Lecture 2F: Hotelling s Model

Transcription

1 Econ 46 Urban Economics Lecture F: Hotelling s Model Instructor: Hiroki Watanabe Spring Hiroki Watanabe / 6 Hotelling s Model Monopoly (N = ) 3 (N = ) 4 Nash Equilibrium 5 Oligopoly (N ) N 4 6 Summary Hiroki Watanabe / 6 Firm s location choice (Lecture D): 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? Hiroki Watanabe 3 / 6

2 Hotelling s model (c.f. any advanced (or some intermediate) microeconomics textbook like Varian Chpt 5 [Var5]). Think of a region in which consumers are uniformly located along a line segment [, ]. Each consumer prefers to travel a shorter distance to a hot dog vendor (homogeneous good). There are N sellers. Where would we expect these sellers to choose their locations? Hiroki Watanabe 4 / 6 Population Denisty (people/mi ) Hiroki Watanabe 5 / 6 Consumer location is given by ( ). n-th vendor s location is given by n ( n ). Mill price of n-th vendor is given by p n. Delivered price of consumer at is given by p n }{{} + n } {{ }. mill price of vendor n distance to vendor n Hiroki Watanabe 6 / 6

3 Each consumer buys one hot dog. Normalize p n = for all n. Demand faced by Liz is D L ( L, K, ) hot dogs. Demand faced by Kenneth is D K ( L, K, ) hot dogs. (Assume the cost is sunk, i.e., the vendor has already made enough hot dogs). Liz s profit is D L ( L, K, ) = D L ( L, K, ) dollars. Kenneth s profit is D K ( L, K, ) = D K ( L, K, ) dollars. Hiroki Watanabe 7 / 6 Hotelling s Model Monopoly (N = ) 3 (N = ) 4 Nash Equilibrium 5 Oligopoly (N ) N 4 6 Summary Hiroki Watanabe 8 / 6 Suppose there is a single vendor (monopolist). Question. (Location of the Monopolist) What is the demand that the monopolist faces? What is the monopolist s profit when she locates =,.5 and.5? 3 Which location maximizes the vendor s profit? 4 Which location minimizes total delivered price payed by the customers? Hiroki Watanabe 9 / 6

4 Liz solves: max L π L ( L ) = D L ( L ). D L ( L ) =. π L ( L ) = D L ( L ) =. Liz s profit is regardless of the location L. Hiroki Watanabe / 6.5 Mill(x)= Delivered(x)= x.75 Hiroki Watanabe / 6.5 Mill(x)= Delivered(x)= x.5.75 Hiroki Watanabe / 6

5 .5 Mill(x)= Delivered(x)= x.5.75 Hiroki Watanabe 3 / 6 Delivered price is location-variant. Total delivered price is vendor s profit and the aggregate sum of L from = to =. TDP( L = ) = +.5 =.5 = 4 6. TDP( L =.5) = = 6. TDP( L =.5) = + 4 = 5 4 = 6. Hiroki Watanabe 4 / 6.5 Total Delivered Price ($) Total Profit TDP(x L ).9.75 Vendor Location (x L ) Hiroki Watanabe 5 / 6

6 What is socially optimal for the economy as a whole is L =.5 (minimizes total cost of trip while maintaining the profitability). What is privately optimal for the vendor is any L in [, ]. Question. (Profit Maximization and Social Welfare) Is there any way that Liz chooses L =.5 by herself? Is political intervention necessary to realize L =.5? Hiroki Watanabe 6 / 6 Hotelling s Model Monopoly (N = ) 3 (N = ) 4 Nash Equilibrium 5 Oligopoly (N ) N 4 6 Summary Hiroki Watanabe 7 / 6 Suppose there are two vendors (N = ). Liz solves: max L π L ( L, K ) = D L ( L, K ). Kenneth solves: max K π K ( L, K ) = D K ( L, K ). Note Liz does not solve: max L, K π L ( L, K ) = D L ( L, K ). (why?) What do D L ( L, K ), D K ( L, K ) look like? Start with ( L, K ) = (, ). Hiroki Watanabe 8 / 6

7 .5 Mill(x)= Delivered L (x)= x Delivered K (x)= x.75 Hiroki Watanabe 9 / 6 D L (, ) =.5 π L (, ) =.5. D K (, ) =.5 π K (, ) =.5. Does Liz improve her profit by moving to the east? Hiroki Watanabe / 6.5 Mill(x)= Delivered L (x)= x. Delivered K (x)= x Hiroki Watanabe / 6

8 D L (., ) = +. =.55 D K (., ) =.55 =.45. Does Kenneth improve his profit by moving to the west? Hiroki Watanabe / 6.5 Mill(x)= Delivered L (x)= x. Delivered K (x)= x Hiroki Watanabe 3 / 6 D L (.,.8) =.+.8 =.45. D K (.,.8) =.45 =.55. Hiroki Watanabe 4 / 6

9 If L < K, Liz takes the consumers from to the midpoint between L and K : D L ( L, K ) = L + K. Kenneth takes the rest: D K ( L, K ) = D L ( L, K ) = L + K. If L > K, Kenneth takes the consumers from to the midpoint between L and K : D K ( L, K ) = L + K. Liz takes the rest: D L ( L, K ) = D K ( L, K ) = L + K. Hiroki Watanabe 5 / 6 3 If L = K, they split the market even: D L ( L, K ) = D K ( L, K ) =.5. Hiroki Watanabe 6 / 6 Hiroki Watanabe 7 / 6

10 An isoprofit curve of Liz indicates the set of location ( L, K ) that results in the same profit. Hiroki Watanabe 8 / 6 For Liz, isoprofit at.4 is a set of location ( L, K ) that results in π L ( L, K ) =.4. If L < K (region above 45 line) π L ( L, K ) = L + K =.4 K = L +.8. If L > K (region below 45 line) π L ( L, K ) = L + K =.4 K = L If L = K (45 line) π L ( L, K ) =.5.4 Hiroki Watanabe 9 / 6.6 Liz s x K x L Hiroki Watanabe 3 / 6.7

11 For Liz, isoprofit at.5 is a set of location ( L, K ) that results in π L ( L, K ) =.5. If L < K (region above 45 line) π L ( L, K ) = L + K =.5 K = L +. If L > K (region below 45 line) π L ( L, K ) = L + K =.5 K = L +. 3 If L = K (45 line) π L ( L, K ) =.5 A crossbuck shape. Hiroki Watanabe 3 / 6 For Kenneth, isoprofit at.4 is a set of location ( L, K ) that results in π K ( L, K ) =.4. If L < K (region below 45 line) π K ( L, K ) = L + K =.4 K = L +.. If L > K (region above 45 line) π K ( L, K ) = L + K =.4 K = L If L = K (45 line) π K ( L, K ) =.5.4 Hiroki Watanabe 3 / 6 Question: where is Kenneth s isoprofit at.5 located? Hiroki Watanabe 33 / 6

12 x K Kenneth s x L Hiroki Watanabe 34 / 6 How does Liz react to Kenneth s location? If Kenneth is at K =, Liz maximizes its profit by locating slightly to the east of K. ( L, ) L = + ε (ε = an infinitesimally small distance). If Kenneth is at K =., Liz maximizes its profit by locating slightly to the east of K. ( L,.) L =. + ε. If Kenneth is at K =.4, Liz maximizes its profit by locating slightly to the east of K. ( L,.4) L =.4 + ε. Hiroki Watanabe 35 / 6 ( L,.5) L =.5. ( L,.6) L =.6 ε. ( L,.8) L =.8 ε. ( L, ) L = ε. Hiroki Watanabe 36 / 6

13 How does Kenneth react to Liz s location? (, K ) K = + ε. (., K ) K =. + ε. (.4, K ) K =.4 + ε. (.5, K ) K =.5. (.6, K ) K =.6 ε. (.8, K ) K =.8 ε. (, K ) K = ε. Hiroki Watanabe 37 / 6 Suppose that vendors take turns in changing locations (in the denomination of. miles). Given ( L, K ) = (, ), Liz locates slightly to the west of Kenneth: (.99, ). 3 Given ( L, K ) = (.99, ), Kenneth locates slightly to the west of Liz: (.99,.98) Given ( L, K ) = (.5,.5), Liz locates slightly to the west of Kenneth: (.5,.5). 6 Given ( L, K ) = (.5,.5), Kenneth locates at.5 instead of.49: (.5,.5). 7 Given ( L, K ) = (.5,.5), Liz does not move: (.5,.5). 8 Given ( L, K ) = (.5,.5), Kenneth does not move: (.5,.5). Hiroki Watanabe 38 / 6 Hotelling s Model Monopoly (N = ) 3 (N = ) 4 Nash Equilibrium 5 Oligopoly (N ) N 4 6 Summary Hiroki Watanabe 39 / 6

14 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). Hiroki Watanabe 4 / 6 Is the Nash equilibrium ( LNE, KNE ) = (.5,.5) socially efficient? Fact: the socially efficient allocation is ( L, K ) = (/4, 3/4) or (3/4, /4). Hiroki Watanabe 4 / 6 Nash Equilibrium Location.5 Mill(x)= Delivered L (x)= x.5 Delivered K (x)= x.5.75 Hiroki Watanabe 4 / 6

15 Efficient Location.5 Mill(x)= Delivered L (x)= x.5 Delivered K (x)= x Hiroki Watanabe 43 / 6 Compare: Location Profit TDP Nash (.5,.5) (.5,.5) +/4 Socially Efficient (.5,.75) (.5,.5) +/8 Socially Efficient (.75,.5) (.5,.5) +/8 Hiroki Watanabe 44 / 6 Liz s.75 x K.75 x L Hiroki Watanabe 45 / 6

16 Question 4. (Socially Inefficient Equilibrium) Why can t vendors choose the socially optimal location, when both Nash equilibrium location and socially efficient location yield the same profits? Hiroki Watanabe 46 / 6 They are indifferent between ( LNE, KNE ) and ( L, K ). Given ( L, K ) = (.5,.75), Liz will move to a location slightly to the west of K =.75. ( L, K ) = (.74,.75) (.74,.73) (.7,.73) (.5,.5). Hiroki Watanabe 47 / 6 Hotelling s Model Monopoly (N = ) 3 (N = ) 4 Nash Equilibrium 5 Oligopoly (N ) N 4 6 Summary Hiroki Watanabe 48 / 6

17 If there are 3 vendors, Liz solves max L π L ( L, K, D ) = D L ( L, K, D ). For, there is no Nash equilibrium. Consider All the vendors locate at different place. L = K = D. 3 L = K D. Hiroki Watanabe 49 / 6 Mill(x)= Delivered L (x)= x. Delivered K (x)= x.3 Delivered D (x)= x.9..6 Hiroki Watanabe 5 / 6 Mill(x)= Delivered L (x)= x. Delivered K (x)= x.3 Delivered D (x)= x.8 5 Hiroki Watanabe 5 / 6

18 Mill(x)= Delivered L (x)= x. Delivered K (x)= x. Delivered D (x)= x.. Hiroki Watanabe 5 / 6 Mill(x)= Delivered L (x)= x. Delivered K (x)= x. Delivered D (x)= x.3 Hiroki Watanabe 53 / 6 Mill(x)= Delivered L (x)= x. Delivered K (x)= x. Delivered D (x)= x.8 Hiroki Watanabe 54 / 6

19 Mill(x)= Delivered L (x)= x. Delivered K (x)= x.3 Delivered D (x)= x.8 5 Hiroki Watanabe 55 / 6 Fact: socially efficient location is ( L, K, D ) = (/6, 3/6, 5/6) (not necessarily in this order). Hiroki Watanabe 56 / 6 Mill(x)= Delivered L (x)= x /6 Delivered K (x)= x 3/6 Delivered D (x)= x 5/6 /3 /3 Hiroki Watanabe 57 / 6

20 N 4 There are Nash equilibria for N 4. Hiroki Watanabe 58 / 6 Hotelling s Model Monopoly (N = ) 3 (N = ) 4 Nash Equilibrium 5 Oligopoly (N ) N 4 6 Summary Hiroki Watanabe 59 / 6 Hotelling s model Social optimality vs individual optimality. Nash equilibrium Hiroki Watanabe 6 / 6

21 References Hal R. Varian. Intermediate Microeconomics. Norton, seventh edition, 5. Hiroki Watanabe 6 / 6 Map du Jour Source Hiroki Watanabe 6 / 6