Location Competition Linear city of length 1 and consumers are uniformly distributed. Two stores sell the same good. These two stores are located at the points respectively where x Xb 20. Therefore store a is to the left of store b The unit cost of the good for eac ch store is c. Consumers incur a transportation cost of t per unit of length s So a consumer living at x=o incurs a cost tx. to go to store a and a cost tx, to go to store b. The customer at x=0 will buy from store a as long as Pa which implies that Pa"Pb t(xb-x )store b will capture the entire market and then store a will have a zero payoff. The customer at point x=l will buy from store b as long as t(1-x3)+Pa>t(1-b)+b which implies that Pb"Pa t(xb-x) (2) So: if Pb-Pa > t(bxa) store a will capture the whole of the market and have a payoff P.-C If inequalities (1)and (2)are both satisfied customer o goes to store a and customer 1 goes to, store b. Let us now find the location x of the
customer who is indifferent between the two stores. First note that if store a attracts customer x, it"also attracts all customers at x> going beyond customers's distances from both sellers increase at the same rate. So we know that if there is an indifferent customer at x, he is situated between x and or this customer t〔 )+p=t(x-×)+p so that Pb"Pa s t(2x -xa-*b) Therefore, 2EI (P1p)+t(x+×1) (3) Since store a keeps all customers between 0 and x, equation (3)is the demand curve for a so long as he does not set his price so far above b's that he looses even customer 0. The payoff of store a will then be ma=(Pa-c)2t(Pb-Pa Store a will choose p so as to maximise its profits The first order condition gives =0 2E|(Pb2)+ 2 2pa"Pb=c+t(xa+b) The demand facing store b is 1-x and by following the same procedure we find that the first order condition for store b is t〔2-x 5 Solving the two first order conditions (4) and (5) for the two prices gives
t(2+x+) t〔4-x Pb (6 Taking into account (6),x in(3)is equal to t(4-x t(2 2 t(xa+xb 2t(1-×-x)3t(x t(2+Xa+3 Consequently 2+X+x)t(2+x 2+X 2+x t 7 Similarly we derive (x飞2 (8) From (7)and (8)we notice that profits are positive and increasing in the transportation cost. The products are differentiated more the higher transportation cost is. On the other hand, when t=o all consumers can go to either store for the same cost (0). The absence of product differentiation leads to the Bertrand result, i. e. from(3)Pa- Pb =c and from (7)and (8) Moreover, if the two firms locate at the same position the consumer will decide to which store to go purely on the basis of prices Inequalities (1)and(2) can not be simultaneously satisfied because (1) collapses, into p<p, and (2) into p,<p. Therefore we have competition in terms of prices for Identical products, thus leading. to the Bertrand result
Absence of phice campetition; the principle of minimal dippenentiation Assume a city of length 1, uniform distribution of consumers, and a price p(>c), determined exogenously. Also assume that firms share demand if the are located identically. Because the prices and the profit marginsare fixed, the firms choose their locations so as to. maximise demand. Let firm 1 locate at point xa and firm 2 at point Xb, where xa0. E>0, firm 2, for instance. would have demand 2-2x-e a >1/2 Thus firms have again an incentive to move towards the centre. Therefore when xa =*b =1/2, neither firm would want to move. Thus, the only equilibrium has both firms located at the centre of the city. In this example, the products are socially close to. each other. Transportati costs could be reduced by having firms move away from the center, but there is no incentive to do so
