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Chapter 7: Product Differentiation Chapter 7: Product Differentiation A1. Firms meet only once in the market. Relax A2. Products are differentiated. A3. No capacity constraints. Timing: 1. firms choose simultaneously their location in the product space, 2. given the location, price competition. Spatial-differentiation model • – Linear city (Hotelling, 1929) – Circular city (Salop, 1979) Vertical differentiation model • – Gabszwicz and Thisse (1979, 1980); – Shaked and Sutton (1982, 1983) Monopolistic competition (Chamberlin, 1933) • Advertising and Informational product differentiation • (Grossman and Shapiro, 1984) 1 1 Spatial Competition 1.1 The linear city (Hotelling, 1929) Linear city of length 1. • Duopoly with same physical good. • Consumers are distributed uniformly along the city, • N =1 Quadratic transportation costs t per unit of length. • They consume either 0 or 1 unit of the good. • If locations are given, what is the NE in price? • Price Competition Maximal differentiation 2 shops are located at the 2 ends of the city, shop 1 is at • x =0andofshop2isatx =1. c unit cost p1 and p2 are the prices charged by the 2 shops. • 2 Price of going to shop 1 for a consumer at x is p1 + tx . • 2 Price of going to shop 2 for a cons. at xp2 + t(1 x) . • − The utility of a consumer located at x is • 2 2 s p1 tx if he buys from shop 1 − − 2 U = s p2 t(1 x) if he buys from shop 2 − − − 0 otherwise Assumption: prices are not too high (2 firms serve the • market) Demands are • p2 p1 + t D (p ,p )= − 1 1 2 2t p1 p2 + t D2(p1,p2)= − 2t and profit • i pj pi + t Π (pi,pj)=(pi c) − − 2t So each firm maximizes its profit and the FOC gives • pj + t c pi = − for each firm i 2 Prices are strategic complements: • ∂2Πi(p ,p ) i j > 0. ∂pi∂pj 3 The Nash equilibrium in price is • pi∗ = pj∗ = c + t The equilibrium profits are • t Π1 = Π2 = 2 Minimal differentiation 2 shops are located at the same location xo. • p1 and p2 are the prices charged by the 2 shops. • Price of going to shop 1 for a consumer at x is • 2 p1 + t(xo x) . − Price of going to shop 2 for a consumer at x is • 2 p2 + t(xo x) . − The consumers compare prices.... Bertrand competition • Nash equilibrium in prices is • pi∗ = pj∗ = c and the equilibrium profits are • Π1 = Π2 =0 4 Different locations 2 shops are located at x = a andofshop2isatx =1 b • where 1 a b 0. − − − ≥ If a = b =0: maximal differentiation • If a + b =1: minimal differentiation • p1 and p2 are the prices charged by the 2 shops. • Price of going to shop 1 for a consumer at x is • 2 p1 + t(x a) . − Price of going to shop 2 for a consumer at x is • 2 p2 + t(1 b x) . − − Thus there exists an indifferent consumer located at x • 2 2 p1 + t(x a) = p2 + t(1 b x) − − − p2 p1 1 b + a x = − + − e ⇒ e2(1 a b) 2 e and thus the demand− for− each firm is • e 1 b a p2 p1 D (p ,p )=a + − − + − 1 1 2 2 2(1 a b) − − 1 b a p1 p2 D2(p1,p2)=b + − − + − 2 2(1 a b) − − 5 The Nash equilibrium in price is • a b p∗(a, b)=c + t(1 a b)(1 + − ) 1 − − 3 b a p∗(a, b)=c + t(1 a b)(1 + − ) 2 − − 3 Profits are • 1 Π (a, b)=[p∗(a, b) c]D1(a, p∗(a, b),p∗(a, b)) 1 − 1 2 2 Π (a, b)=[p∗(a, b) c]D2(b, p∗(a, b),p∗(a, b)) 2 − 1 2 Product Choice Timing: 1. firms choose their location simultaneously 2. given the location, they simultaneously choose prices Firm 1 chooses a that maximizes Π1(a, b) a(b) • ⇒ Firm 2 chooses b that maximizes Π2(a, b) b(a) • ⇒ andthen(a∗,b∗). • What is the optimal choice of location? • 6 dΠ1(a, b) ∂Π1(a, b)∂p ∂Π1(a, b) ∂Π1(a, b)∂p = 1 + + 2 da ∂p1 ∂a ∂a ∂p2 ∂a ∂Π1(a, b)∂p where 1 =0(due to envelope theorem) ∂p1 ∂a We can rewrite • 1 dΠ (a, b) ∂D1(.) ∂D2(.)∂p2 =[p1∗(a, b) c]( + ) da − ∂a ∂p2 ∂a ∂D1(.) 3 5a b where = − − Demand Effect (DE) ∂a 6(1 a b) − − ∂D2(.)∂p2 2+a and = − < 0 Strategic Effect (SE) ∂p2 ∂a 3(1 a b) − − dΠ1(a, b) 1 3a b Thus, =[p∗(a, b) c](− − − ) < 0 da 1 − 6(1 a b) − − As a decreases, Π1(a, b) increases. • Result The Nash Equilibrium is such that there is maximal differentiation,i.e.(a∗ =0,b∗ =0) 7 2 effects may work in opposite direction • –SE<0 (always): price competition pushes firms to locate as far as possible. –DEcan be >0ifa 1/2,toincreasemarket share, given prices, pushes≤firms toward the center. – But overall SE>DE, and they locate at the two extremes. 1.1.1 Social planner Minimizes the average transportation costs. • Thus locates firms at • 1 1 as = ,bs = 4 4 Result Maximal differentiation yields too much product differentiation compared to what is socially optimal. 8 1.2 The circular city (Salop, 1979) circular city • large number of identical potential firms • Free entry condition • consumers are located uniformly on a circle of perimeter • equal to 1 Density of unitary around the circle. • Each consumer has a unit demand • unit transportation cost • gross surplus s. • f fixed cost of entry • marginal cost is c • Firm i’s profitis • (pi c)Di f if entry Πi = − − ( 0 otherwise How many firms enter the market? (entry decision) • 9 Timing: two-stage game 1. Potential entrants simultaneously choose whether or not to enter (n). They are automatically located equidistant from one another on the circle. 2. Price competition given these locations. Price Competition Equilibrium is such that all firms charge the same price. • Firm i has only 2 real competitors, on the left and right. • Firm i charges pi. • Consumer indifferent is located at x (0, 1/n) from i • 1 ∈ pi + tx = p + t( x) n − Thus demand for i is t • p + pi D (p ,p)=2x = n − i i t Firm i maximizes its profit • (pi c)Di(pi,p) f − − Because of symmetry pi = p, and the FOC gives • t p = c + n 10 How many firms? Because of free entry condition 1 t (pi c) f =0 f =0 − n − ⇒ n2 − which gives t 1 n =( )2 ∗ f and the price is 1 p∗ = c +(tf)2 Remark: p c>0 but profit =0.... • − If f increases, n decreases, and p c increases. • − If t increases, n increases, and p c increases. • − If f 0, n and p c (competitive market). • → →∞ → Average transportation cost is • 1 2n t 2n xtdx = 0 4n From a social viewpointZ • 1 2n Min[nf +2n xtdx] n Z0 s 1 n = n∗ <n∗ ⇒ 2 11 Result The market generates too many firms. Firms have too much an incentive to enter: incentive is • stealing the business of other firms. Natural extensions: • – location choice – sequential entry – brand proliferation 12 1.3 Maximal or minimal differentiation Spatial or vertical differentiation models make important • prediction about business strategies Firms want to differentiate to soften price compe- tition In some case: maximal product differentiation. Opposition to maximal differentiation • – Be where the demand is (near the center of linear city) – Positive externalities between firms (many firms may locate near a source of raw materials for instance) – Absence of price competition (prices of ticket airline before deregulation) 13 2 Vertical differentiation Gabszwicz and Thisse (1979, 1980); Shaked and Sutton • (1982, 1983) Timing: two-stage game 1. Simultaneously choice of quality. 2. Price competition given these qualities. Duopoly • Each consumer consumes 0 or 1 unit of a good. • N =1consumers. • A consumer has the following preferences: • θs p if he buys the good of quality s at price p u = − ( 0 if he does not buy where θ > 0 is a taste parameter. θ is uniformly distributed between θ and θ = θ +1; • density f(θ)=1; cumulative distribution F (θ) [0, ) ∈ ∞ F (θ): fraction of consumers with a taste parameter < θ. • 2 qualities s2 >s1 • 14 Quality differential: ∆s = s2 s1 • − Unit cost of production: c • Assumptions • A1. θ > 2θ (insure demand for the two qualities) θ 2θ p1∗ A2. c + − ∆s<θs1 (insure that < θ) 3 s1 Firms choose p1 and p2 • Price competition (given qualities) p2 p1 There exists an indifferent consumer: θ = − . s2 s1 • – A consumer with θ θ buys the quality− 2 (θ p2 p1 ≥ ≥ − ). The proportion of consumers who will buy s2 s1 e − good of quality 2 is F (θe) F (θ). − – A consumer with θ < θ and θ θ > p1 buys low s1 quality 1. So the proportion ofe consumers≥ who will p2 p1 buy good of quality 1 iseF ( − ) F (θ). s2 s1 − – if θ < θ no purchase. − Then demands are • p2 p1 D1(p1,p2)= − θ ∆s − p2 p1 D2(p1,p2)=θ − − ∆s 15 Each firm maximizes its profit • Max(pi c)Di(pi,pj) pi − The reaction functions are • 1 R1(p2)=p1 = [p2 + c θ∆s] 2 − 1 R (p )=p = [p + c + θ∆s] 2 1 2 2 1 The Nash equilibrium is • θ 2θ p∗ = c + − ∆s (A2.) 1 3 2θ θ p∗ = c + − ∆s>p∗ 2 3 1 Demands are • θ 2θ D∗ = − (A1.) 1 3 2θ θ D∗ = − 2 3 and profits • (θ 2θ)2 Π1(s ,s )= − ∆s 1 2 9 (2θ θ)2 Π2(s ,s )= − ∆s 1 2 9 16 High quality firm charges a higher price than low quality • firm.
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