Bertrand Hotelling
Assume: Many Buyers Few Sellers ðEach firm faces downward-sloping demand because each is a large producer compared to the total market size ðThere is no one dominant model of oligopoly… we will review several. 2
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. 3
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 4
How will each firm set price? ðHomogeneity implies that consumers will buy from the low-price seller. ðFurther, each firm realizes that the demand that it faces depends both on its own price and on the price set by other firms ðSpecifically, any firm charging a higher price than its rivals will sell no output. ðAny firm charging a lower price than its rivals will obtain the entire market demand. 5
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 6
Price Example: Residual Demand Curve, Price Setting Market Demand • 0 Residual Demand Curve (thickened line segments) Quantity 7
·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. 8
Example: Reaction Functions, Price Setting and Homogeneous Products 45° line Price charged by firm 2 Reaction function of firm 1 Reaction function of firm 2 p 2* 0 • p 1* Price charged by firm 1 9
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 10
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! 11
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 (!) 12
Bertrand Competition n n n Homogenous good market / perfect substitutes Demand q=15 -p Constant marginal cost MC=c=3 It always pays to undercut Only equilibrium where price equals marginal costs Equilibrium good for consumers Collusion must be ruled out 13
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Sample result: Bertrand Two Firms Fixed Partners Random Partners Five Firms Random Partners “I “Some that collusion can take place in a learnt people are undercutting bastards!!! competitive though, even without any actual how the Seriously market it was interesting to see meeting taking place practise. ” two parties. ” theory is shown in between the 19
Hotelling’s (1929) linear city n Why do all vendors locate in the same spot? n 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? n This can be explained by firms trying to get the most customers. 20
Hotelling (voting version) Voters vote for the closest party. R L Party A Party B If Party A shifts to the right then it gains voters. R L Party A Party B Each has incentive to locate in the middle. 21
Hotelling Model R L Party A Party B Average distance for voter is ¼ total. This isn’t “efficient”! R L Party A Party B Most “efficient” has average distance of 1/8 total. 22
Further considerations: Hotelling n Firms choose location and then prices. n Consumers care about both distance and price. n If firms choose close together, they will eliminate profits as in Bertrand competition. n If firms choose further apart, they will be able to make some profit. n Thus, they choose further apart. 23
Price competition with differentiated goods n n n n Prices p. A and p. B Zero marginal costs Transport cost t V value to consumer Consumers on interval [0, 1] Firms A and B at positions 0 and 1 Consumer indifferent if n n Residual demand q. A=(p. B- p. A+t)/2 t for firm A Residual demand q. B=(p. A- p. B+t)/2 t for firm B V-tx- p. A= V-t(1 -x)- p. B 24
Price competition with differentiated goods Residual demand q. A=(p. B- p. A+t)/2 t for firm A n Residual demand q. B=(p. A- p. B+t)/2 t for firm B n Residual inverse demands n p. A=-2 t q. A +p. B+t, p. B=-2 t q. B +p. A+t n Marginal revenues must equal MC=0 MRA=-4 t q. A +p. B+t=0, MRB=-4 t q. B +p. A+t=0 MRA=-2(p. B- p. A+t)+p. B+t=0, MRB=-2(p. A- p. B+t)+p. A+t=0 MRA=2 p. A-p. B-t=0, MRB=2 p. B-p. A-t=0 p. A=2 p. B-t; 4 p. B-2 t-p. B-t=0; p. B=p. A=t n Profits t/2 25
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 26
Example: Q 1 = 100 - 2 P 1 + P 2 "Coke's demand" Q 2 = 100 - 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 110 = 100 - 2 P 1 + 10 = 110 - 2 P 1 Q 10 = 100 - 2 P 1 + 0 = 100 - 2 P 1 27
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 100 0 Pepsi’s price = $0 for D 0 and $10 for D 10 MR 0 Coke’s quantity 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 110 100 Pepsi’s price = $0 for D 0 and $10 for D 10 D 0 0 Coke’s quantity 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 Pepsi’s price = $0 for D 0 and $10 for D 10 Coke’s price 110 100 MR 10 0 MR 0 D 10 D 0 Coke’s quantity 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 Pepsi’s price = $0 for D 0 and $10 for D 10 Coke’s price 110 100 D 10 5 0 MR 10 MR 0 D 0 Coke’s quantity 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 Pepsi’s price = $0 for D 0 and $10 for D 10 Coke’s price 110 100 30 27. 5 D 10 MR 10 5 0 MR 0 45 50 D 0 Coke’s quantity 32
Example: MR 110 = 55 - Q 110 = 5 ðQ 110 = 50 ðP 110 = 30 Therefore, firm 1's best response to a price of $10 by firm 2 is a price of $30 33
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 34
And, using the demand curve, we have: P 1 = 50 + P 2/2 - 45/2 - P 2/4 …or… P 1 = 27. 5 + P 2/4…reaction function 35
Pepsi’s price (P 2) P 2 = 27. 5 + P 1/4 (Pepsi’s R. F. ) 27. 5 Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products Coke’s price (P 1) 36
P 1 = 27. 5 + P 2/4 (Coke’s R. F. ) Pepsi’s price (P 2) P 2 = 27. 5 + P 1/4 (Pepsi’s R. F. ) • 27. 5 P 1 = 110/3 Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products Coke’s price (P 1) 37
P 1 = 27. 5 + P 2/4 (Coke’s R. F. ) Pepsi’s price (P 2) Bertrand Equilibrium P 2 = 110/3 • 27. 5 P 1 = 110/3 P 2 = 27. 5 + P 1/4 (Pepsi’s R. F. ) Example: Equilibrium and Reaction Functions, Price Setting and Differentiated Products Coke’s price (P 1) 38
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. . . 39
Example: Firm 1 and firm 2, continued P 1 = 27. 5 + P 2/4 P 2 = 27. 5 + 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* = 2005. 55 = 4011. 10 40
Notice that 1. profits are positive in equilibrium since both prices are above marginal cost! ðEven if we have no capacity constraints, and constant marginal cost, a firm cannot capture all demand by cutting price… ðThis blunts price-cutting incentives and means that the firms' own behavior does not mimic free entry 41
ðOnly 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" 42
Back to Cournot n n Inverse demand P=260 -Q 1 -Q 2 Marginal costs MC=20 3 possible predictions Price=MC, Symmetry Q 1=Q 2 260 -2 Q 1=20, Q 1=120, P=20 n Cournot duopoly: MR 1=260 -2 Q 1 -Q 2=20, Symmetry Q 1=Q 2 260 -3 Q 1=20, Q 1=80, P=100 n Shared monopoly profits: Q=Q 1+Q 2 MR=260 -2 Q=20, Q=120, Q 1=Q 2=60, P=140 43
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Bertrand with Compliments (? !*-) n Q=15 -P 1 -P 2, MC 1=1. 5, MC 2=1. 5, MC=3 n Monopoly: P=15 -Q, MR=15 -2 Q=3, Q=6, P=P 1+P 2=9, Profit (9 -3)*6=36 n Bertrand: P 1=15 -Q-P 2, MR 1=15 -2 QP 2=1. 5 15 -2(15 -P 1 -P 2)-P 2=-15+2 P 1+P 2=1. 5 Symmetry P 1=P 2; 3 P 1=16. 5, P 1=5. 5, Q=4<9 P 1+P 2=11>9, both make profit (11 -3)*4=32<36 Competition makes both firms and consumers worse off! 46
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The Capacity Game GM DNE Ford Expand 18 20 Expand 18 15 20 15 * 16 16 What is the equilibrium here? Where would the companies like to be? 49
War Mars Not Shoot Venus Shoot -5 -15 -15 * -10 -1 -10 50
Repeated games 1. if game is repeated with same players, then there may be ways to enforce a better solution to prisoners’ dilemma n 2. suppose PD is repeated 10 times and people know it n – then backward induction says it is a dominant strategy to cheat every round n 3. suppose that PD is repeated an indefinite number of times – then it may pay to cooperate n 4. Axelrod’s experiment: tit-for-tat 51
Continuation payoff n n Your payoff is the sum of your payoff today plus the discounted “continuation payoff” 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 Your trade-off short run against long run gains. 52
Infinitely repeated PD n Discounted payoff, 0<d<1 discount factor (d 0=1) n Normalized payoff: (d 0 u 0+ d 1 u 1+ d 2 u 2+… +dtut+…)(1 -d) n Geometric series: (d 0+ d 1+ d 2+… +dt+…)(1 -d) =(d 0+ d 1+ d 2+… +dt+…) -(d 1+ d 2+ d 3+… +dt+1+…)= d 0=1 53
Infinitely repeated PD n Constant “income stream” u 0= u 1=u 2=… =u each period yields total normalized income u. n Grim Strategy: Choose “Not shoot” until someone chooses “shoot”, always choose “Shoot” thereafter 54
n Payoff if nobody shoots: (-5 d 0 - 5 d 1 -5 d 2 -… -5 dt+…)(1 -d)=-5 =-5(1 -d)-5 d n Maximal payoff from shooting in first period (-15<-10!): (-d 0 -10 d 1 -10 d 2 -… -10 dt-…)(1 -d) =-1(1 -d)-10 d n -1(1 -d)-10 d< -5(1 -d)-5 d iff 4(1 -d)<5 d or 4<9 d d>4/9 0. 44 n Cooperation can be sustained if d> 0. 45, i. e. if players weight future sufficiently highly. 55
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