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Two sets of problems: • How does product differentiation influence the firms’ price competition? Bertrand model where demanded quantity is reduced in own price and increased in the other’s price • How do firms differentiate their products to weaken price competion? Types of differentiation: Horizontal differentiation Vertical differentiation Bertrand with differentiated products Each firm’s demand increases in the other’s price: Di ( pi , p j ) a bpi dp j If production costs equal zero, then profit equals: i ( pi , p j ) a bpi dp j pi i ( pi , p j ) / pi a 2bpi dp j 0 a dp j Best response fn: pi f i ( p j ) 2b Best response fns are 2 2 i d i increasing; prices are f i ( p j ) 0 2 pi p j pi 2b strateg. complements: FOC: Bertrand with production costs Nash equilibrium in the Bertrand model p2 Are there prices for the two firms so that no firm will regret its own choice when getting to know the other firm’s choice? 1's best resonse fn: p1 ( a dp2 ) / 2b 2's best response fn: p2 (a dp1 ) / 2b p1b p2b a /( 2b d ) q1b q2b ab / (2b d ) 1 ( p1b , p2b ) 2 ( p1b , p2b ) a 2 b /(2b d ) 2 p1 Observations about Bertrand competition • Best response curves have positive slope p2 • The ”reaction story” p2 is stabile. • Increased c1 shifts 1’s curve outwards. • Increased c1 leads to inpp1 1 creased p1 and increased p2. • Increased c1 is an indirect advantage for both. 1 ( p1 , p2 ) a bp1 dp2 ( p1 c1 ) p2 Firm 1’s best response fn Firm 2’s best response fn 2 ( p1 , p2 ) a bp2 dp1 ( p2 c2 ) FOC: p1 a dp2 bc1 / 2b p2 a dp1 bc2 / 2b 2b( a bc1 ) d ( a bc2 ) b p1 ( 2b d )( 2b d ) 2b( a bc2 ) d ( a bc1 ) b p2 ( 2b d )( 2b d ) What happens when firm 1’s costs increase? p1 • Horizontal product differentiation Eks: Different locations of stores. Different time slots for airline departures. The optimal choice for identical prices depends on the consumer taste. • Vertical product price differentiation Ex: Hyundai, BMW For identical prices, all wish to own a BMW. The optimal choice for identical prices is same for everyone. For different prices, consumer make different choices. 1 Choice of location, followed by price comp. Horizontal differentiation: Linear city 0 A consumer Store 1 • Stage 1: Simultaneously firms decide where to locate (a og b). • Stage 2: Given location, firms compete in price. 1 Store 2 x a b An SPE is found by determining the Nash equilibrium in stage 2 for all locations. • Consumers distributed uniformly between 0 and 1. • Two stores located a og b from the end points. When firms choose locations in stage 1, they take into account the consequences in stage 2. Motivation: Show how firms in stage 1 choose maximal differentiation to weaken price comp. • With quadratic transport costs, consumer at x will buy from store1 if and only if p1 t ( x a ) 2 p2 t (1 b x ) 2 Nash equilibrium in prices in stage 2 Demand For given firm locations, a og b, and prices, p1 og p2, where does the indifferent consumer live? p t( ~ x a ) 2 p t (1 b ~ x )2 1 2 p p1 t (1 a b)[2a (1 a b)] ~ x 2 2t (1 a b) Consumers to the left of ~ x buy from firm 1. 1 a 2b p2 p1 ~ 2 2 p1 Dt 1 ~ x( p1, p22a)~ xxaap2 t ~ x 2(1 b) ~ x (1 b) 2 2 2t (1 a b) ~ 2 from Consumers to ptheright a 2 ) firm 2. p tof ((1 xb)buy ~ x 2 ~ 1 1 a b p1 p2 D2 ( p1 , p2 ) 1 x 2bt (1 a b) 2 2t (1 a b) 2 ( a, b) D2 ( a , b, p1c ( a , b), p2c ( a, b))( p2c ( a, b) c ) 0 p 1 D1 D1 p c D ( p1 c ) D1 1 ( p1c c ) a a p2 a p1 a 1 ( a , b) 0 a p1 12 c p2 t (1 a b)(1 a b) p2 12 c p1 t (1 b a )(1 b a ) a b p1c ( a, b) c t (1 a b)1 3 ba p2c ( a , b) c t (1 b a )1 3 What happens when firm 1 gets close to 0; i.e. reduces a? p1 Firm 1’s best response fn Firm 2’s best response fn Welfare analysis 1 ( a , b) D1 ( a, b, p1c ( a , b), p2c ( a , b))( p1c ( a, b) c ) Demand effect 2 D2 ( p1 , p2 )( p2 c ) p2 Consumption is given, all buy. Therefore: Welfare max. by min. transport costs. Location choice in stage 1 c 2 1 D1 ( p1 , p2 )( p1 c ) Store 2 Store 1 1 c 1 Strategic effect a 14 b 14 Transport costs in equilibrium 0 Max differentiation Differentiation to weaken price competition. a0 b0 a 0 and b 0 2 0 tx dx x 0 Transport costs in social optimum a 14 and b 14 1 2 2 2t 3 3 1 2 t 12 4 0 tx 2 dx 43t x 3 0 48t 1 4 1 4 2