HOTELLING LOCATION MODEL

Transcription

1 HOTELLING LOCATION MODEL THE LINEAR CITY MODEL The Example of Choosing only Locaion wihou Price Compeiion Le a be he locaion of rm and b is he locaion of rm. Assume he linear ransporaion cos equal o d, where > 0. If he good is free, he only cos consumers have o pay is he ransporaion cos. We have already seen in he miderm ha he Nash equilibrium of his model is ha boh rms are locaed in he middle of he line and ge marke share. I is easily seen ha he oal ransporaion cos of all people residing in his linear ciy is =4: Is his a Pareo e cien soluion? Of course no. To nd he Pareo Opimal soluion, he social planner mus minimize he ransporaion cos. Consider he gure on page. If a and b are o be on arbirary poin on he uniary line, we know ha 0 < a < b <. Thus, consumers o he lef of a have he ransporaion coss equal o aa = a ; consumers beween a and a+b have he ransporaion coss equal o a + b a + b a a = b a ; consumers beween a + b a b ; and nally, and b have he ransporaion coss equal o consumers beween b and have he ransporaion coss equal o ( b) : Social planner minimizes wih respec o a and b of he erm

2 0 / 0 a b a + b a + a b + ( b) : This resuls in wo equaions and wo unknowns so i should be your ask o work on o nd ou he value of a and b. The ransporaion coss of he whole ciy decrease from /4 o /8. (You should also verify his.) The Price Compeiion Model of Linear Ciy There are wo rms or sores, which sell he same idednical good. For simpliciy, suppose ha hese wo sores are locaed a he exremes of he ciy; sore is a x = 0 and sore is a x =. Assume here is no marginal cos wih each rm. Consumers incur a ransporaion cos per uni of lengh. Thus, a consumer living a x incurs a cos of x o go o sore and a cos of ( x) o go o sore. Also assume ha each consumer has a uniary demand. A consumer indi eren beween he wo rms is locaed a x, where x is demand given by his consumer in equilibrium, p + x = p + ( x)

3 p + x = p + x x = p p + ) x = p p + ; x = p p + : Suppose we wan o nd he pro for rm. Since by assumpion ha each consumer has a uniary demand, rm choose price insead of quaniy. Hence, rm max p F.O.C. : p x = max p p p p + p = p : p p Firm hen max p p p p + F.O.C. : p p + p p p = max + p p + = 0 =) p p + = 0 = p =) p p + = p =) p + = p p + = 4p =) p = =) ) p = p = : x = p p + = ; Pro = : Consider a more generalized version where rm is locaed a poin a > 0 and rm is locaed a poin > b > a: A raional consumer should be beween a and b, bu we are no sure where excaly he is. To deermine his locaion and he rm s price and pro, we noe ha in equilibrium, a consumer mus be indi eren beween he cos obaining a good so ha

4 p + (x a) = p + (b x) p + x a = p + b x x = p p + (a + b) x = p p + (a + b) : Thus, demand for rm is x = p p + (a + b) x = + p p (a + b) p p p Firm max + (a + b) p p F.O.C. : p p p p = (a + b) + a + b = 0 p = p (a + b) Firm max p p + p p p (a + b) p F.O.C. : + p p = a + b + p 4p + (a + b) = (a + b) + (a + b) = p ) p = ( + a + b) ; p = ( + a + b) (a + b) = 4 (a + b) Since boh rms have zero marginal cos, i mus be ha rm is geing demand equal o ha of rm. This means a = b: Hence, p = p = ; x = ; = 4

5 p i THE CIRCULAR CITY MODEL Consider he circular ciy where consumers are locaed uniformly on a circle wih a perimeer equal o. Densiy is uniary around he ciy, and all consumers ravel along he perimeer of he circle (no across he circle). Think of he ciy around he circular lake, for example. We sill reain he assumpion ha each consumer buys one uni of he good, has a uni ranspor cos, and is willing o buy a he smalles ransporaion cos. The rms have zero marginal cos. Le n denoe he number of enering rms. Those rms do no choose heir locaions, bu raher are auomaically locaed equidisan from one anoher on he circle. Then rms compee in prices given hese locaions. Also assume free enry. Since n rms enered he marke and were locaed symmerically around he circle, i is reasonable o consider an equilibrium in which hey all charge he same price p since he good is idenical. In pracice, we can consider only rm i and i has only wo real rivals; namely hose around i. Consider he nex gure. In equilibrium, a consumer locaed a he disance x 0; n from rm i mus be indi eren beween purchasing from rm i and purchasing from anoher rm besides i. Hence, p i + x = + x! p i + x = + x! x = p i + n n n : 5

6 x /n p i /n x Noe ha x is he demand for each rm since one x accouns for he lef hand side and anoher x accouns for he righ hand side. pj p i Therefore, rm i max p i + p i p i = max + p i p i n p i n : F.O.C. : p i + n = 0! p i = n! p i = n : Since he model is symmeric, you can show ha we will derive anoher equaion, namely p i = n : Therefore, in equilibrium, p i = = n : Each consumer should be locaed a he disance x = : We can see ha n each rm s pro decreases wih he number of rms. If we are o impose he xed cos in enering he marke for each rm, say f. Wih free enry and exi, we can deermined he equilibrium number of rms as oal zero pro condiion. n f = 0! n = f! n eq = r f : Wih higher xed cos, he oal number of rms will decrease. Anoher poin o address is he socially opimal number of rms. Is n eq a socially opimum? Is n eq oo much or oo lile a number of rms? We need o consider he social welfare of he ciy. We see ha he oal cos of raveling is he area of he riangle as in he rs example. each riangle has he area of x n = n n : 6

7 Bu we have oally n riangles (why?) Thus, he oal cos of raveling is : The social planner minimizes he oal ransporaion coss plus oal 4n xed coss. min n 4n + nf! F:O:C: : f = 4n! n = 4f! n opimal = r f < n eq: Noe ha couning he number of riangle like he rs example can be done wih only linear ransporaion cos. If he cos is no linear; i.e., he general funcion of x, namely, f(x). The ransporaion cos mus be he n Z n 0 f (x) dx; so he social planner should minimize nf+n Z n 0 f (x) dx: A rivial poin bu of some imporance of his circular model is ha rms price above marginal cos and ye do no make pro s. Thus an empirical nding ha rms do no make more han normal pro s in an indusry should no lead one o conclude ha rms do no have marke power, where marke power is de ned as pricing above marginal cos. 7