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Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation - Solutions
1. Consider a market where there are two differentiated goods. The demand for good 1 is
given by q1 = a − bp1 + δ p2 and the demand for good 2 is given by q2 = a − bp2 + δ p1 q2
where a > 0 b > δ > 0 . The production cost of each good is zero. Suppose that both
goods are produced by the same firm (a monopolist). Compute the prices set by the
monopolist. Suppose now that each good is produced by a different firm and the firms
choose prices simultaneously. Compute the Bertrand-Nash equilibrium prices and
confirm that they are lower than the monopoly prices. Now assume that each good is
produced by a different firm but the firms set prices sequentially; in particular, firm 2 can
observe the price set by firm 1 before setting its own price. Compute the subgame-perfect
equilibrium price of firm 1 in this two-stage game.
Solution:
Monopoly
π = p1q1 + p2 q2 = p1 (α − β p1 + δ p2 ) + p2 ( a − bp2 + δ p1 )
Thus:
∂π
= a − 2bp1 + 2δ p2 = 0
∂ p1
⇒
p1 =
1
( a + 2δ p2 )
2b
∂π
= a − 2bp2 + 2δ p1 = 0
∂ p2
⇒
p2 =
1
( a + 2δ p1 )
2b
Thus:
1
Industrial Organisation (ES30044)
p1 =
Seminar Three: Horizontal Product Differentiation
⎡1
⎤⎫
1 ⎧
⎨ a + 2δ ⎢ ( a + 2δ p1 ) ⎥ ⎬
2b ⎩
⎣ 2b
⎦⎭
⇒
2bp1 = a +
δ
( a + 2δ p1 )
b
⇒
2b2 p1 = ab + δ ( a + 2δ p1 )
⇒
2b2 p1 = ab + aδ + 2δ 2 p1
⇒
p1 =
a (b + δ )
(
2 b −δ
2
2
=
a (b + δ )
) 2 ( b + δ )( b − δ )
=
a
2(b − δ )
Thus:
p2 =
⎡
⎤ ⎫⎪
1 ⎧⎪
a
a
+
2
δ
⎢
⎥⎬
⎨
2b ⎪
⎢⎣ 2 ( b − δ ) ⎥⎦ ⎭⎪
⎩
⇒
⎡ a ⎤
2bp2 = a + δ ⎢
⎥
⎢⎣ ( b − δ ) ⎥⎦
⇒
2b ( b − δ ) p2 = a ( b − δ ) + aδ
⇒
(
)
2 b2 − bδ p2 = ab − aδ + aδ
⇒
p2 =
(
ab
2 b − bδ
2
)
=
a
2(b − δ )
Thus:
p1 = p2 = p m =
a
2(b − δ )
Bertrand
Max π 1 = p1 ( a − bp1 + δ p2 )
p1
2
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
∂π 1
= a − 2bp1 + δ p2 = 0
∂ p1
⇒
p1 =
1
( a + δ p2 )
2b
Max π 2 = p2 ( a − bp2 + δ p1 )
p2
∂π 2
= a − 2bp2 + δ p1 = 0
∂ p2
⇒
p2 =
1
( a + δ p1 )
2b
Thus:
p1 =
⎡1
⎤⎫
1 ⎧
⎨ a + δ ⎢ ( a − δ p1 ) ⎥ ⎬
2b ⎩
⎣ 2b
⎦⎭
⇒
⎡1
⎤
2bp1 = a + δ ⎢ ( a − δ p1 ) ⎥
⎣ 2b
⎦
⇒
4b2 p1 = 2ab + aδ − δ 2 p1
⇒
( 4b
2
⇒
p1 =
)
+ δ 2 p1 = a ( 2b + δ )
a ( 2b + δ )
( 4b
2
+δ
2
=
a ( 2b + δ )
) ( 2b + δ )( 2b − δ )
=
a
2b − δ
And:
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Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
⎛ a ⎞⎤
1 ⎡
⎢a + δ ⎜
⎥
2b ⎣
⎝ 2b − δ ⎟⎠ ⎦
p2 =
⇒
2b ( 2b − δ ) p2 = a ( 2b − δ ) + aδ
⇒
( 4b
2
)
− 2bδ p2 = 2ab − aδ + aδ
⇒
p2 =
2ab
ab
a
= 2
=
4b − 2bδ 2b − bδ 2b − δ
2
Thus
p1 = p2 = p b =
a
2b − δ
Note that p m > p b if δ > 0 vis:
pm =
a
a
a
=
>
= pb
2b
−
2
δ
2b
−
δ
2(b − δ )
Stackelberg
Max π 2 = p1 ( a − bp2 + δ p1 )
p2
∂π 2
= a − 2bp2 + δ p1 = 0
∂ p2
⇒
p2 =
1
( a + δ p1 )
2b
Max π 1 = p1 ( a − bp1 + δ p2 )
p1
st
p2 =
1
( a + δ p1 )
2b
4
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
∂π 1
δ 2 p1
= a − 2bp1 + δ p2 +
=0
∂ p1
2b
⇒
a2b − 2b2 p1 + 2bδ p2 + δ 2 p1 = 0
⇒
(
)
p1 4b2 − δ 2 = 2ab + 2bδ p2
⇒
⎡1
⎤
p1 4b2 − δ 2 = 2ab + 2bδ ⎢ ( a + δ p1 ) ⎥
⎣ 2b
⎦
⇒
(
)
(
)
p1 4b2 − δ 2 = 2ab + δ ( a + dp1 )
⇒
(
)
p1 4b2 − δ 2 = 2ab + aδ + δ 2 p1
⇒
(
)
p1 4b2 − 2δ 2 = a ( 2b + δ )
⇒
p1 =
a ( 2b + δ )
( 4b
2
− 2δ 2
a ( 2b + δ )
=
) 2 ( 2b
2
−δ 2
)
And:
⎧
⎡ a ( 2b + δ )
1 ⎪
p2 =
⎨a + δ ⎢
2b ⎪
⎢ 2 2b2 − δ 2
⎣
⎩
⇒
(
⎡ a ( 2b + δ )
2bp2 = a + δ ⎢
⎢ 2 2b2 − δ 2
⎣
⇒
(
(
)
(
)
)
⎤ ⎫⎪
⎥⎬
⎥⎪
⎦⎭
⎤
⎥
⎥
⎦
)
4b 2b2 − δ 2 p2 = 2 2b2 − δ 2 a + aδ ( 2b + δ )
⇒
(8b − 4bδ ) p
3
2
2
= 4ab2 − 2aδ 2 + 2abδ + aδ 2
⇒
p2 =
4ab2 − aδ 2 + 2abδ
(8b − 4bδ )
3
2
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Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
2. Suppose that there are only two firms (Firm 1 and Firm 2) selling coffee. Let α i denote
the advertising level of firm i, i = 1, 2. Assume that the profits of the two firms are
affected by advertising as follows:
π 1 (α 1 , α 2 ) = 4α 1 + 3α 1α 2 − α 1α 1
π 2 (α 1 , α 2 ) = 2α 2 + α 1α 2 − α 2α 2
(a) Calculate and draw the best-response function of each firm – i.e. for any given level
of advertising of firm j, find the profit maximising advertising level of firm i;
(b) Infer whether the strategies are strategic complements or strategic substitutes;
(c) Find the Nash equilibrium advertising levels and calculate the firms’ Nash
equilibrium level of profits.
Solution:
max π 1 (α 1 , α 2 ) = 4α 1 + 3α 1α 2 − α 1α 1
α1
⇒
∂π 1
(α1,α 2 ) = 4 + 3α 2 − 2α1∗ = 0
∂α 1
⇒
3
α 1∗ = 2 + α 2
2
And:
max π 2 (α 1 , α 2 ) = 2α 2 + α 1α 2 − α 2α 2
α2
⇒
∂π 1
(α1,α 2 ) = 2 + α1 − 2α 2∗ = 0
∂α 2
⇒
1
α 2∗ = 1+ α 1
2
Thus:
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Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
α2
3
R1 : α1∗ = 2 + α 2
2
1
R2 : α 2∗ = 1+ α1
2
αb
8
1
2
0
2
α
14
-4/3
Figure 1: Advertising Best-Response Functions
(b) Infer whether the strategies are strategic complements or strategic substitutes;
The two best-response functions are upward sloping and are hence strategic
complements.
(c) Find the Nash equilibrium advertising levels and calculate the firms’ Nash
equilibrium level of profits.
3
α 1b = 2 + α 2b
2
⇒
3⎛ 1 ⎞
α 1b = 2 + ⎜ 1+ α 1b ⎟
2⎝ 2 ⎠
⇒
α 1b = 2 +
3 3 b
+ α1
2 4
⇒
1 b 7
α1 =
4
2
⇒
28
α 1b =
= 14
2
7
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
And:
1
α 2b = 1+ α 1b
2
⇒
1
α 2b = 1+ (14 )
2
⇒
α 2b = 8
Thus:
π 1b (α 1b , α 2b ) = 4α 1b + 3α 1bα 2b − α 1bα 2b
⇒
π 1b (14,8 ) = 4 ⋅14 + 3⋅14 ⋅ 8 − 14 2
⇒
π 1b (14,8 ) = 14 2
And:
π 2b (α 1b , α 2b ) = 2α 2b + α 1bα 2b − α 2bα 2b
⇒
π 1b (14,8 ) = 2 ⋅ 8 + 14 ⋅ 8 − 8 2
⇒
π 1b (14,8 ) = 8 2
⇒
π 1b (14,8 ) = 14 2
3. Pultney Bridge is best described by the interval [0,1]. Two fast-food restaurants serving
identical food are located at the edges of the road, so that Restaurant 1 is located at the
most left-hand side and Restaurant 2 is located at he most right hand side. Consumers are
uniformly distributed on the unit interval [0,1], where at each point on the interval lives
one consumer. Each consumer buys one meal from the restaurant in which price plus the
transportation cost is the lowest. Along Pultney Bridge, the wind blows from right to left
such that the transportation cost for a consumer who travels to the right (left) is $R ($1)
per unit of distance:
(a) Let pi denote the price of a meal at restaurant i, i = 1, 2. Assume that p1 and p2 are
given and satisfy:
0 < p1 − R < p2 < 1+ p1
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Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
Denote by x̂ the location of a consumer who is indifferent as to whether he eats at
Restaurant 1 or Restaurant 2 and calculate x̂ as a function of p1 , p2 and R;
(b) Suppose that the given prices satisfy p1 = p2 . What is the minimal value of the
parameter R such that all consumers will only eat at Restaurant 1?
Solution:
(a) Let pi denote the price of a meal at restaurant i, i = 1, 2. Assume that p1 and p2 are
given and satisfy:
0 < p1 − R < p2 < 1+ p1
Denote by x̂ the location of a consumer who is indifferent as to whether he eats at
Restaurant 1 or Restaurant 2 and calculate x̂ as a function of p1 , p2 and R;
The indifferent consumer, x̂ , is defined implicitly by:
x̂ ∗1+ p1 = (1− x̂ ) ∗ R + p2
⇒
x̂ =
R + ( p2 − p1 )
1+ R
(b) Suppose that the given prices satisfy p1 = p2 . What is the minimal value of the
parameter R such that all consumers will only eat at Restaurant 1?
Substituting p1 = p2 into x̂ yields:
x̂ =
R
1+ R
Thus:
1 ⎞
⎛ R ⎞
⎛ 1+ R − 1 ⎞
⎛
lim x̂ = lim ⎜
= lim ⎜
= lim ⎜ 1−
⎟
⎟
⎟ =1
R→∞
R→∞ ⎝ 1+ R ⎠
R→∞ ⎝ 1+ R ⎠
R→∞ ⎝
1+ R ⎠
Therefore, all consumers will eat in Restaurant 1 (i.e. x! = 1 ) only if the transportation cost for
travelling to the east is infinite. Otherwise, some consumers will always eat at Restaurant 2.
4. Two firms (Firm 1 and Firm 2) produce differentiated products and compete over prices.
Their respective demand curves are given by:
q1 =
1
(120 − 4 p1 + 2 p2 )
2
q2 =
1
(120 − 2 p2 + p1 )
4
9
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
Both firms have constant average costs of £10. Calculate the Nash equilibrium (prices,
quantities and profits), illustrate it in a diagram, and explain your results
Solution:
⎡1
⎤
π 1 = ( p1 − c ) q1 = ( p1 − 10 ) ⎢ (120 − 4 p1 + 2 p2 ) ⎥ = 80 p1 − 2 p12 + p1 p2 − 10 p2 − 600
⎣2
⎦
⎡1
⎤
1
π 2 = ( p2 − c ) q2 = ( p2 − 10 ) ⎢ (120 − 2 p2 + p1 ) ⎥ = 70 p2 − p22 + p1 p2 − 5p1 − 600
2
⎣4
⎦
Thus:
∂π 1
1
= 80 − 4 p1 + p2 = 0 ⇒ p1∗ = 20 + p2
∂ p1
4
∂π 2
1
1
= 70 − 2 p2 + p1 = 0 ⇒ p2∗ = 35 + p1
∂ p2
2
4
Thus:
⎛
1⎛
1 ⎞
1 ⎞
15
460 92
p1∗ = 20 + ⎜ 35 + p1∗ ⎟ ⇒ 4 p1∗ = 80 + ⎜ 35 + p1∗ ⎟ ⇒ p1∗ = 115 ⇒ p1∗ =
=
4⎝
4 ⎠
4 ⎠
4
15
3
⎝
1 92 420 + 92 512 128 2
p2∗ = 35 + ⋅ =
=
=
42
4 3
12
12
3
3
Check:
1 128 240 + 128 368 92
2
p1∗ = 20 + ⋅
=
=
=
= 30
4 3
12
12
3
3
Thus:
2 92
p1n = 30 =
3 3
2 128
p2n = 42 =
3
3
Thus:
10
Industrial Organisation (ES30044)
q1 =
⇒
Seminar Three: Horizontal Product Differentiation
1
⎡⎣120 − 4 ( 92 3 ) + 2 (128 3 ) ⎤⎦
2
1 ⎛ 3⋅120 − 4 ⋅ 92 + 2 ⋅128 ⎞
q1 = ⎜
⎟⎠
2⎝
3
⇒
1 ⎛ 360 − 368 + 256 ⎞
q1 = ⎜
⎟⎠
2⎝
3
⇒
q1 =
248 124
=
6
3
And:
q2 =
1
⎡120 − 2 (128 3 ) + 92 3 ⎤
⎦
4⎣
⇒
1 ⎛ 3⋅120 − 2 ⋅128 + 92 ⎞
q2 = ⎜
⎟⎠
4⎝
3
⇒
1 ⎛ 360 − 256 + 92 ⎞
q2 = ⎜
⎟⎠
4⎝
3
⇒
q2 =
196 49
=
12
3
Thus:
π 1 = ( p1 − c ) q1
⇒
⎛ 92
⎞ 124
π 1 = ⎜ − 10⎟
⎝ 3
⎠ 3
⇒
π1 =
62 124 7688
⋅
=
3 3
9
And:
11
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
π 2 = ( p2 − c ) q2
⇒
⎛ 128
⎞ 49
π2 = ⎜
− 10⎟
⎝ 3
⎠ 3
⇒
π2 =
98 49 4802
⋅ =
3 3
9
p2
R1 : p1∗ = 20 +
1
p
4 2
R2 : p2∗ = 35 +
1
p
4 1
pb
128/3
35
-140
0
20
p1
92/3
-80
Figure 2
Note: If firms’ reaction functions are upward (downward) sloping, then we say that their
strategies are strategic complements (substitutes) for one another.
5. Consider a market in which there are two similar, but not identical, goods with inverse
demand pi = α − β qi − γ q j≠i , where i, j = 1,2 , α > 0 and β > γ > 0 . The cost of producing
good i is given by Ci = cqi , i = 1,2 .
(a) Assume that the goods are produced by two different firms (e.g. Firm 1 and Firm 2)
who set their outputs simultaneously. Compute the Nash equilibrium in quantities.
(b) Assume now that the goods are produced by a single monopolist. Compute and
comment upon the outputs set by the monopolist. How do these compare to those set
by the competing firms in (a) above?
(c) How would your answer to (a) change if outputs were set simultaneously, but Firm 1
had the option of credibly committing to a particular output. Would Firm 1 always
exercise such an option? Why or why not?
12
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
(d) Using your analysis in (a)-(c) above, in what sense is it true to assert that product
differentiation alleviates competition?
Solution:
(a) Assume that the goods are produced by two different firms (e.g. Firm 1 and Firm 2) who
set their prices simultaneously. Compute the Nash equilibrium in quantities.
Firm 1 chooses q1 to maximise:
π 1 = (α − β q1 − γ q2 ) q1 − cq1
taking q2 as given, while Firm 2 chooses q2 to to maximise:
π 2 = (α − β q2 − γ q1 ) q2 − cq2
taking q1 as given.
Thus:
∂π 1
= α − c − 2β q1* − γ q2 = 0
∂q1
⇒
q1* =
1⎡ ⎛γ ⎞ ⎤
q ⎥
⎢θ −
2 ⎣ ⎜⎝ β ⎟⎠ 2 ⎦
And:
∂π 2
= α − c − 2β q2* − γ q1 = 0
∂q2
⇒
q2* =
1⎡ ⎛γ ⎞ ⎤
q⎥
⎢θ −
2 ⎣ ⎜⎝ β ⎟⎠ 1 ⎦
where θ = (α − c ) β . Thus:
13
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
⎛γ ⎞ 1⎡ ⎛γ ⎞ ⎤
2q1* = θ − ⎜ ⎟ ⎢θ − ⎜ ⎟ q1* ⎥
⎝β⎠ 2⎣ ⎝β⎠ ⎦
⇒
⎛
γ γ⎞
γ
q1* ⎜ 2 −
⋅ ⎟ =θ −
θ
2β β ⎠
2β
⎝
⇒
⎛
γ2 ⎞ ⎛
γ ⎞
q1* ⎜ 2 − 2 ⎟ = ⎜ 1−
θ
2β ⎠ ⎝ 2β ⎟⎠
⎝
⇒
⎛ 4β 2 − γ 2 ⎞ ⎛ 2β − γ
q1* ⎜
2
⎟⎠ = ⎜⎝ 2β
⎝ 2β
⎞
⎟⎠ θ
⇒
⎡
⎤ ⎛ 2β − γ
2β 2
q =⎢
⎥⎜
⎢⎣ ( 2β + γ ) ( 2β − γ ) ⎥⎦ ⎝ 2β
⇒
α −c
q1n =
2β + γ
n
1
⎛ β
⎞
⎟⎠ θ = ⎜⎝ 2β + γ
⎞
β
α −c
⎟⎠ θ = 2β + γ ⋅ β
And so:
q2n =
α −c
2β + γ
(b) Assume now that the goods are produced by a single monopolist. Compute and comment
upon the outputs set by the monopolist. How do these compare to those set by the
competing firms in (a) above?
The monopoly will choose q1 and q2 to maximise:
π = p1q1 + p2 q2 − c ( q1 + q2 )
⇒
π = (α − β q1 − γ q2 ) q1 + (α − β q2 − γ q1 ) q2 − c ( q1 + q2 )
⇒
(
)
π = (α − c ) ( q1 + q2 ) − β q12 + q22 − 2γ q1q2
Thus:
∂π
= (α − c ) − 2β q1m − 2γ q2 = 0
∂q1
14
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
And:
∂π
= (α − c ) − 2β q2m − 2γ q1 = 0
∂q2
Thus:
q1m =
θ γ m
− q
2 β 2
And:
q2m =
θ γ m
− q
2 β 1
Thus:
q1m =
θ γ m θ γ ⎛ θ γ m⎞
− q = −
− q
2 β 2 2 β ⎜⎝ 2 β 1 ⎟⎠
⇒
q1m =
θ⎛ γ ⎞ γ2 m
1−
+ q
2 ⎜⎝ β ⎟⎠ β 2 1
⇒
⎛ γ2⎞ ⎛ γ ⎞θ
q1m ⎜ 1− 2 ⎟ = ⎜ 1− ⎟
⎝ β ⎠ ⎝ β⎠ 2
⇒
⎛ β2 −γ 2 ⎞ ⎛ β −γ ⎞ θ
q1m ⎜
2
⎟⎠ = ⎜⎝ β ⎟⎠ 2
⎝ β
⇒
q1m =
⎛ β −γ ⎞ θ ⎛ β ⎞ θ
β2
=
( β + γ )( β − γ ) ⎜⎝ β ⎟⎠ 2 ⎜⎝ β + γ ⎟⎠ 2
⇒
⎛ β
q1m = ⎜
⎝ β +γ
⎞ ⎛ α − c⎞
⎟⎠ ⎜⎝ 2β ⎟⎠
⇒
q1m =
α −c
2(β + γ )
And:
15
Industrial Organisation (ES30044)
q2m =
Seminar Three: Horizontal Product Differentiation
θ γ m
− q
2 β 1
⇒
q2m =
θ γ ⎛ β ⎞θ
−
2 β ⎜⎝ β + γ ⎟⎠ 2
⇒
⎡ ⎛ γ
q2m = ⎢1− ⎜
⎣ ⎝ β +γ
⇒
⎞ ⎤θ
⎟⎠ ⎥ 2
⎦
⎛ β +γ −γ ⎞ θ
q2m = ⎜
⎝ β + γ ⎟⎠ 2
⇒
⎛ β ⎞θ
q2m = ⎜
⎝ β + γ ⎟⎠ 2
⇒
⎛ β ⎞ ⎛ α − c⎞
q2m = ⎜
⎝ β + γ ⎟⎠ ⎜⎝ 2β ⎟⎠
⇒
q2m =
α −c
2(β + γ )
By symmetry
(c) How would your answer to (a) change if outputs were set simultaneously, but Firm 1 had
the option of credibly committing to a particular output?
We are essentially assuming now that Firm 1 is the Stackelberg market leader and makes it’s
decision over output before Firm 2. Thus:
π 1 ( q1 ,q2 ) = p1q1 = α q1 − β q12 − γ q1q2 − cq1
π 2 ( q1 ,q2 ) = p2 q2 = α q2 − β q22 − γ q1q2 − cq2
As before, we solve the model recursively vis:
∂π 2 ( q1;q2 )
∂ q2
= α − 2β q2∗ − γ q1 − c = 0
⇒
1⎛
γ ⎞
q2∗ = ⎜ θ − q1 ⎟
2⎝
β ⎠
16
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
Firm 1’s problem is thus:
max π 1 ( q1 ,q2 ) = p1q1 = α q1 − β q12 − γ q1q2 − cq1
q1
s.t.
1⎛
γ ⎞
q2 = q2∗ = ⎜ θ − q1 ⎟
2⎝
β ⎠
Thus:
dπ 1 ( q1;q2 )
dq1
=
∂π 1 ( q1;q2 )
∂q1
+
∂π 1 ( q1;q2 ) dq2
⋅
=0
∂q2
dq1
⇒
(49)
(α − 2β q
∗
1
⎛γ ⎞
− γ q2 − c + ⎜ ⎟ q1∗ = 0
⎝ 2β ⎠
)
2
Thus:
⎛
γ2⎞ ∗
∗
⎜⎝ 2β − 2β ⎟⎠ q1 = (α − c ) − γ q2
⇒
⎛ 4β 2 − γ 2 ⎞ s
γ⎛
γ s⎞
⎜⎝ 2β ⎟⎠ q1 = (α − c ) − 2 ⎜⎝ θ − β q1 ⎟⎠
⇒
(50)
⎡
γ ⎤
2β ⎢(α − c ) − θ ⎥
2 ⎦
⎣
q1s =
2
2
4β − 2γ
⇒
q1s =
(α − c )( 2β − γ )
4β 2 − 2γ 2
Note that:
lim q1∗ =
γ →β
α −c
2β
(51)
which equates to the Stackelberg leader’s equilibrium level of output when output is
homogenous. Now:
17
Industrial Organisation (ES30044)
Seminar Three: Horizontal Product Differentiation
1⎛
γ ⎞
q2s = ⎜ θ − q1 ⎟
2⎝
β ⎠
⇒
q2s =
⇒
q2s =
1 ⎧⎪
γ ⎡ (α − c ) ( 2β − γ ) ⎤ ⎫⎪
θ
−
⎢
⎥⎬
⎨
2⎪
β ⎢⎣ 4β 2 − 2γ 2 ⎥⎦ ⎪
⎩
⎭
1 ⎧⎪⎛ α − c ⎞ γ ⎡ (α − c ) ( 2β − γ ) ⎤ ⎫⎪
− ⎢
⎥⎬
⎨
2 ⎪⎜⎝ β ⎟⎠ β ⎢⎣ 4β 2 − 2γ 2 ⎥⎦ ⎪
⎩
⎭
⇒
(
)
⎡ (α − c ) 4β 3 − 2βγ 2 − (α − c ) ( 2β − γ ) βγ
q =⎢
8β 4 − 4β 2γ 2
⎢⎣
⇒
s
2
s
2
q
⇒
s
2
q
⇒
s
2
q
(α − c )( 4β
=
(α − c )( 4β
=
3
− 2βγ 2 − 2β 2γ + βγ 2
8β − 4β γ
4
3
2
8β − 4β γ
(α − c )( 4β
=
2
2
)
2
− 2βγ − γ 2
8β − 4βγ
3
)
2
− 2β 2γ − βγ 2
4
⎤
⎥
⎥⎦
)
2
Note that:
lim q2s =
γ →β
α −c
4β
which equates the Stackelberg follower’s equilibrium level of output when output is
homogenous.
It can be shown that the Stackelberg leader (follower) in outputs makes more (less)
profit than it does in Cournot competition such that the leader would always exert the option
to commit to a certain level of output. Indeed, it follows that the leader can do no worse than
it could do under Cournot competition because it always has the option of committing to the
Cournot equilibrium level of output.
(d) Using your analysis in (a)-(c) above, in what sense is it true to assert that product
differentiation alleviates competition?
Product differentiation alleviates competition because firms become monoplies in their own
brands.
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