4.4. Vertical Differentiation Matilde Machado Industrial - - PDF document

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Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

Matilde Machado

Industrial Organization- Matilde Machado

Vertical Differentiation

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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)

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Industrial Organization- Matilde Machado

Vertical Differentiation

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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

a b 1

Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

Utility of consumer x is: Two-stage game: 1st stage: Firms select their locations (i.e. their product quality) 2nd stage: Firms compete in prices simultaneously)

( )

A x B

ax p i A U i bx p i B − = ⎧ = ⎨ − = ⎩

Consumer’s location in [0,1] firm For a given position x, the gross consumer surplus of buying from B is higher (bx>ax) so willing to pay a higher price. Notice that as x increases the consumer has a higher valuation for both goods ∂U/∂ x>0

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Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

We solve the game backwards. 2nd stage: Suppose there is an indifferent consumer between A and B : Consumers to the right of buy from B and to the left buy from A:

ˆ x

ˆ ˆ

ˆ ˆ ( ) ( ) ˆ

x A B x B A

U A ax p bx p U B p p x b a = − = − = − ⇔ = −

ˆ x

If prices are equal all consumers buy from B (i.e. the indifferent consumer is located at zero)

Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

Therefore demand for A is and for B is ˆ

x ˆ 1 x −

Ux(A) Ux(B) x Ux(A) Ux(B)

• pA
• pB

1

ˆ x

Demand for B a b Because b>a (b is higher quality), we must have PB>PA. Demand for A z If we require U≥0 for the consumer to buy, then the demand for A is only [z, ] that is -z , where z=pA/a

ˆ x

ˆ x

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Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

Note that if pA>pB 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.

Ux(A) Ux(B)

• pB
• pA

Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

Let’s suppose c=0, the problem of firm A is:

( )

ˆ , , , 1 FOC: 2 for 2

A

B A A A B A A p A B A A A B B A A

p p Max a b p p p x p b a p p p p b a b a p p p p b a b a b a π π − ⎛ ⎞ = = ⎜ ⎟ − ⎝ ⎠ ∂ − = ⇔ − = ∂ − − ⇔ − = ⇔ = ≠ − −

Firm A’s reaction function

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Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

El problema de la empresa B es:

( )

ˆ , , , (1 ) 1 1 FOC: 1 2 2 1 1 for 2

B

B A B A B B B p B B A B B B B A A A B

p p Max a b p p p x p b a p p p p b a b a p p b a p p b a b a b a b a b a p p b a π π − ⎛ ⎞ = − = − ⎜ ⎟ − ⎝ ⎠ ∂ − = ⇔ − − = ∂ − − − + ⇔ − + = ⇔ = − − − − − + ⇔ = ≠

Firm B’s reaction function

Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

The equilibrium is the solution of the system:

2 2 2 2 3 2( ) >0 and 4 2 3 3

B A B A B B B A

p p p b a b a p p b a b a b a p p p MC ⎧ = ⎪ ⎪ ⎨ − + ⎪ − + = = ⎪ ⎩ − − − ⇔ = ⇔ = = > =

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.

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Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

Demands in the 2nd stage are:

( )

( , ) ( , ) 1 ˆ ( , ) 3 ( , ) ( , ) 2 ˆ ( , ) 1 1 3

B A A B A B

p a b p a b D a b x b a p a b p a b D a b x b a − ⎧ ⎛ ⎞ = = = ⎜ ⎟ ⎪ − ⎪ ⎝ ⎠ ⇔ ⎨ − ⎛ ⎞ ⎪ = − = − = ⎜ ⎟ ⎪ − ⎝ ⎠ ⎩

Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

Profits are:

( ) ( ) ( )

1 ˆ ( , ) 3 3 2( ) ˆ ( , ) 1 1 3 1 2( ) ( , ) 3 3 3 9 2 4( ) ( , ) ( ) ( , ) 3 9

B A A A B A B A B B A B B A A

b a p p a b p x p p b a b a p p a b p x b a b a b a b a a b b a a b b a p p a b π π π π π − − ⎧ ⎛ ⎞ = = = − ⎜ ⎟ ⎪ − ⎪ ⎝ ⎠ ⇔ ⎨ − − ⎛ ⎞ ⎪ = − = − ⎜ ⎟ ⎪ − ⎝ ⎠ ⎩ − − − ⎧ ⎛ ⎞ = − = ⎜ ⎟ ⎪ ⎪ ⎝ ⎠ ⎨ − ⎪ = − − + = > ⎪ ⎩

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Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

1st stage:

* *

( , ) 9 1 FOC: 9 4( ) ( , ) 9 4 FOC: 1 9

A a A B b B

b a Max a b a a b a Max a b b b π π π π − ⎧ = ⎪ ⎪ ⎨ ∂ ⎪ = − < ⇒ = ⎪ ∂ ⎩ − ⎧ = ⎪ ⎪ ⎨ ∂ ⎪ = > ⇒ = ⎪ ∂ ⎩

Principle of maximum differentiation. Intuition: With vertical differentiation, firms specialize in a given quality niche (high valuation consumers and low valuation consumers). The higher is the difference in quality the higher is the market power of each firm

Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

* * * * * * *

Because 0, =1, 1 4 , 9 9 1 2 , 3 3 1 ˆ 3

A B A B

a b p p x π π = = = = = =

Conclusion: Firms look for maximum differentiation from their rivals. Although qualities here have the same cost (c=0), still firm A prefers to produce an inferior good in order to differentiate from the rival. If firms choose to locate sequentially, then the first one to enter would select b=1 where profits are higher.

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Industrial Organization- Matilde Machado

Vertical Differentiation

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4.4. Vertical Differentiation

If consumers may choose not to buy

Ux(A) Ux(B) x Ux(A) Ux(B)

• pA
• pB

1

ˆ x

Demand for B a b Consumers between 0 and z=pA/a would be better off if they do not buy. The indifferent consumer is the same as before. Demand for A z

ˆ x

( , ) ( , ) 1 1

B A A A A B B A B A B

p p p D p p x z b a a p p D p p x b a − = − = − − − = − = − −