4.4. Vertical Differentiation Matilde Machado Industrial - PDF document

4.4. Vertical Differentiation Matilde Machado Industrial Organization- Matilde Machado Vertical Differentiation 1 4.4. Vertical Differentiation The Hotelling model studies situations of horizontal differentiation since for equal prices there are always consumers that prefer A to B and others B to A. Let’s modify the Hotelling model to incorporate quality differences between the goods (i.e. vertical differentiation) Industrial Organization- Matilde Machado 2 Vertical Differentiation 1

4.4. Vertical Differentiation Firms and consumers are locates in the interval [0,1]. All consumers prefer a good close to 1. Consumers are uniformly distributed along [0,1]. 2 firms A and B located in a and b respectively. W.lo.g 0 ≤ a ≤ b ≤1 0 a b 1 Industrial Organization- Matilde Machado Vertical Differentiation 3 4.4. Vertical Differentiation Utility of consumer x is: For a given position x, the gross − = ⎧ consumer surplus of ax p i A = ⎨ A U ( ) i buying from B is − = x ⎩ bx p i B higher (bx>ax) so B willing to pay a higher price. Notice Consumer’s that as x increases firm location in [0,1] the consumer has a higher valuation for both goods ∂ U/ ∂ x>0 Two-stage game: 1st stage: Firms select their locations (i.e. their product quality) 2nd stage: Firms compete in prices simultaneously) Industrial Organization- Matilde Machado 4 Vertical Differentiation 2

4.4. Vertical Differentiation We solve the game backwards. 2nd stage: Suppose there is an indifferent consumer between A and B : x ˆ = − = − = ˆ ˆ U ( ) A ax p bx p U ( ) B x ˆ A B x ˆ − p p If prices are equal all consumers buy ⇔ = B A x ˆ from B (i.e. the indifferent consumer is − b a located at zero) Consumers to the right of buy from B and to the x ˆ left buy from A: Industrial Organization- Matilde Machado Vertical Differentiation 5 4.4. Vertical Differentiation − Therefore demand for A is and for B is ˆ 1 x ˆ x Because b>a (b U x (A) U x (B) U x (A) is higher U x (B) quality), we must have P B >P A . z 0 ˆ x 1 x a -p A Demand for Demand for If we require U ≥ 0 for the A B b consumer to buy, then the -p B demand for A is only [z, ] x ˆ x ˆ that is -z , where z=p A /a Industrial Organization- Matilde Machado 6 Vertical Differentiation 3

4.4. Vertical Differentiation Note that if p A >p B all consumers would buy from B, (higher quality and lower prices), the indifference curves would not cross because there would not be an indifferent consumer . U x (B) U x (A) -p B -p A Industrial Organization- Matilde Machado Vertical Differentiation 7 4.4. Vertical Differentiation Let’s suppose c=0, the problem of firm A is: − ⎛ ⎞ p p ( ) π = = ˆ ⎜ B A ⎟ Max a b p , , , p p x p − A A B A A ⎝ ⎠ b a p A ∂ π − p p 1 = ⇔ − = A B A FOC: 0 p 0 ∂ − − A p b a b a A p 2 p ⇔ − = ⇔ = ≠ B B p 0 p for b a − − A A b a b a 2 Firm A’s reaction function Industrial Organization- Matilde Machado 8 Vertical Differentiation 4

4.4. Vertical Differentiation El problema de la empresa B es: − ⎛ ⎞ p p ( ) π = − = − B A Max a b p , , , p p (1 x ˆ ) p ⎜ 1 ⎟ − B A B B B ⎝ ⎠ b a p B ∂ π − p p 1 = ⇔ − − = B B A FOC: 0 1 p 0 ∂ − − B p b a b a B − + 2 p 1 2 p b a p ⇔ − + = ⇔ = B B A 1 p 0 − − − − A b a b a b a b a − + b a p ⇔ = ≠ A p for b a B 2 Firm B’s reaction function Industrial Organization- Matilde Machado Vertical Differentiation 9 4.4. Vertical Differentiation The equilibrium is the solution of the system: ⎧ p = B p ⎪ A ⎪ 2 ⎨ p − + ⎪ − + b a B b a p = = 2 A ⎪ p ⎩ B 2 2 − − − 3 b a 2( b a ) >0 and b a ⇔ = ⇔ = = > = p p p 0 MC B B A 4 2 3 3 The firm with the highest quality charges a higher price but both firms charge above marginal cost. The higher is the difference in quality (i.e. the higher is the distance (b-a)) the higher are both prices. Industrial Organization- Matilde Machado 10 Vertical Differentiation 5

4.4. Vertical Differentiation Demands in the 2nd stage are: ⎧ − ⎛ ⎞ p ( , ) a b p ( , ) a b 1 = = = ˆ ⎜ B A ⎟ ⎪ D ( , ) a b x − ⎪ A ⎝ ⎠ b a 3 ⇔ ⎨ − ⎛ ⎞ ⎪ p ( , ) a b p ( , ) a b 2 ( ) = − = − = ˆ ⎜ B A ⎟ D ( , ) a b 1 x 1 ⎪ − B ⎝ ⎠ ⎩ b a 3 Industrial Organization- Matilde Machado Vertical Differentiation 11 4.4. Vertical Differentiation Profits are: ⎧ − − ⎛ ⎞ b a p p 1 ( ) π = = = − ˆ ⎜ B A ⎟ ⎪ ( , ) a b p x p p − ⎪ A A ⎝ ⎠ B A 3 b a 3 ⇔ ⎨ − − ⎛ ⎞ ⎪ 2( b a ) p p ( ) π = − = − ˆ ⎜ B A ⎟ ( , ) a b p 1 x 1 ⎪ − B B ⎝ ⎠ ⎩ 3 b a ⎧ − − − ⎛ ⎞ 1 2( b a ) b a b a π = − = ⎜ ⎟ ⎪ ( , ) a b ⎪ A ⎝ ⎠ 3 3 3 9 ⎨ − ⎪ 2 4( b a ) ( ) π = − − + = > π ( , ) a b ( b a ) p p ( , ) a b ⎪ ⎩ B B A A 3 9 Industrial Organization- Matilde Machado 12 Vertical Differentiation 6

4.4. Vertical Differentiation 1st stage: − ⎧ b a π = Max ( , ) a b ⎪ ⎪ A Principle of maximum 9 a ⎨ differentiation. ∂ π ⎪ 1 = − < ⇒ = * A FOC: 0 a 0 Intuition: With vertical ⎪ ⎩ ∂ a 9 differentiation, firms specialize in a given − ⎧ 4( b a ) quality niche (high π = Max ( , ) a b ⎪ ⎪ valuation consumers B 9 b ⎨ and low valuation ∂ π 4 consumers). The higher ⎪ = > ⇒ = * B FOC: 0 b 1 is the difference in ⎪ ⎩ ∂ b 9 quality the higher is the market power of each firm Industrial Organization- Matilde Machado Vertical Differentiation 13 4.4. Vertical Differentiation = * * Because a 0, b =1, 1 4 π = π = Conclusion: Firms look for maximum * * , A B differentiation from their rivals. Although 9 9 qualities here have the same cost (c=0), 1 2 still firm A prefers to produce an inferior = = * * p , p good in order to differentiate from the rival. A B 3 3 If firms choose to locate sequentially, then the first one to enter would select b=1 1 = * x ˆ where profits are higher. 3 Industrial Organization- Matilde Machado 14 Vertical Differentiation 7

4.4. Vertical Differentiation If consumers may choose not to buy Consumers between 0 and U x (A) U x (B) U x (A) z=p A /a would be U x (B) better off if they do not buy. The indifferent consumer is the same as before. x ˆ z 0 x ˆ 1 x a -p A Demand for Demand for − A B p p p = − = − � B A A D ( p , p ) x z b A A B − b a a -p B − p p = − = − � B A D ( p , p ) 1 x 1 − B A B b a Industrial Organization- Matilde Machado Vertical Differentiation 15 8