 Bertrand with Compliments (?!*-) Q=15-P1-P2, MC1=1.5, MC2=1.5, MC=3 Q=15-P1-P2, MC1=1.5, MC2=1.5, MC=3 Monopoly: P=15-Q, MR=15-2Q=3, Q=6, P=P1+P2=9, Profit (9-3)*6=36 Monopoly: P=15-Q, MR=15-2Q=3, Q=6, P=P1+P2=9, Profit (9-3)*6=36 Bertrand: P1=15-Q-P2, MR1=15-2Q- P2=1.5 Bertrand: P1=15-Q-P2, MR1=15-2Q- P2= (15-P1-P2)-P2=-15+2P1+P2=1.5 Symmetry P1=P2; 3P1=16.5, P1=5.5, Q=4<9 P1+P2=11>9, both make profit (11-3)*4=32<36 Competition makes both firms and consumers worse off!

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49 The Capacity Game GM Ford 20 Expand DNE Expand DNE What is the equilibrium here? Where would the companies like to be? *

50 Mars Venus Shoot Not Shoot Shoot Not Shoot War *

51 Repeated games 1. if game is repeated with same players, then there may be ways to enforce a better solution to prisoners’ dilemma 1. if game is repeated with same players, then there may be ways to enforce a better solution to prisoners’ dilemma 2. suppose PD is repeated 10 times and people know it 2. suppose PD is repeated 10 times and people know it – then backward induction says it is a dominant strategy to cheat every round 3. suppose that PD is repeated an indefinite number of times 3. suppose that PD is repeated an indefinite number of times –then it may pay to cooperate 4. Axelrod’s experiment: tit-for-tat 4. Axelrod’s experiment: tit-for-tat

52 Continuation payoff Your payoff is the sum of your payoff today plus the discounted “continuation payoff” Your payoff is the sum of your payoff today plus the discounted “continuation payoff” Both depend on your choice today Both depend on your choice today If you get punished tomorrow for bad behaviour today and you value the future sufficiently highly, it is in your self-interest to behave well today If you get punished tomorrow for bad behaviour today and you value the future sufficiently highly, it is in your self-interest to behave well today Your trade-off short run against long run gains. Your trade-off short run against long run gains.

53 Infinitely repeated PD Discounted payoff, 0<d<1 discount factor (d 0 =1) Discounted payoff, 0<d<1 discount factor (d 0 =1) Normalized payoff: (d 0 u 0 + d 1 u 1 + d 2 u 2 +… +d t u t +…)(1-d) Normalized payoff: (d 0 u 0 + d 1 u 1 + d 2 u 2 +… +d t u t +…)(1-d) Geometric series: Geometric series: (d 0 + d 1 + d 2 +… +d t +…)(1-d) (d 0 + d 1 + d 2 +… +d t +…)(1-d) =(d 0 + d 1 + d 2 +… +d t +…) -(d 1 + d 2 + d 3 +… +d t+1 +…)= d 0 =1

54 Infinitely repeated PD Constant “income stream” u 0 = u 1 =u 2 =… =u each period yields total normalized income u. Constant “income stream” u 0 = u 1 =u 2 =… =u each period yields total normalized income u. Grim Strategy: Choose “Not shoot” until someone chooses “shoot”, always choose “Shoot” thereafter Grim Strategy: Choose “Not shoot” until someone chooses “shoot”, always choose “Shoot” thereafter

55 Payoff if nobody shoots: Payoff if nobody shoots: (-5d 0 - 5d 1 -5d 2 -… -5d t +…)(1-d)=-5 =-5(1-d)-5d Maximal payoff from shooting in first period (-15<-10!): Maximal payoff from shooting in first period (-15<-10!): (-d 0 -10d 1 -10d 2 -… -10d t -…)(1-d) =-1(1-d)-10d -1(1-d)-10d 4/9  (1-d)-10d 4/9  0.44 Cooperation can be sustained if d> 0.45, i.e. if players weight future sufficiently highly. Cooperation can be sustained if d> 0.45, i.e. if players weight future sufficiently highly.