[PDF] HOTELLING'S MODEL - UC Davis





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HOTELLING'S MODEL - UC Davis

population i e 300 consumers) Two firms offer the same product (e g milk) Firm 1 is located at point x1 and firm 2 is located at point x2 (let firm 1 be to the left of firm 2 so that 0 ? x1 ? x2 ? 1)



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200C Micro Theory Professor Giacomo Bonanno

HOTELLING'S MODEL

Cournot's model assumes that the products of all the firms in the industry are identical, that is, all consumers view them as perfect substitutes. It is a very usef ul model in that it enables us to

prove in a simple way such claims as: "the larger the number of firms in an industry the stronger the

competition among them", "consumers benefit from the entry of new firms in an industry", "perfect competition can be thought of as an approximation to what happ ens in industries where the number of firms is very large", "one of the factors that determine the number of firms in an industry is the size of the fixed cost", etc. Bertrand in his review (1883) of Cournot's book criticized the assumption that firms choose output levels, maintaining that in reality the main decision firms have to make is what price to charge for their products. He showed that if output competition is replaced by price competition, while maintaining the assumption of product homogeneity, then the only equilibrium is one where price is equal to marginal cost. Another criticism of Cournot's model concerns the assum ption of product homogeneity. Very few products can be considered identical. Sugar, milk, cement, might be good examples of

products that do not differ across firms. Yet the local convenience store might charge a bit more for

a carton of milk than the nearest supermarket and some consumers might be willing to pay the

small premium associated with convenience and proximity. As a matter of fact, it is very difficult to

think of examples of truly homogeneous products. Furthermore, firms seem to spend a lot of money

and effort in the attempt to differentiate their products from those of their competitors. Think, for

example, of the frequen t-flyer programs introduced by airlines. A firm can either add something

"real" to the product in order to differentiate it from its competitors' products or it can try to add -

usually through advertising - a "perception" that the product is different, even though it is not. The first model of product differentiation is due to Hotelling (1929). Imagine a town with a Main Street of length 1. There are N consumers living on this street and they are uniformly distributed along the street, that is, on a segment of the street of length x there are xN consumers (thus, if N = 900 and we take a segment of length 1/3 then on this segment lives 1/3 of the

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population, i.e. 300 consumers). Two firms offer the same product (e.g. milk). Firm 1 is located at point x 1 and firm 2 is located at point x 2 (let firm 1 be to the left of firm 2, so that 0 x 1 x 2

Firm 1 charges (mill) price p

1 , while firm 2 charges (mill) price p 2 . Consumers have a transportation

cost of per unit of distance. In the following diagram, the vertical lines represent mill prices, while

the oblique lines represent "delivered" prices, i.e. mill price plus transportation cost. delivered price 0 1 xx 12 firm firm 1 2 p 1 p 2 z The consumer located at z is indifferent between the two firms. Those located to the left of z prefer firm 1, while those located to the right of z prefer firm 2. Hence firm 1 will serve the market segment [0,z], while firm 2 will serve the market segment [z,1]. Obviously, the location of the indifferent consumer depends on both prices. Given p 1 and p 2 , the indifferent consumer is given by the solution to: p 1 + (z x 1 ) = p 2 + (x 2 z). Thus z = p 2 p 1 2 x 2 + x 1 2

Hence the demand functions are given by:

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D 1 (p 1 ,p 2 ) = z N = p 2 p 1 2 x 2 + x 1 2 N and D 2 (p 1 ,p 2 ) = ()1 z N = 1 p 2 p 1 2 x 2 + x 1 2 N.

To see firms' incentive to differentiate their products, first consider the case where both firms are

located at 1 2 , i.e. x 1 = x 2 1 2 . Assume that both firms have the same cost function:

TC = c q

(with c > 0) and that = 1. Then the unique Nash equilibrium in prices (Bertrand-Nash equilibrium) is given by p 1 = p 2 = c. Hence both firms make zero profits in equilibrium. Consider now the case where x 1 = 0 and x 2 = 1 (that is, the two firms are located at the extremes). The profit function of firm 1 is (recall that = 1): 1 (p 1 ,p 2 ) = p 1 p 2 p 1 2 1 2

N c

p 2 p 1 2 1 2 N while the profit function of firm 2 is given by: 2 (p 1 ,p 2 ) = p 2 1 2 p 2 p 1 2 N c 1 2 p 2 p 1 2 N As usual, the Nash equilibrium is given by the solution to the system of equations: 1 p 1 = 0 and 2 p 2 = 0. The solution is: p 1 = p 2 = c + 1 1 2 1 2 N .

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