Lecture 1D: Hotelling s Model

Transcription

1 Econ 4935 Urban Economics Lecture 1D: Hotelling s Model Instructor: Hiroki Watanabe Fall 2012 Watanabe Econ D Hotelling s Model 1 / 54

2 1 Hotelling s Model 2 Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N 2) 5 Now We Know Watanabe Econ D Hotelling s Model 2 / 54

3 1 Hotelling s Model Hot Dog Vendors in Rockefeller Plaza Vendor s Decision Consumer s Decision 2 Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N 2) 5 Now We Know Watanabe Econ D Hotelling s Model 3 / 54

4 Hot Dog Vendors in Rockefeller Plaza Firm s location choice (Lecture 1B): Firm A s location choice affects Firm B via bid rent. Production level ȳ (total size of office space) was given. What if ȳ is endogenous and firm s location choice affects profit of other firms? Watanabe Econ D Hotelling s Model 4 / 54

5 Hot Dog Vendors in Rockefeller Plaza Firm s location choice (Lecture 1B): Firm A s location choice affects Firm B via bid rent. Production level ȳ (total size of office space) was given. What if ȳ is endogenous and firm s location choice affects profit of other firms? Watanabe Econ D Hotelling s Model 4 / 54

6 Hot Dog Vendors in Rockefeller Plaza Firm s location choice (Lecture 1B): Firm A s location choice affects Firm B via bid rent. Production level ȳ (total size of office space) was given. What if ȳ is endogenous and firm s location choice affects profit of other firms? Watanabe Econ D Hotelling s Model 4 / 54

7 Hot Dog Vendors in Rockefeller Plaza Firm s location choice (Lecture 1B): Firm A s location choice affects Firm B via bid rent. Production level ȳ (total size of office space) was given. What if ȳ is endogenous and firm s location choice affects profit of other firms? Watanabe Econ D Hotelling s Model 4 / 54

8 Hot Dog Vendors in Rockefeller Plaza Hampton at Gravois, St. Louis, MO Watanabe Econ D Hotelling s Model 5 / 54

9 Hot Dog Vendors in Rockefeller Plaza Hotelling s model 1 Consumers are spread evenly along a line segment [0, 1] (Rockefeller Plaza). Each consumer purchases one hot dog. 1 Cf Varian Chpt 25 [Var05]. Watanabe Econ D Hotelling s Model 6 / 54

10 Hot Dog Vendors in Rockefeller Plaza Hotelling s model 1 Consumers are spread evenly along a line segment [0, 1] (Rockefeller Plaza). Each consumer purchases one hot dog. 1 Cf Varian Chpt 25 [Var05]. Watanabe Econ D Hotelling s Model 6 / 54

11 Hot Dog Vendors in Rockefeller Plaza Hotelling s model 1 Consumers are spread evenly along a line segment [0, 1] (Rockefeller Plaza). Each consumer purchases one hot dog. 1 Cf Varian Chpt 25 [Var05]. Watanabe Econ D Hotelling s Model 6 / 54

12 Hot Dog Vendors in Rockefeller Plaza Population Denisty (people/mi) Location (miles) Watanabe Econ D Hotelling s Model 7 / 54

13 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

14 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

15 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

16 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

17 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

18 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

19 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

20 Vendor s Decision There are 1, 2 or 3 vendors: 1 Liz alone (monopoly) 2 Liz and Kenneth (duopoly) 3 Liz, Kenneth and Jack (oligopoly) Assume that the cost is sunk. Liz s profit is 1 sales volume = her market share. Same for Kenneth. Watanabe Econ D Hotelling s Model 8 / 54

21 Consumer s Decision Vendor s location is given by L and K. Definition 1.1 (Mill & Delivered Price) 1 Mill price is the on-site price of a hot dog. Assume that hot dogs are $1 each. 2 Delivered price is the overall cost that a consumer pays for a hot dog, including trip cost. A consumer at (0 1) pays: }{{} 1 + L or K. } {{ } mill price distance to the closest vendor Watanabe Econ D Hotelling s Model 9 / 54

22 Consumer s Decision Vendor s location is given by L and K. Definition 1.1 (Mill & Delivered Price) 1 Mill price is the on-site price of a hot dog. Assume that hot dogs are $1 each. 2 Delivered price is the overall cost that a consumer pays for a hot dog, including trip cost. A consumer at (0 1) pays: }{{} 1 + L or K. } {{ } mill price distance to the closest vendor Watanabe Econ D Hotelling s Model 9 / 54

23 Consumer s Decision Vendor s location is given by L and K. Definition 1.1 (Mill & Delivered Price) 1 Mill price is the on-site price of a hot dog. Assume that hot dogs are $1 each. 2 Delivered price is the overall cost that a consumer pays for a hot dog, including trip cost. A consumer at (0 1) pays: }{{} 1 + L or K. } {{ } mill price distance to the closest vendor Watanabe Econ D Hotelling s Model 9 / 54

24 Consumer s Decision Vendor s location is given by L and K. Definition 1.1 (Mill & Delivered Price) 1 Mill price is the on-site price of a hot dog. Assume that hot dogs are $1 each. 2 Delivered price is the overall cost that a consumer pays for a hot dog, including trip cost. A consumer at (0 1) pays: }{{} 1 + L or K. } {{ } mill price distance to the closest vendor Watanabe Econ D Hotelling s Model 9 / 54

25 Consumer s Decision Mill Price and Delivered Price ($) Mill(x)=1 Delivered(x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 10 / 54

26 1 Hotelling s Model 2 Monopoly (Liz Alone) What Is Best for Liz What Is Best for Everyone 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N 2) 5 Now We Know Watanabe Econ D Hotelling s Model 11 / 54

27 What Is Best for Liz Mill Price and Delivered Price ($) Mill(x)=1 Delivered(x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 12 / 54

28 What Is Best for Liz Mill Price and Delivered Price ($) Mill(x)=1 Delivered(x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 13 / 54

29 What Is Best for Liz Mill Price and Delivered Price ($) Mill(x)=1 Delivered(x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 14 / 54

30 What Is Best for Everyone L Liz s profit Liz s optimum anywhere 1 Social optimum Appendix Watanabe Econ D Hotelling s Model 15 / 54

31 What Is Best for Everyone L Liz s profit Liz s optimum anywhere 1 Social optimum Appendix Watanabe Econ D Hotelling s Model 15 / 54

32 1 Hotelling s Model 2 Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) What Is Best for Everyone What Is Best for Liz & Kenneth Nash Equilibrium vs Social Optimum Example 4 Oligopoly (N 2) 5 Now We Know Watanabe Econ D Hotelling s Model 16 / 54

33 What Is Best for Everyone Fact: the social optimal outcome is... Watanabe Econ D Hotelling s Model 17 / 54

34 What Is Best for Everyone Mill Price and Delivered Price ($) 2.00 Mill(x)= Delivered L (x)=1+ x.25 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 18 / 54

35 What Is Best for Liz & Kenneth Exercise 3.1 (Duopolistic Competition) Starting from ( L, K ) = (0, 1), take turns switching your locations to maximize your profit. In case of L = K, split the market equally in half. No threatening or cartels. 1 Where will you end up and how much will you earn? 2 Do we get ( L, K ) = (.25,.75) or (.75,.25) in the end? Watanabe Econ D Hotelling s Model 19 / 54

36 What Is Best for Liz & Kenneth Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)=1+ x 0 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 20 / 54

37 What Is Best for Liz & Kenneth Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)=1+ x 0 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 21 / 54

38 What Is Best for Liz & Kenneth Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)=1+ x.1 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 22 / 54

39 What Is Best for Liz & Kenneth Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)=1+ x.5 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 23 / 54

40 What Is Best for Liz & Kenneth Watanabe Econ D Hotelling s Model 24 / 54

41 What Is Best for Liz & Kenneth The end result is called a Nash equilibrium. Watanabe Econ D Hotelling s Model 25 / 54

42 Nash Equilibrium vs Social Optimum Definition 3.2 (Nash Equilibrium) is the location ( LNE, KNE ) such that none of the vendor can profit by unilaterally changing its location. For duopoly, the Nash equilibrium ( LNE, KNE ) = (.5,.5). Watanabe Econ D Hotelling s Model 26 / 54

43 Nash Equilibrium vs Social Optimum Discussion 3.3 (Social Optimal Choice) 1 How much do Kenneth and Liz earn with Nash equilibrium (.5,.5) and with social optimal outcome (.25,.75)? 2 Can they reach (.25,.75) on their own? If not, why not? Watanabe Econ D Hotelling s Model 27 / 54

44 Nash Equilibrium vs Social Optimum Discussion 3.3 (Social Optimal Choice) 1 How much do Kenneth and Liz earn with Nash equilibrium (.5,.5) and with social optimal outcome (.25,.75)? 2 Can they reach (.25,.75) on their own? If not, why not? Watanabe Econ D Hotelling s Model 27 / 54

45 Nash Equilibrium vs Social Optimum Compare: Location Profit Sum of Delivered Price Social Optimum (.25,.75) Social Optimum (.75,.25) (.5,.5) Nash Equilibrium (.5,.5) Deadweight loss of $.125. Watanabe Econ D Hotelling s Model 28 / 54

46 Nash Equilibrium vs Social Optimum Compare: Location Profit Sum of Delivered Price Social Optimum (.25,.75) Social Optimum (.75,.25) (.5,.5) Nash Equilibrium (.5,.5) Deadweight loss of $.125. Watanabe Econ D Hotelling s Model 28 / 54

47 Nash Equilibrium vs Social Optimum Why can t Liz and Kenneth choose (.25,.75) on their own? Watanabe Econ D Hotelling s Model 29 / 54

48 Nash Equilibrium vs Social Optimum Mill Price and Delivered Price ($) 2.00 Mill(x)= Delivered L (x)=1+ x.25 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 30 / 54

49 Nash Equilibrium vs Social Optimum Mill Price and Delivered Price ($) 2.00 Mill(x)= Delivered L (x)=1+ x.25 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 31 / 54

50 Nash Equilibrium vs Social Optimum Mill Price and Delivered Price ($) 2.00 Mill(x)= Delivered L (x)=1+ x.5 Delivered K (x)=1+ x Location (miles) Watanabe Econ D Hotelling s Model 32 / 54

51 Example Example 3.4 (Rockefeller Square) Rockefeller Square is a 10-by-10 mile square and there are 100 consumers uniformly distributed over the Square. Location in the Square is identified by (, y), where (0 10) measures longitude and y (0 y 10) measures latitude. There are two hot-dog vendors, Liz and Kenneth. The mill price is $1 for both. Production cost is sunk and already off their plates. 1 Suppose that ( L, y L ; K, y K ) = (0, 0; 10, 10). Show their market shares on a map. 2 Suppose that ( L, y L ; K, y K ) = (3, 1; 1, 5). Show their market shares on a map. 3 Find the Nash equilibrium. 4 Show that ( L, y L ; K, y K ) = (0, 0; 10, 10) yields the same profit as in ➌ but it is not sustainable. Watanabe Econ D Hotelling s Model 33 / 54

52 Example Latitude y (miles) Longitude x (miles) Watanabe Econ D Hotelling s Model 34 / 54

53 Example Watanabe Econ D Hotelling s Model 35 / 54

54 Example Watanabe Econ D Hotelling s Model 36 / 54

55 1 Hotelling s Model 2 Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N 2) N = 3 N 4 5 Now We Know Watanabe Econ D Hotelling s Model 37 / 54

56 N = 3 3 vendors: Liz, Kenneth and Jack Donaghy. Fact: socially efficient location is ( L, K, D ) = (1/6, 3/6, 5/6) 2. 2 not necessarily in this order. Watanabe Econ D Hotelling s Model 38 / 54

57 N = 3 3 vendors: Liz, Kenneth and Jack Donaghy. Fact: socially efficient location is ( L, K, D ) = (1/6, 3/6, 5/6) 2. 2 not necessarily in this order. Watanabe Econ D Hotelling s Model 38 / 54

58 N = 3 Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)= x 1/6 Delivered K (x)= x 3/6 Delivered D (x)= x 5/ /3 2/3 1 Location (miles) Watanabe Econ D Hotelling s Model 39 / 54

59 N = 3 Exercise 4.1 (Oligopolistic Competition) Starting from ( L, K, D ) = (0, 0, 0), take turns switching your locations to maximize your profit. 1 Where will you end up and how much will you earn? 2 Is ( L, K, D ) = (1/6, 3/6, 5/6) sustainable? Watanabe Econ D Hotelling s Model 40 / 54

60 N = 3 3 vendors: Liz, Kenneth and Jack. Consider 1 All the vendors locate at different place. 2 L = K = D. 3 L = K D. Watanabe Econ D Hotelling s Model 41 / 54

61 N = 3 3 vendors: Liz, Kenneth and Jack. Consider 1 All the vendors locate at different place. 2 L = K = D. 3 L = K D. Watanabe Econ D Hotelling s Model 41 / 54

62 N = 3 3 vendors: Liz, Kenneth and Jack. Consider 1 All the vendors locate at different place. 2 L = K = D. 3 L = K D. Watanabe Econ D Hotelling s Model 41 / 54

63 N = 3 3 vendors: Liz, Kenneth and Jack. Consider 1 All the vendors locate at different place. 2 L = K = D. 3 L = K D. Watanabe Econ D Hotelling s Model 41 / 54

64 N = 3 3 vendors: Liz, Kenneth and Jack. Consider 1 All the vendors locate at different place. 2 L = K = D. 3 L = K D. Watanabe Econ D Hotelling s Model 41 / 54

65 N = 3 Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)= x.1 Delivered K (x)= x.3 Delivered D (x)= x Location (miles) Watanabe Econ D Hotelling s Model 42 / 54

66 N = 3 Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)= x.2 Delivered K (x)= x.3 Delivered D (x)= x Location (miles) Watanabe Econ D Hotelling s Model 43 / 54

67 N = 3 Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)= x.2 Delivered K (x)= x.2 Delivered D (x)= x Location (miles) Watanabe Econ D Hotelling s Model 44 / 54

68 N = 3 Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)= x.2 Delivered K (x)= x.2 Delivered D (x)= x Location (miles) Watanabe Econ D Hotelling s Model 45 / 54

69 N = 3 Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)= x.2 Delivered K (x)= x.2 Delivered D (x)= x Location (miles) Watanabe Econ D Hotelling s Model 46 / 54

70 N = 3 Mill Price and Delivered Price ($) 2 Mill(x)=1 Delivered L (x)= x.2 Delivered K (x)= x.3 Delivered D (x)= x Location (miles) Watanabe Econ D Hotelling s Model 47 / 54

71 N = 3 For N = 3, there is no Nash equilibrium. Watanabe Econ D Hotelling s Model 48 / 54

72 N 4 There are Nash equilibria for N 4. Watanabe Econ D Hotelling s Model 49 / 54

73 1 Hotelling s Model 2 Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N 2) 5 Now We Know Watanabe Econ D Hotelling s Model 50 / 54

74 Hotelling s model Social optimality vs individual optimality. Nash equilibrium Watanabe Econ D Hotelling s Model 51 / 54

75 References [Var05] Hal R. Varian. Intermediate Microeconomics. Norton, seventh edition, Watanabe Econ D Hotelling s Model 52 / 54

76 Map du Jour Source Watanabe Econ D Hotelling s Model 53 / 54

77 Airline du Jour Today s color theme is provided by courtesy of Thai Airways Watanabe Econ D Hotelling s Model 54 / 54

78 Index deadweight loss, 28 delivered price, 9 duopoly, 17 mill price, 9 monopoly, 12 Nash equilibrium, 25, 26 oligopoly, 34 Watanabe Econ D Hotelling s Model 55 / 54