Math 152: Applicable Mathematics and Computing

Transcription

1 Math 152: Applicable Mathematics and Computing May 22, 2017 May 22, / 19

2 Bertrand Duopoly: Undifferentiated Products Game (Bertrand) Firm and Firm produce identical products. Each firm simultaneously decides on the price per unit of their product. Whichever firm sets a smaller price captures the entire market. The total demand is Q = (a P) +, where P is the smaller of the two prices and a is a constant. f both firms set equal prices, they split the market equally. The cost of producing one unit is c. This model was studied by Joseph Bertrand in 1883, as an alternative approach to Cournot. (Recall: n the Cournot model, firms set production volume and price is derived from this. n Bertrand, firms set price and production is derived from this.) May 22, / 19

3 Bertrand Duopoly: Undifferentiated Products Game (Bertrand) Firm and Firm produce identical products. Each firm simultaneously decides on the price per unit of their product. Whichever firm sets a smaller price captures the entire market. The total demand is Q = (a P) +, where P is the smaller of the two prices and a is a constant. f both firms set equal prices, they split the market equally. The cost of producing one unit is c. The payoff function for Firm is: (p 1 c)(a p 1 ) + p 1 < p 2 u 1 (p 1, p 2 ) = (p 1 c)(a p 1 ) + /2 p 1 = p 2 0 p 1 > p 2 May 22, / 19

4 Bertrand Duopoly: Undifferentiated Products Game (Bertrand) Firm and Firm produce identical products. Each firm simultaneously decides on the price per unit of their product. Whichever firm sets a smaller price captures the entire market. The total demand is Q = (a P) +, where P is the smaller of the two prices and a is a constant. f both firms set equal prices, they split the market equally. The cost of producing one unit is c. n Cournot, the relationship between price and production volume was given by P = (a Q) +. Whenever profits are positive, this is the same as Q = (a P) +. n particular, in a monopoly situation, the analysis for the Bertrand and Cournot models are exactly the same. May 22, / 19

5 Bertrand Duopoly: Undifferentiated Products What are the strategic equilibria in this model? Notice that if p 1 > c, then Firm s best response is to set p 2 = p 1 ε, thereby capturing the entire market. f either p 1 < c or p 2 < c, then the lower-price firm will increase their profits by setting the price to c. So in an equilibrium, p 1 c and p 2 c. f p 1 > c, then p 2 = p 1 ε > c. But then p 1 should be p 2 ε. This contradiction implies p 1 c and p 2 c. Hence p 1 = p 2 = c. May 22, / 19

6 Bertrand: Duopoly vs Monopoly Comparison Monopoly Amount Produced Price Payoff Payoff a c 2 a + c 2 (a c) 2 Duopoly a c c The result here is counterintuitive: in a dupoly, neither firm will make a profit? One alternative approach is due to Francis Edgeworth (1889): firms may be unwilling or unable to meet all demand if the price is too low. n this case, if p 1 = p 2 = c, then both firms have an incentive to increase price slightly, as they will produce sell more goods. May 22, / 19

7 Bertrand Duopoly: Differentiated Products Game (Bertrand, Differentiated Products) Firm and Firm produce different, but similar, products. Each firm simultaneously decides on the price per unit of their product, denoted p 1 and p 2 respectively. The demands for each product are given by q 1 (p 1, p 2 ) = (a p 1 + bp 2 ) + q 2 (p 1, p 2 ) = (a p 2 + bp 1 ) + where a, b are positive constants with b 1. The cost of producing one unit is c. May 22, / 19

8 Bertrand Duopoly: Differentiated Products Game (Bertrand, Differentiated Products) Firm and Firm produce different, but similar, products. Each firm simultaneously decides on the price per unit of their product, denoted p 1 and p 2 respectively. The demands for each product are given by q 1 (p 1, p 2 ) = (a p 1 + bp 2 ) + q 2 (p 1, p 2 ) = (a p 2 + bp 1 ) + where a, b are positive constants with b 1. The cost of producing one unit is c. The payoff functions in this case are given by: u 1 (p 1, p 2 ) = (a p 1 + bp 2 ) + (p 1 c) u 2 (p 1, p 2 ) = (a p 2 + bp 1 ) + (p 2 c) May 22, / 19

9 Bertrand Duopoly: Differentiated Products We want to find the strategic equilibria of the game with payoff functions u 1 (p 1, p 2 ) = (a p 1 + bp 2 ) + (p 1 c) u 2 (p 1, p 2 ) = (a p 2 + bp 1 ) + (p 2 c) We fix p 2, and then maximize u 1 as p 1 varies. u 1 p 1 (p 1, p 2 ) = So the best response strategy for Firm is { 0 a p 1 + bp 2 < 0 a 2p 1 + bp 2 + c a p 1 + bp 2 0 p 1 = a + bp 2 + c 2 May 22, / 19

10 Bertrand Duopoly: Differentiated Products We have deduced that By symmetry, we have Solving this, we get p 1 = a + bp 2 + c 2 p 2 = a + bp 1 + c 2 p 1 = p 2 = a + c 2 b Note: f b = 0, this means neither product is a replacement for the other. This situation degenerates into two monopolies, and we have that p 1 = (a + c)/2 which agrees with what we found before. May 22, / 19

11 Bertrand Duopoly: Differentiated Products We have found a strategic equilibrium with p 1 = p 2 = a + c 2 b n this case, the demand for both products is gien by q 1 = q 2 = a c + bc 2 b The profit for each company is given by ( a c + bc u 1 (p 1, p 2 ) = u 2 (p 1, p 2 ) = 2 b ) 2 May 22, / 19

12 Stackelberg Duopoly Game (Heinrich von Stackelberg, 1934) Firm and Firm produce identical products. Firm will decide its production amount q 1 for some fiscal period, and announce this publically. Firm will then decide its production amount q 2. The price of a unit of this product is then given by (a q 1 q 2 ) +, for some constant a. The cost of producing one unit of the product is c. May 22, / 19

13 Stackelberg Duopoly Game (Heinrich von Stackelberg, 1934) Firm and Firm produce identical products. Firm will decide its production amount q 1 for some fiscal period, and announce this publically. Firm will then decide its production amount q 2. The price of a unit of this product is then given by (a q 1 q 2 ) +, for some constant a. The cost of producing one unit of the product is c. The payoffs here are the same as under Cournot: u 1 (q 1, q 2 ) = q 1 (a q 1 q 2 ) + cq 1 u 2 (q 1, q 2 ) = q 2 (a q 1 q 2 ) + cq 2 The difference with Cournot is that now Firm knows Firm s strategy. May 22, / 19

14 Games of Perfect nformation Recall: A game of perfect information is any game where every information set consists of exactly one vertex. n a finite game of perfect information, we can solve it by iteratively removing dominated strategies, starting from the bottom (so this is an instance of backwards induction). The Stackelberg model is a game of perfect information. May 22, / 19

15 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A B C D E F (5,4) (3,2) G H J K (1,2) (4,1) (0,0) (3,1) (2,2) May 22, / 19

16 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A B C D E F (5,4) (3,2) G H J (1,2) (4,1) (3,1) May 22, / 19

17 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A B C D E F (5,4) (3,2) (3,1) G H (1,2) (4,1) May 22, / 19

18 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A B C D E F (5,4) (3,2) (3,1) H (4,1) May 22, / 19

19 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A B C D E F (4,1) (5,4) (3,2) (3,1) May 22, / 19

20 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A B D E (5,4) (3,2) May 22, / 19

21 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A B (5,4) (3,2) May 22, / 19

22 Games of Perfect nformation: Example For example, we solve the game of perfect information below. A (5,4) May 22, / 19

23 Games of Perfect nformation: Example For example, we solve the game of perfect information below. (5,4) May 22, / 19

24 Let us return to Stackelberg. We have a game of perfect information, where Firm moves first with strategy q 1 and then Firm chooses their strategy q 2. The payoffs are: u 1 (q 1, q 2 ) = q 1 (a q 1 q 2 ) + cq 1 u 2 (q 1, q 2 ) = q 2 (a q 1 q 2 ) + cq 2 Firm will choose a best response to q 1. By calculus, just like for Cournot, this best response is q 2 = a c q 1 2 Firm knows this is what Firm will do. So Firm knows their payoff is ( u 1 (q 1 ) = q 1 a q 1 a c q ) + 1 cq 1 2 May 22, / 19

25 Firm knows their payoff is ( u 1 (q 1 ) = q 1 a q 1 a c q ) + ( 1 a q1 + c cq 1 = q So they will maximize this over q 1. We have u 1 q 1 = The max occurs when q 1 = (a c)/2. This gives q 2 = (a c)/4. { a 2q1 c 2 a q 1 + c 0 0 a q 1 + c < 0 ) + cq 1 May 22, / 19

26 Bertrand: Duopoly vs Monopoly Comparison Cournot Stackelberg Amount Produced Price Payoff Payoff 2(a c) 3 3(a c) 4 a + 2c 3 a + 3c 4 (a c) 2 9 (a c) 2 8 (a c) 2 9 (a c) 2 16 So under Stackelberg, Firm does better than under Cournot. Additionally, the consumer is better off. Remark: t seems that Firm has an advantage in Stackelberg, since they know Firm s strategy. But Firm knows that Firm knows Firm s strategy. This gives Firm control. May 22, / 19

27 Hedge Fund nterview Question Game (Quant nterview Question) Two players each draw a uniform random number from the interval [0, 1]. f a player does not like their number, they can redraw a new number, but only once. Neither player knows the other s number, or whether their opponent has redrawn. The player with the higher number wins 1. What are the optimal strategies for each player? The answer may be a little counterintuitive. (This won t be on an exam). May 22, / 19