lecture 1f: hotelling’s model - university of minnesota...

Model Monopoly Duopoly Nash Equilibrium Oligopoly

Econ 460 Urban Economics

Lecture 1F: Hotelling’s Model

Instructor: Hiroki Watanabe

Spring 2012

© 2012 Hiroki Watanabe 1 /64


1 Hotelling’s Model

2 Monopoly (N = 1)

3 Duopoly (N = 2)

4 Nash Equilibrium

5 Oligopoly (N ≥ 2)

6 Now We Know



1 Hotelling’s ModelHot Dog Vendors in Rockefeller Plaza

2 Monopoly (N = 1)

3 Duopoly (N = 2)

4 Nash Equilibrium


6 Now We Know



Hot Dog Vendors in Rockefeller Plaza

Firm’s location choice (Lecture 1D):Firm A’s location choice affects Firm B via bid rent.Production level y was given.

What if y is endogenous and firm’s location choiceaffects profit of other firms?




Hotelling’s model (c.f. Varian Chpt 25 [Var05]).Think of a region in which consumers are uniformlylocated along a line segment [0,1].Each consumer prefers to travel a shorter distanceto a hot dog vendor (homogeneous good).There are N sellers.Where would we expect these sellers to choosetheir locations?




−0.2 0 0.2 0.4 0.6 0.8 1 1.2

0

1

Location (miles)

Pop

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Den

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(pe

ople

/mi)

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

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1

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Pop

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Den

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(pe

ople

/mi)




Consumer location is given by (0 ≤ ≤ 1).n-th vendor’s location is given by n (0 ≤ n ≤ 1).

Definition 1.1 (Mill & Delivered Price)

1 is the on-site price of a hot dog. The mill price of then-th vendor is given by pn.

2 is the overall cost that a consumer pays for a hot dog,including trip cost. A consumer at pays

pn︸︷︷︸

mill price of vendor n

+ |− n|︸︷︷︸

distance to vendor n

.




Each consumer buys one hot dog.Normalize pn = 1 for all n.Demand faced by Liz is DL(L, K , · · · ) hot dogs.Demand faced by Kenneth is DK(L, K , · · · ) hotdogs.(Assume the cost is sunk, i.e., the vendor hasalready made enough hot dogs).Liz’s profit is 1 ·DL(L, K , · · · ) = DL(L, K , · · · )dollars.Kenneth’s profit is 1 ·DK(L, K , · · · ) = DK(L, K , · · · )dollars.




2 Monopoly (N = 1)

3 Duopoly (N = 2)

4 Nash Equilibrium


6 Now We Know



Suppose there is a single vendor (monopolist).

Question 2.1 (Location of the Monopolist)

1 What is the demand that the monopolist faces?

2 What is the monopolist’s profit when she locates = 0, .25 and .5?

3 Which location maximizes the vendor’s profit?

4 Which location minimizes total delivered price payed bythe customers?



Liz solves:

max0≤L≤1πL(L) = 1 ·DL(L).

DL(L) = 1.πL(L) = 1 ·DL(L) = 1.Liz’s profit is 1 regardless of the location L.



0 0.25 0.5 0.75 10

0.5

1

1.5

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Location (miles)

Mill

Pric

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Mill(x)=1Delivered(x)=|x−0|

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Mill(x)=1Delivered(x)=|x−.25|

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0 0.25 0.5 0.75 10

0.5

1

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Location (miles)

Mill

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Pric

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)

Mill(x)=1Delivered(x)=|x−.5|

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Delivered price is location-variant.

Definition 2.2 (Total Delivered Price (TDP))Total delivered price measures the overall price thatconsumers paid to purchase their hot dogs. It isequivalent to vendor’s profit and the aggregate sum of|− L| from = 0 to = 1.

TDP(L = 0) = 1+ .5 = 1.5 = 2416 .

TDP(L = .25) = 1+ 516 =

2116 .

TDP(L = .5) = 1+ 14 =

54 =

2016 .



0 0.25 0.5 0.75 10.9

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Vendor Location (xL)

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Total Profit

TDP(xL)

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What is socially optimal for the economy as a wholeis L = .5 (minimizes total cost of trip whilemaintaining the profitability).What is privately optimal for the vendor is any L in[0,1].

Question 2.3 (Profit Maximization and Social Welfare)

Is there any way that Liz chooses L = .5 by herself?Is political intervention necessary to realize L = .5?




2 Monopoly (N = 1)

3 Duopoly (N = 2)DuopolyIsoprofit

4 Nash Equilibrium


6 Now We Know



Duopoly

Suppose there are two vendors (N = 2).Liz solves:

max0≤L≤1πL(L, K) = 1 ·DL(L, K).

Kenneth solves:

max0≤K≤1πK(L, K) = 1 ·DK(L, K).

Note Liz does not solve:

max0≤L≤1,0≤K≤1πL(L, K) = 1 ·DL(L, K).

(why?)What do DL(L, K), DK(L, K) look like?Start with (L, K) = (0,1).



Duopoly

0 0.25 0.5 0.75 10

0.5

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Location (miles)

Mill

Pric

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Pric

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Mill(x)=1

DeliveredL(x)=|x−0|

DeliveredK(x)=|x−1|

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Duopoly

DL(0,1) = .5⇒ πL(0,1) = .5.DK(0,1) = .5⇒ πK(0,1) = .5.Does Liz improve her profit by moving to the east?



Duopoly

0 0.1 0.450.5 0.8 10

0.5

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Location (miles)

Mill

Pric

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d D

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Pric

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Mill(x)=1

DeliveredL(x)=|x−.1|

DeliveredK(x)=|x−.8|

0 0.1 0.450.5 0.8 10

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1

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Duopoly

DL(.1,1) = 1+.12 = .55

DK(.1,1) = 1− .55 = .45.Does Kenneth improve his profit by moving to thewest?



Duopoly

0 0.1 0.450.5 0.8 10

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Mill

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Mill(x)=1



0 0.1 0.450.5 0.8 10

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Duopoly

DL(.1, .8) = .1+.82 = .45.

DK(.1, .8) = 1− .45 = .55.



Duopoly

1 If L < K ,Liz takes the consumers from 0 to the midpointbetween L and K :

DL(L, K ) =L + K

2.

Kenneth takes the rest:

DK (L, K ) = 1−DL(L, K ) = 1−L + K

2.



Duopoly

2 If L > K ,Kenneth takes the consumers from 0 to themidpoint between L and K :

DK (L, K ) =L + K

2.

Liz takes the rest:

DL(L, K ) = 1−DK (L, K ) = 1−L + K

2.

3 If L = K , they split the market even:

DL(L, K) = DK(L, K) = .5.



Isoprofit



Isoprofit

An isoprofit curve of Liz indicates the set of location(L, K) that results in the same profit.



Isoprofit

For Liz, isoprofit at .4 is a set of location (L, K)that results in πL(L, K) = .4.

1 If L < K (region above 45◦ line)

πL(L, K ) =L + K

2= .4⇒ K = −L + .8.

2 If L > K (region below 45◦ line)

πL(L, K ) = 1−L + K

2= .4⇒ K = −L + 1.2.

3 If L = K (45◦ line)

πL(L, K ) = .5 , .4



Isoprofit

0 0.2 0.4 0.6 0.8 10

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Liz′s Isoprofit



Isoprofit

For Liz, isoprofit at .5 is a set of location (L, K)that results in πL(L, K) = .5.

1 If L < K (region above 45◦ line)

πL(L, K ) =L + K

2= .5⇒ K = −L + 1.

2 If L > K (region below 45◦ line)

πL(L, K ) = 1−L + K

2= .5⇒ K = −L + 1.

3 If L = K (45◦ line)

πL(L, K ) = .5

A crossbuck shape.© 2012 Hiroki Watanabe 32 /64


Isoprofit

For Kenneth, isoprofit at .4 is a set of location(L, K) that results in πK(L, K) = .4.

1 If L < K (region below 45◦ line)

πK (L, K ) = 1−L + K

2= .4⇒ K = −L + 1.2.

2 If L > K (region above 45◦ line)

πK (L, K ) =L + K

2= .4⇒ K = −L + .8.

3 If L = K (45◦ line)

πK (L, K ) = .5 , .4



Isoprofit

Question: where is Kenneth’s isoprofit at .5 located?



Isoprofit

0 0.2 0.4 0.6 0.8 10

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Kenneth′s Isoprofit



Isoprofit

How does Liz react to Kenneth’s location?If Kenneth is at K = 0, Liz maximizes its profit bylocating slightly to the east of K .(L,0)⇒ L = 0+ ε (ε = an infinitesimally smalldistance).If Kenneth is at K = .2, Liz maximizes its profit bylocating slightly to the east of K .(L, .2)⇒ L = .2+ ε.If Kenneth is at K = .4, Liz maximizes its profit bylocating slightly to the east of K .(L, .4)⇒ L = .4+ ε.



Isoprofit

(L, .5)⇒ L = .5.(L, .6)⇒ L = .6− ε.(L, .8)⇒ L = .8− ε.(L,1)⇒ L = 1− ε.



Isoprofit

How does Kenneth react to Liz’s location?(0, K)⇒ K = 0+ ε.(.2, K)⇒ K = .2+ ε.(.4, K)⇒ K = .4+ ε.(.5, K)⇒ K = .5.(.6, K)⇒ K = .6− ε.(.8, K)⇒ K = .8− ε.(1, K)⇒ K = 1− ε.



Isoprofit

Turn L K

Initial 0 1Liz .99 1

Kenneth .99 .98Liz .97 .98...

Kenneth .52 .51Liz .50 .51

Kenneth .50 .50Liz .50 .50

Kenneth .50 .50




2 Monopoly (N = 1)

3 Duopoly (N = 2)

4 Nash EquilibriumNash Equilibrium & Socially Efficient Location


6 Now We Know



Nash Equilibrium & Socially Efficient Location

Definition 4.1 (Nash Equilibrium)

Nash equilibrium is the location (LNE, KNE) such thatnone of the vendor can profit by unilaterally changingits location.

For duopoly, the Nash equilibrium(LNE, KNE) = (.5, .5).




Is the Nash equilibrium (LNE, KNE) = (.5, .5)socially efficient?Fact: the socially efficient allocation is(L∗, K∗) = (1/4,3/4) or (3/4,1/4).




0 0.25 0.5 0.75 10

0.5

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Location (miles)

Mill

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Nash Equilibrium Location

Mill(x)=1



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Nash Equilibrium Location




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Efficient Location

Mill(x)=1



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Efficient Location




Compare:

Location Profit TDP

Nash (.5, .5) (.5, .5) 1+1/4Socially Efficient (.25, .75) (.5, .5) 1+1/8Socially Efficient (.75, .25) (.5, .5) 1+1/8




0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

xL

xK

Liz′s Isoprofit




Question 4.2 (Socially Inefficient Equilibrium)

Why can’t vendors choose the socially optimal location,when both Nash equilibrium location and sociallyefficient location yield the same profits?




They are indifferent between (LNE, KNE) and(L∗, K∗).Given (L, K) = (.25, .75), Liz will move to alocation slightly to the west of K = .75.(L, K) = (.74, .75)→ (.74, .73)→ (.72, .73)→· · · → (.5, .5).




2 Monopoly (N = 1)

3 Duopoly (N = 2)

4 Nash Equilibrium

5 Oligopoly (N ≥ 2)N = 3N ≥ 4

6 Now We Know



N = 3

If there are 3 vendors, Liz solves

max0≤L≤1πL(L, K , J) = 1 ·DL(L, K , J).

For N = 3, there is no Nash equilibrium.Consider

1 All the vendors locate at different place.2 L = K = J.3 L = K , J.



N = 3

0 0.2 0.6 11

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Location (miles)

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0 0.2 0.6 11

2

Location (miles)

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Pric

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Mill(x)=1



DeliveredD(x)=|x−.9|



N = 3

0 0.25 0.55 11

2

Location (miles)

Mill

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0 0.25 0.55 11

2

Location (miles)

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Mill(x)=1






N = 3

0 0.2 11

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0 0.2 11

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Location (miles)

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Mill(x)=1






N = 3

0 0.25 11

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Location (miles)

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0 0.25 11

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Location (miles)

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Mill(x)=1






N = 3

0 0.5 11

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Location (miles)

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0 0.5 11

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Location (miles)

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Mill(x)=1






N = 3

0 0.25 0.55 11

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Location (miles)

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0 0.25 0.55 11

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Location (miles)

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0 0.25 0.55 11

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Mill(x)=1






N = 3

Fact: socially efficient location is(L, K , J) = (1/6,3/6,5/6) (not necessarily in thisorder).



N = 3

0 1/3 2/3 11

2

Location (miles)

Mill

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0 1/3 2/3 11

2

Location (miles)

Mill

Pric

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Pric

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Mill(x)=1

DeliveredL(x)=|x−1/6|

DeliveredK(x)=|x−3/6|

DeliveredD(x)=|x−5/6|



N ≥ 4

There are Nash equilibria for N ≥ 4.




2 Monopoly (N = 1)

3 Duopoly (N = 2)

4 Nash Equilibrium


6 Now We Know



Hotelling’s modelSocial optimality vs individual optimality.Nash equilibrium



References

Hal R. Varian.Intermediate Microeconomics.Norton, seventh edition, 2005.



Map du Jour

Source http://www.worldmapper.org/


http://www.worldmapper.org/

http://www.worldmapper.org/


D, demand, 8Delivered price, 7duopoly, 19isoprofit, 29Mill price, 7monopoly, 10N, number of sellers, 5n, vendor, 7Nash equilibrium, 41oligopoly, 50p, price, 7

π, profit, 11, 19private optimum, 17profit, 8social optimum, 17socially efficient allocation,42TDP, see total deliveredpricetotal delivered price, 15, location, 7


lecture 1f: hotelling’s model - university of minnesota...

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