Lecture 1D: Hotelling s Model

Transcription

1 Econ 460 Urban Economics Lecture D: Hotelling s Model Instructor: Hiroki Watanabe Summer 0 0 Hiroki Watanabe / 47 Hotelling s Model Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N ) 5 Now We Know 0 Hiroki Watanabe / 47 Hotelling s Model Hot Dog Vendors in Rockefeller Plaza Vendor s Decision Consumer s Decision Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N ) 5 Now We Know 0 Hiroki Watanabe 3 / 47

2 Hot Dog Vendors in Rockefeller Plaza Firm s location choice (Lecture B): Firm A s location choice affects Firm B via bid rent. Production level ȳ was given. What if ȳ is endogenous and firm s location choice affects profit of other firms? 0 Hiroki Watanabe 4 / 47 Hot Dog Vendors in Rockefeller Plaza Hampton at Gravois, St. Louis, MO 0 Hiroki Watanabe 5 / 47 Hot Dog Vendors in Rockefeller Plaza Hotelling s model (c.f. Varian Chpt 5 [Var05]). Consumers are spread evenly along a line segment [0, ] (Rockefeller Plaza). Each consumer purchases one hot dog. There are N sellers. 0 Hiroki Watanabe 6 / 47

3 Hot Dog Vendors in Rockefeller Plaza Population Denisty (people/mi) Hiroki Watanabe 7 / 47 Vendor s Decision Assume that the cost is sunk. Liz s profit is sales volume = her market share. Same for Kenneth. 0 Hiroki Watanabe 8 / 47 Consumer s Decision Vendor s location is given by L and K. Definition. (Mill & Delivered Price) Mill price is the on-site price of a hot dog. Assume that hot dogs are $ each. Delivered price is the overall cost that a consumer pays for a hot dog, including trip cost. A consumer at (0 ) pays: }{{} + L or K. } {{ } mill price distance to the closest vendor 0 Hiroki Watanabe 9 / 47

4 Consumer s Decision Delivered(x)=+ x Hiroki Watanabe 0 / 47 Hotelling s Model Monopoly (Liz Alone) What Is Best for Liz What Is Best for Everyone 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N ) 5 Now We Know 0 Hiroki Watanabe / 47 What Is Best for Liz Delivered(x)=+ x Hiroki Watanabe / 47

5 What Is Best for Liz Delivered(x)=+ x Hiroki Watanabe 3 / 47 What Is Best for Liz Delivered(x)=+ x Hiroki Watanabe 4 / 47 What Is Best for Everyone L Liz s profit Liz s optimum anywhere Social optimum.5 Appendix 0 Hiroki Watanabe 5 / 47

6 Hotelling s Model Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) What Is Best for Everyone What Is Best for Liz & Kenneth Nash Equilibrium vs Social Optimum 4 Oligopoly (N ) 5 Now We Know 0 Hiroki Watanabe 6 / 47 What Is Best for Everyone Fact: the social optimal outcome is... 0 Hiroki Watanabe 7 / 47 What Is Best for Everyone Delivered L (x)=+ x.5 Delivered K (x)=+ x Hiroki Watanabe 8 / 47

7 What Is Best for Liz & Kenneth Exercise 3. (Duopolistic Competition) Starting from ( L, K ) = (0, ), take turns switching your locations to maximize your profit. Kenneth moves first. In case of L = K, split the market equally in half. No threatening or cartels. Where will you end up and how much will you earn? Do we get ( L, K ) = (.5,.75) or (.75,.5) in the end? 0 Hiroki Watanabe 9 / 47 What Is Best for Liz & Kenneth Delivered L (x)=+ x 0 Delivered K (x)=+ x Hiroki Watanabe 0 / 47 What Is Best for Liz & Kenneth Delivered L (x)=+ x 0 Delivered K (x)=+ x Hiroki Watanabe / 47

8 What Is Best for Liz & Kenneth Delivered L (x)=+ x. Delivered K (x)=+ x Hiroki Watanabe / 47 What Is Best for Liz & Kenneth Delivered L (x)=+ x.5 Delivered K (x)=+ x Hiroki Watanabe 3 / 47 What Is Best for Liz & Kenneth 0 Hiroki Watanabe 4 / 47

9 What Is Best for Liz & Kenneth The end result is called a Nash equilibrium. 0 Hiroki Watanabe 5 / 47 Nash Equilibrium vs Social Optimum Definition 3. (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). 0 Hiroki Watanabe 6 / 47 Nash Equilibrium vs Social Optimum Compare: Location Profit Sum of Delivered Price Social Optimum (.5,.75) (.5,.5).5 Social Optimum (.75,.5) (.5,.5).5 Nash Equilibrium (.5,.5) (.5,.5).50 0 Hiroki Watanabe 7 / 47

10 Nash Equilibrium vs Social Optimum Why can t Liz and Kenneth choose (.5,.75) on their own? 0 Hiroki Watanabe 8 / 47 Nash Equilibrium vs Social Optimum Delivered L (x)=+ x.5 Delivered K (x)=+ x Hiroki Watanabe 9 / 47 Nash Equilibrium vs Social Optimum Delivered L (x)=+ x.5 Delivered K (x)=+ x Hiroki Watanabe 30 / 47

11 Nash Equilibrium vs Social Optimum Delivered L (x)=+ x.5 Delivered K (x)=+ x Hiroki Watanabe 3 / 47 Hotelling s Model Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N ) N 4 5 Now We Know 0 Hiroki Watanabe 3 / 47 3 vendors: Liz, Kenneth and Jack. For, there is no Nash equilibrium. Consider All the vendors locate at different place. L = K = D. 3 L = K D. 0 Hiroki Watanabe 33 / 47

12 Delivered L (x)= x. Delivered K (x)= x.3 Delivered D (x)= x Hiroki Watanabe 34 / 47 Delivered L (x)= x. Delivered K (x)= x.3 Delivered D (x)= x Hiroki Watanabe 35 / 47 Delivered L (x)= x. Delivered K (x)= x. Delivered D (x)= x Hiroki Watanabe 36 / 47

13 Delivered L (x)= x. Delivered K (x)= x. Delivered D (x)= x Hiroki Watanabe 37 / 47 Delivered L (x)= x. Delivered K (x)= x. Delivered D (x)= x Hiroki Watanabe 38 / 47 Delivered L (x)= x. Delivered K (x)= x.3 Delivered D (x)= x Hiroki Watanabe 39 / 47

14 Fact: socially efficient location is ( L, K, D ) = (/6, 3/6, 5/6) (not necessarily in this order). 0 Hiroki Watanabe 40 / 47 Delivered L (x)= x /6 Delivered K (x)= x 3/6 Delivered D (x)= x 5/6 0 /3 /3 0 Hiroki Watanabe 4 / 47 N 4 There are Nash equilibria for N 4. 0 Hiroki Watanabe 4 / 47

15 Hotelling s Model Monopoly (Liz Alone) 3 Duopoly (Liz & Kenneth) 4 Oligopoly (N ) 5 Now We Know 0 Hiroki Watanabe 43 / 47 Hotelling s model Social optimality vs individual optimality. Nash equilibrium 0 Hiroki Watanabe 44 / 47 References [Var05] Hal R. Varian. Intermediate Microeconomics. Norton, seventh edition, Hiroki Watanabe 45 / 47

16 Map du Jour Source 0 Hiroki Watanabe 46 / 47 Index delivered price, 9 mill price, 9 N, number of sellers, 6 Nash equilibrium, 5, 6 0 Hiroki Watanabe 47 / 47