gradio05lecset3 - Markets with Dierentiated Products...

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Markets with Di f erentiated Products. Oligopoly: Horizontal Product Di f erentiation. The Linear City Model: This is the basic model of horizontal product di f erentiation where the prod- ucts are separated on one (horizontal) dimension or attribute. It was developed as a (spatial) model of location choice by Hotelling (1929) and has been co-opted by several distinct areas in economics. There is a linear city of length one, the [0,1] interval. A unit mass of con- sumers are uniformly distributed on this interval. Consumers are identical ex- cept for their location. Each consumer has unit demand. If a consumer buys from a f rm located at distance d at price p , she incurs a transport cost t ( d ) and her net surplus is S p t ( d ) , where S is the gross surplus from consuming the good, t (0) = 0 and t ( d ) is strictly increasing. There are two f rms in the market that produce the good. Firms are located on the unit interval. The unit cost of production is identical across f rms and is independent of location. Theproductso fthetwo f rms are horizontally di f erentiated by their loca- tion. Other things being equal, each consumer prefers the product of the f rm located closest to her. Interpret [0 , 1] as the product space. The location of a f rm on this interval indicates the type of product it supplies. The location of a consumer indicates her most preferred product type. When a consumer buys from a f rm whose product is not her most preferred type, she incurs a psychological cost that depends on how far removed the purchased product is from her most preferred type; the transport cost function captures this psychological cost. Socially optimal solution: Firms locate at 1 4 , 3 4 so as to minimize the total transport cost of society (and serve all consumers as long as S is large enough). 1. Product Choice with No Price Competition: Assume prices are f xed so that there is no price competition in the market: c p 1 = p 2 = e p<S + t (1) In this case, all consumers buy, no matter where f rms locate. A consumer strictly prefers to buy from the f rm closest to it. Firms decide on location simultaneously. Note that payo f of each f rm is directly proportional to its market share. Suppose that f rm 1 locates at a point a and f rm 2 located at a point 1 b. Without loss of generality, assume a 1 b, i.e., f rm 1 locates to the left of f rm 2. 1
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The market share of f rm 1 is a +1 b 2 and the market share f rm 2 is 1 a b 2 = b a 2 . Unique NE: Minimal Di f erentiation i.e., a = b =0 . 5 . Proof: First, we show that a = b . 5 is a NE. To see this note that if a . 5 and f rm 2 sets b< 0 . 5 , then only consumers to the right of the mid point of the interval [0 . 5 , 1 b ] buy from f rm 2 so that its market share is 1 0 . 5+(1 b ) 2 . 25 + b 2 < 0 . 5 . Next,wenotethat a< 1 b cannot be a NE. That it because f rm 1 can increase market share by locating at 1 b ±>a instead, where ±> 0 is small. Finally, note that a =1 b 6 . 5 cannot be a NE. In such a situation, all consumers are indi f erent between the f rms and so each f rms gets market share equal to 0 . 5 .
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gradio05lecset3 - Markets with Dierentiated Products...

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