Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
Industrial Organisation (ES30044)
Topic Four:
Horizontal Product Differentiation (ii)
Outline:
Introduction
The Linear Approach – Hotelling (1929)
Discussion
Salop’s Circular City
1.
I.
II.
III.
IV.
Introduction
While the Nash models of differentiation yield some insights, they do not explain how
differentiation arises. The extent to which one firm’s product is different from another’s is
partly a matter of choice. A common way of analyzing differentiation is to think of the
properties of products as being located in a space where each dimension represents a
characteristic of the product. Firms producing homogeneous goods are then seen as located at
the same point in the space. Products that are far from each other are highly differentiated.
This device means that the theory of industrial location and the theory of differentiation are
the same, and that they run into the same problems.
In what follows we therefore consider two interpretations of ‘location’. First, location
can mean the physical location of a particular consumer, in which case the consumer
observes the prices charges by all seller and then chooses to purchase from the seller at which
the price plus the transportation cost is minimised. Second, location may be interpreted as the
distance between the brand characteristic that particular consumer views as ideal and the
characteristics of the brand actually purchased. That is, we might view a space (e.g. a line
interval) as measuring the degree of sweetness of a candy bar. Consumers located towards
the left are those who prefer high-sugar bars, and those located towards the right prefer lowsugar bars. In this case, the distance between a consumer and a seller can measure the
consumer’s disutility from buying a less-than-ideal brand. This disutility is equivalent to the
transportation cost in the previous interpretation.
2.
The Linear Approach
Hotelling (1929) consider consumers who reside on a linear street with a length L > 0.
Assume consumers are uniformly distributed on this interval, so that at each point lays a
single consumer. Hence, the total number of consumers in the economy is L. Each consumer
is indexed by x ∈ [0, L], so x is just a name of a consumer (located at point x from the origin).
We first take the locations of the sellers as given (afterwards, we will determine locations
endogenously) and assume that firms compete in prices.
Price Game with Fixed Locations
Suppose there are two firms selling a product that is identical in all respects except one
characteristic vis. the location where it is sold. Figure 1 shows that Firm A is located a units
of distance from point 0. Firm B is located to the right of Firm A, b units of distance from
point L. Assume for simplicity that production is costless and that each consumer buys one
unit of the product. To go to a store (i.e. a firm) a consumer has to incur transportation costs
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
of t per unit of distance.1 Thus, a consumer located at some point x has to pay transportation
costs of t x − a for shopping at Firm A, or t x − ( L − b ) for shopping at Firm B.2
Define the utility function of a consumer located at point x as:
⎧⎪− p − t x − a
Ux ≡ ⎨ A
⎪⎩ − pB − t L − b − x
a
0
if he buys from A
if he buys from B
A
(1)
B
x̂
a
L-b
b
L
Figure 1
We define the indifferent consumer, x̂ , as the consumer who derives equal utility from
purchasing the good from Firm A or Firm B. Formally, if a < xˆ < L − b then it must be the
case that:
− p A − t ( xˆ − a ) = − pB − t ( L − b − xˆ )
⇒
(2)
1 ⎛ p − pA
⎞
xˆ = ⎜ B
+ L −b+ a⎟
2⎝
t
⎠
Note that (2) is the demand function faced by Firm A. The demand function faced by firm B
is:
1
See Appendix A1 for an analysis based on non-linear (i.e. quadratic) transport costs.
Note that distance here can be interpreted as the divergence between the actual and desired characteristics of a
particular product. For example, the individual located at x̂ desires x̂ degree of sweetness from a bar of
2
chocolate, and buying either chocolate bar A or B implies a utility loss of t x − a and t x − (L − b )
respectively.
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
1 ⎛ p − pB
⎞
L − xˆ = ⎜ A
+ L +b− a⎟
2⎝
t
⎠
(3)
We can envisage a number of scenarios. For example, if p A = pB and a = b , then each firm
will supply one-half of the market, with the ‘indifferent’ consumer being located at:
1 ⎛ p − pA
⎞ L
xˆ = ⎜ B
+ L −b + a⎟ =
2⎝
t
⎠ 2
(4)
That is, halfway along the line. This scenario is illustrated in Figure 2 following:
pA
pB
t
pA
0
A
a
x̂
B t
pB
L-b
L
Figure 2: Firms A and B Share the Market
If pA is set sufficiently below pB, then it may be the case that Firm A takes all the market – see
Figure 3 following. Another alternative is that Firm A takes only its ‘hinterland’ i.e. the part
of the line to the left of Firm A’s location, a. This requires pB to be less than pA – see Figure 4
following:
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
pA
pB
B
pB
A
pA
0
a
L-b
L
Figure 3: Firm A Takes All The Market
pA
pB
A
pA
B
0
a
L-b
Figure 4: Firm A Takes Only its Hinterland
4
pB
L
Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
More generally, given (2) we can look for a Nash-Bertrand equilibrium in price strategies.
Assuming for simplicity that production is costless, this requires Firm A taking pB as given
and choosing p A to solve:
⎤
1 ⎛ p − pA
1 ⎡ p p − pA2
⎞
ˆ A= ⎜ B
max π A = xp
+ L − b + a ⎟ pA = ⎢ B A
+ ( L − b + a ) pA ⎥
pA
2⎝
t
2⎣
t
⎠
⎦
(5)
The first-order condition is given by:3
⎞
∂π A 1 ⎛ pB − 2 p Ah
= ⎜
+ L −b + a⎟ = 0
∂p A 2 ⎝
t
⎠
⇒
1
p Ah = ⎡⎣ pB + t ( L − b + a )⎤⎦
2
(6)
Expression (6) denotes Firm A’s reaction function vis. the optimal price Thus, p Ah denotes
the Hoteling optimal price. Firm B takes p A as given and chooses pB to:
⎤
1 ⎛ p − pB
1 ⎡ p p − pB2
⎞
max π B = ⎜ A
+ L + b − a ⎟ pA = ⎢ A B
+ ( L + b − a ) pB ⎥
pB
2⎝
t
2⎣
t
⎠
⎦
(7)
The first-order condition is given by:
⎞
∂π B 1 ⎛ p A − 2 pBh
= ⎜
+ L +b − a⎟ = 0
∂pB 2 ⎝
t
⎠
⇒
1
pBh = ⎡⎣ p A + t ( L + b − a )⎤⎦
2
(8)
Of course, because of symmetry, the second-order condition is again satisfied such that in
equilibrium we have:
p Ah =
1 h
1 ⎧1
⎫
⎡⎣ pB + t ( L − b + a )⎤⎦ = ⎨ ⎡⎣ p Ah + t ( L + b − a )⎤⎦ + t ( L − b + a )⎬
2
2 ⎩2
⎭
⇒
p Ah =
(9)
t
( 3L − b + a )
3
And:
3
( )
Note that the second-order condition for a profit maximum is satisfies vis. ∂ 2 π B ∂ pB2 = − 1 t < 0
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Industrial Organisation (ES30044)
pBh =
Topic Four: Horizontal Product Differentiation
1 h
1 ⎡t
⎤
⎡⎣ p A + t ( L + b − a )⎤⎦ = ⎢ ( 3L − b + a ) + t ( L + b − a )⎥
2
2 ⎣3
⎦
⇒
pBh =
(10)
t
( 3L + b − a )
3
Note that if there is no differentiation (i.e. t = 0 ) then we return to Bertrand with both firms
setting rice equal to (marginal-average cost) zero. Note also the difference in the Hotelling
equilibrium prices:
Δp h = pAh − pBh =
t
t
2
(3L − b + a ) − (3L + b − a ) = t ( a − b )
3
3
3
(11)
Here, note that if there is no differentiation (i.e. t = 0 ) or if the firms are located at equal
distances from the end-points (i.e. a = b) then there is no differentiation in prices. Moreover,
note that:
∂Δp h 2t
= >0
∂a
3
(12)
∂Δp h
2t
=− <0
∂b
3
(13)
>
∂Δp h 2
= (a − b) 0
<
∂t
3
(14)
Intuitively, Firm A’s price will be relatively higher than Firm B’s price the further is Firm A
located along the line toward the end point L. And conversely, Firm A’s price will be
relatively lower the further is Firm B located away from the end point L. An increase in
transportation costs has an ambiguous effect on relative prices depending on the locations of
the two firms.
Equations (2) and (11) imply an equilibrium market share for Firm A of:
⎞ 1 ⎡ 2t
1 ⎛ pBh − p Ah
⎤
xˆ = ⎜
+ L − b + a ⎟ = ⎢ (b − a ) + L − b + a ⎥
2⎝
t
⎦
⎠ 2 ⎣ 3t
h
⇒
xˆ h =
(15)
1
( 3L − b + a )
6
Whilst equations (3) and (11) imply an equilibrium market share for Firm B of:
L − xˆ h = L −
1
1
(3L − b + a ) = (3L + b − a )
6
6
(16)
Note that if a = b (i.e. the firms are located the same distance from either end of the line) then
p Ah = pBh = tL and xˆ h = L − xˆ h = L 2 .
Firm A’s profits from (9) and (15) are given by:
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Industrial Organisation (ES30044)
π A = pAh x h =
Topic Four: Horizontal Product Differentiation
t
1
t
2
(3L − b + a ) ⋅ (3L − b + a ) = (3L − b + a )
3
6
18
(17)
Note that:
∂π A 1
2
= ( 3L − b + a ) > 0
∂t 18
(18)
That is, the profit of each brand-producing firm increases with the transportation cost
parameter, t. This follows from our previous analysis of Cournot and Bertrand oligopolies
with heterogeneous products in which we found that firms obtained higher profits when the
brands they produced became more differentiated. Indeed, as Hotelling argued:
These particular merchants would do well, instead of organising improvement clubs and
booster associations to better the roads, to make transportation as difficult as possible.
[Hotelling (1929), p. 50)].
Essentially, the higher is t then the more differentiated the good are from the
perspective of the consumers and, so, the higher is the market power of the firms
since it is more costly for a consumer to turn to the competition. This allows the firms
increase prices and profits.
3.
Discussion
It is apparent from (17) that:
∂π A t
= ( 3L − b + a ) > 0
∂a 9
(19)
In words, for any locations a and b, Firm A always has an incentive to locate closer to Firm B
in order to gain a higher market share. This inevitably brings it to the region where no pricing
equilibrium exists, unless it locates at the same point, in which case profits are zero. This
case, whereby firms move to the centre, is commonly referred to in the literature as the
principle of minimum differentiation, since by moving towards the centre the firms produce
less-differentiate products. As Hotelling argued:
Buyers are confronted everywhere with an excessive sameness. When a new merchant or
manufacturer sets up shop he must not produce something exactly like what is already on ht
market or he will risk a price war of the type discussed by Bertrand…. But there is an
incentive to make a new product very much like to old, applying some slight change which
will seem an improvement to as many buyers as possible without ever going far in this
direction … (a) tendency to make only slight deviations in order to have for the new
commodity as many buyers of the old as possible, to get … between one’s competition and a
mass of customers. [Hotelling (1929), pp. 481-482].
Introspection and casual empiricism would seem to confirm this: It is not unusual to observe
suppliers of similar products operating form nearby outlets. Indeed, Hotelling makes
plausible analogies between differentiation in location, politics, religion and product quality
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
and his work has had a fundamental influence on the way scholars in many different fields
think about endogenous differentiation.4
But, as we have seen, if Firm A moves too close to Firm B, then an equilibrium will
not exist, and if it moves to the same point as Firm B, then its profit will drop to zero,
implying that it is better off moving back to the left. Indeed, it can be shown that in the
Hotelling linear-city game, there is no equilibrium for the game in which firms use both
prices and locations as strategies.
Indeed, it is somewhat striking, therefore, that the result Hotelling claimed is false!
The details underpinning this failure are somewhat technical – and were not discovered until
50 years after Hotelling’s paper - see d’Aspremont et al. (1979) –testimony indeed to the
pioneering nature of Hotelling’s work.
In address / location models, sales quantities are varied by changing location, so that
firms can effectively shift their demand curves by location choice. This means that analyzing
price and quantity competition separately is not sensible, since both depend on the location of
the demand curve. We essentially analyze the situation as a two-stage game in which location
decisions are made in the first stage, and price decisions are made in the second. Our analysis
in Section 2 was based on the presumption that an equilibrium where firms charge positive
prices always exists. More generally, d’Aspremont et al. (1979) show that:
1.
If both firms are located at the same point (i.e. a + b = L , implying that the products
are homogenous), then pA = pB = 0 is a unique equilibrium in prices;
2.
If a + b < L , then a unique equilibrium in prices exists and is described by
equation (9) and (10) previously iff firms are ‘sufficiently distant’ from each
other with;
(i)
a − b ⎞ 4 L ( a + 2b )
⎛
a >b⇒⎜L+
⎟ ≥
3 ⎠
3
⎝
(ii)
b − a ⎞ 4 L ( a + 2b )
⎛
a <b⇒⎜L+
⎟ ≥
3 ⎠
3
⎝
2
2
These results mean that equilibrium in the second stage pricing game exists if firms are
located at the same place, or if they are separated by some minimum distance. Indeed,
d’Aspremont et al. (1979) show that there are, in fact four different regimes:
a. Firms that are located in the same place are in a Bertrand equilibrium with
zero profits;
b. Firms that are ‘too close’ can never establish an equilibrium because they
continuously undercut each other;
c. More differentiated firms establish an equilibrium in which they share the
market;
4
‘The competition for votes between the Republican and Democrat parties does not lead to a clear drawing of
issues, an adoption of two strongly contrasting positions between which the voter may chose. Instead, each
party strives to make its platform as much like the other’s as possible.’ [Hotelling (1929), p. 482]. And:
‘Methodists and Presbyterian churches are too much alike; cider is too homogenous.’ [Hotelling (1929), p. 484].
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
d. And finally, if locations are so far apart that there exists a consumer who buys
from neither firm, then the firms operate as local monopolies.
5.
Salop’s Circular City Model5
An alternative to Hotelling’s linear model is developed by Salop (1979) who assumes a
circular city of circumference 1. As with Hotelling, this location model can be given an
interpretation of describing differentiated products that differ from the physical-location
interpretation. Consider, for example, airline, bus and train firms that can provide a roundthe-clock service. If we treat the circle as twenty-four hours, each brand can be interpreted as
the time when such a firm schedules as departure.
Firms
The model does not explicitly model how firms choose where to locate. It assumes, instead, a
monopolistic-competition market structure in which the number of firms, N, is endogenously
determined. All of the (potentially infinite) firms have the same technology. Let F denote the
fixed cost of production, c the marginal cost of production, and qi and π i ( qi ) the output and
profit level of the firm producing brand i. We assume:
⎧( pi − c ) qi − F if
−F
if
⎩
π i ( qi ) ⎨
qi > 0
qi = 0
(36)
Consumers
Consumers are uniformly distributed on the unit circle. We again denote by t the consumer’s
transportation cost per unit of distance, and we assume that each consumer purchase one unit
of he brand that minimises the sum of the price and transportation cost.
Assuming that each of the N firms is located at an equal distance from one another
implies that the distance between any two firms is 1 N . Consider Firm 1 setting p1 in Figure
5. Firm 1 faces Firm 2 to its right and Firm N to its left, and can attract potential customers of
each by setting its price accordingly. If we assume that Firms 2 and N set a uniform price, p,
then the consumer who is indifferent between buying from Firms 1 and 2 is located at point
x̂ on the circle, where x̂ is defined implicitly through:
⎛1
⎞
p1 + txˆ = p + t ⎜ − xˆ ⎟
⎝N
⎠
⇒
xˆ =
(37)
p − p1
1
+
2t
2N
And since Firm 1 has potential customers on its left and right, its demand function is given
by:
q1 ( p1 , p ) = 2 xˆ =
5
p − p1 1
+
t
N
(37)
This section is not examinable.
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
Given this demand, and assuming that the other N − 1 firms set a price p j = p o , ∀j ≠ i , then
Firm 1 chooses p1 to:
⎛ p o − p1 1 ⎞
max π1 ( p1 , p ) = p1q1 ( p1 , p ) − ( F + cq1 ) = ( p1 − c ) ⎜
+ ⎟−F
p1
N⎠
⎝ t
o
o
(38)
The first order condition for Firm 1’s profit maximisation problem is:
∂π 1 ( p1 , p o )
∂p1
⎛ p o − p1 ⎞ ⎛ 1 ⎞ ⎛ p1 − c ⎞
=⎜
⎟+⎜ ⎟−⎜
⎟=0
⎝ t
⎠ ⎝N⎠ ⎝ t ⎠
⇒
p o − 2 p1 + c 1
+ =0
t
N
⇒
(39)
1⎛
t ⎞
p1 = ⎜ p o + c + ⎟
2⎝
N⎠
Consumers buying from Firm 1
p1
pN = p
x
1/N
p2 = p
p3 = p
Figure 5: Salop’s Circular City
In a symmetric equilibrium p1 = p o such that:
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
1⎛
t ⎞
po = ⎜ po + c + ⎟
2⎝
N⎠
⇒
2 po = po + c +
t
N
(40)
⇒
po = c +
t
N
And:
xˆ =
po − po
1
1
+
=
2t
2N 2N
(41)
Assuming free entry of firms (i.e. brands), profits must approach zero in long run equilibrium
such that:
⎛ po − po 1 ⎞
+ ⎟−F = 0
t
N⎠
⎝
π i ( po , po ) = ( po − c ) ⎜
⇒
π i ( po , po ) =
1 o
( p − c) − F = 0
N
(42)
⇒
π i ( po , po ) =
t
−F =0
N2
Thus, the long run equilibrium number of brands is given by:
No = t F
(43)
Such that that long run equilibrium price is given by:
po = c +
t
t F
(44)
The long run equilibrium quantity supplied by Firm i is given by:
qi ( p o , p o ) =
po − po 1
1
+ =
t
N
t F
(45)
A trivial, but important point of models of this type is that firms price above marginal cost
and yet do not make profits. Thus, an empirical finding that firms do not make supernormal
11
Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
profits should not lead one to conclude that firms do not have market power, where ‘market
power’ is defined as pricing above marginal cost.6
It is apparent from (43) and (44) that an increase in fixed costs causes a decrease in
the number of firms and an increase in the profit margin, po − c and that an increase in the
(
)
transportation cost increases the profit margin and (therefore) the number of firms – firms see
that there is an increased possibility of differentiation. Formally:
t F
∂N o
=
>0
∂t
2t
(46)
t F
∂N o
=−
<0
∂F
2F
(47)
∂ ( po − c )
∂t
∂ ( po − c )
∂F
=
1
>0
2 t F
(48)
=
t F
>0
2
(49)
Finally, consider the average consumer who in long run equilibrium is located half way
between the indifferent consumer, xˆ = 1 2 N o [recall (41)], and a firm. On average such a
consumer will have to travel:
1
2N
1
−0
2N
∫
0
⎡ 1 2 N o )2
⎤
2 2N
1
o ⎡x ⎤
o ⎢(
xdx = 2 N ⎢ ⎥ = 2 N
− 0⎥ =
⎢
⎥ 4N 0
2
⎣ 2 ⎦0
⎣
⎦
1
(50)
Thus, the long run equilibrium transportation cost (ATC) of the average consumer is given
by:
ATC =
t
tF
=
o
4N
4
(51)
Note that this does note increase as rapidly as t:
∂ATC
F
=
∂t
8 tF
(52)
It is apparent from (43) and (44) that when the entry cost or fixed production cost F
converges to zero, the number of entering firms tends to infinity and the market price tends
toward marginal cost. Similarly, an increase in consumer density, with the fixed cost held
constant, would increase the number of entering firms and push prices towards marginal cost.
6
Note that an economist’s definition of market power differs form a policy maker’s definition. The latter
generally means pricing above average cost. For this second meaning, the firms in our free-entry model have no
market power.
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
Thus, with very low entry costs, each consumer purchases a product very close to his
preferred product and the market is approximately competitive.
Welfare
An interesting and pertinent question is whether the ‘free-market’ produces a larger or
smaller variety than the ‘optimal’ level of variety. Before defining the economy’s welfare
function, we calculate the economy’s aggregate transportation cost, denoted by T. It is
apparent from Figure 5 that in equilibrium, all consumers purchasing from, say, Firm 1 are
located between 0 and 1 2N units of distance from Firm 1 (on each side). Since there are
2N such intervals, the economy’s aggregate transportation cost is given by:
1
T ( N ) = 2Nt ∫
1
2N
0
⎡ ( 1 )2
⎤
⎡ x2 ⎤ 2N
t
2N
x dx = 2Nt ⎢ ⎥ = 2Nt ⎢
− 0⎥ =
⎢ 2
⎥ 4N
⎣ 2 ⎦0
⎣
⎦
(53)
We define the economy’s loss function, L ( F , t , N ) , as the sum of the fixed costs paid by the
producing firms and the economy’s aggregate transportation cost. Formally, the ‘Social
Planner’ chooses the optimal number of brands, N ∗ , to:
min L ( F , t , N ) = NF + T ( N ) = NF +
N
t
4N
(54)
Thus:
∂T ( N )
∂N
⇒
N∗ =
=F−
t
=0
4N 2
(55)
1
2
t F < t F = No
Therefore, in a free-entry location model, too many brands are produced. Notice that there is
a welfare trade-off between economies of scale and the aggregate transportation cost. That is,
a smaller number of brands is associated with lower average production costs but higher
aggregate transportation costs (because of fewer firms). A large number of brands means a
lower scale of production (higher average cost) but with a lower aggregate transportation
cost. Indeed, equation (55) shows that it is possible to raise the economy’s welfare by
reducing the number of brands.
References
Capozza, D. R. and R. Van Order (1982). ‘Product Differentiation and the Consistency of Monopolistic
Competition: A Spatial Perspective.’ Journal of Industrial Economics, 31(1-2), pp. 27-39.
D’Aspremont, C., J. Gabszewicz, and J. Thisse. (1979). ‘On Hotellling's Stability in Competition.’
Econometrica, 17, pp. 1145-1151.
Hotelling. H. H. (1929). ‘Stability in Competition.’ Economic Journal, 39, pp. 41-57.
Martin, S. (1993). Advanced Industrial Economics, Blackwell, Cambridge: USA
Salop, S. (1979). ‘Monopolistic Competition with Outside Goods.’ Bell Journal of Economics, 10, pp. 141-156.
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Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
Appendix
A1.
Quadratic Transport Costs7
Treating transport costs as linear in distance is a simple, but perhaps unreasonable,
assumption. Indeed, if transportation cost is regarded as a proxy for the disutility that arises
from purchasing a less preferred product, it is perhaps more realistic to assume that unit
‘transportation cost’ (i.e. disutility) rises with distance. In what follows we will focus on
Martin’s (1993) reworking of d’Aspremont et al. (1979) and Capozza and Van Order (1982).
Assuming now that transportation costs rise with the square of distance implies an
indifferent consumer, x̂ , via:
− pA − t ( xˆ − a ) = − pB − t ( L − b − xˆ )
2
2
(A1)
Thus:
− p A − t ⎡⎣( xˆ − a )( xˆ − a )⎤⎦ = − pB − t ⎡⎣( L − b − xˆ )( L − b − xˆ )⎤⎦
⇒
− p A − t ( xˆ 2 − 2axˆ + a 2 ) = − pB − t ⎡⎣( L − b − xˆ )( L − b − xˆ )⎤⎦
(A2)
⇒
− p A − t ( xˆ 2 − 2axˆ + a 2 ) = − pB − t ( L2 + b 2 + xˆ 2 − 2 Lb − 2 Lxˆ + 2bxˆ )
Such that:
− p A − txˆ 2 + 2taxˆ − ta 2 = − pB − tL2 − tb 2 − txˆ 2 + 2tLb + 2 Lxˆ − 2bxˆ
⇒
( pB − p A ) + tL2 + tb 2 − ta 2 − 2tLb = txˆ 2 − 2taxˆ − txˆ 2 + 2Lxˆ − 2bxˆ
⇒
( pB − p A ) + tL2 − ta 2 + tb 2 − 2tLb = txˆ 2 − 2taxˆ − txˆ 2 + 2Lxˆ − 2bxˆ
⇒
(A3
( pB − p A ) + t ( L
2
+ b − a − 2 Lb ) = 2txˆ ( L − b − a )
2
2
⇒
( pB − p A ) + t ( L − b + a )( L − a − b ) = 2txˆ ( L − b − a )
⇒
( pB − p A ) + t L − b + a = 2txˆ
(
)
(L − b − a)
This implies:
⎤
1 ⎡ p − pA
xˆ = ⎢ B
+ L − b + a⎥
2 ⎣t (L − b − a)
⎦
7
(A4)
Note, you should be aware of this result but you do not need to work through the derivations.
14
Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
Note that (23) is the demand function faced by Firm A – c.f. (2) above. The demand function
faced by firm B is:
⎤
1 ⎡ p − pB
L − xˆ = ⎢ A
+ L + b − a⎥
2 ⎣t (L − b − a)
⎦
(A4)
We now look for a Nash-Bertrand equilibrium in price strategies. That is, Firm A takes pB as
given and chooses p A to:
⎤
⎤
1 ⎡ p − pA
1 ⎡ p p − pA2
max π A = ⎢ B
+ L − b + a ⎥ pA = ⎢ B A
+ ( L − b + a ) pA ⎥
pA
2 ⎣t (L − b − a)
2 ⎣t (L − b − a)
⎦
⎦
(A5)
The first-order condition is given by:
⎤
∂π A 1 ⎡ pB − 2 p A
= ⎢
+ ( L − b + a )⎥ = 0
∂p A 2 ⎣ t ( L − b − a )
⎦
⇒
1
p Ah = ⎡⎣ pB + t ( L − b − a )( L − b + a )⎤⎦
2
(A6)
Firm B takes p A as given and chooses pB to:
⎤
⎤
1 ⎡ p A − pB
1 ⎡ pB pA − pB2
max π B = ⎢
+ L + b − a ⎥ pB = ⎢
+ ( L + b − a ) pB ⎥
pB
2 ⎣t (L − b − a)
2 ⎣t (L − b − a)
⎦
⎦
(A7)
The first-order condition is given by:
⎤
∂π B 1 ⎡ p A − 2 pB
= ⎢
+ ( L + b − a )⎥ = 0
∂pB 2 ⎣ t ( L − b − a )
⎦
⇒
1
pBh = ⎡⎣ p A + t ( L − b − a )( L + b − a )⎤⎦
2
(A8)
Hence in equilibrium:
15
Industrial Organisation (ES30044)
p Ah =
Topic Four: Horizontal Product Differentiation
1 h
⎡ pB + t ( L − b − a )( L − b + a )⎤⎦
2⎣
⇒
2 p Ah =
1 h
⎡ pa + t ( L − b − a )( L + b − a )⎤⎦ + ⎡⎣t ( L − b − a )( L − b + a )⎤⎦
2⎣
⇒
4 p Ah = pah + t ( L − b − a )( L + b − a ) + 2t ( L − b − a )( L − b + a )
⇒
(A9)
3 p = t ( L − b − a ) ⎡⎣( L + b − a ) + 2 ( L − b + a )⎤⎦
h
A
⇒
3 p Ah = t ( L − b − a )( 3L − b + a )
⇒
b−a⎞
⎛
p Ah = t ( L − b − a ) ⎜ L −
⎟
3 ⎠
⎝
And:
pBh =
1 h
⎡ p A + t ( L − b − a )( L + b − a )⎤⎦
2⎣
⇒
a −b ⎞
⎛
2 pBh = t ( L − b − a ) ⎜ L +
⎟ + t ( L − b − a )( L + b − a )
3 ⎠
⎝
⇒
⎡⎛
a −b ⎞
⎤
2 pBh = t ( L − b − a ) ⎢⎜ L +
⎟ + ( L + b − a )⎥
3 ⎠
⎣⎝
⎦
⇒
(A10)
2b − 2a ⎞
⎛
2 pBh = t ( L − b − a ) ⎜ 2 L +
⎟
3 ⎠
⎝
⇒
b−a⎞
⎛
pBh = t ( L − b − a ) ⎜ L +
⎟
3 ⎠
⎝
Note that:
⎡
b − a ⎞⎤ ⎡
b − a ⎞⎤
⎛
⎛
p Ah − pBh = ⎢t ( L − b − a ) ⎜ L −
⎟ ⎥ − ⎢t ( L − b − a ) ⎜ L +
⎟
3 ⎠⎦ ⎣
3 ⎠ ⎥⎦
⎝
⎝
⎣
⇒
2
Δp h = p Ah − pBh = t ( L − b − a )( a − b )
3
Equilibrium market shares are given by:
16
(31)
Industrial Organisation (ES30044)
xˆ =
Topic Four: Horizontal Product Differentiation
⎤
1 ⎡ pB − p A
+ L − b + a⎥
⎢
2 ⎣t (L − b − a)
⎦
⇒
xˆ h =
⎫⎪
1 ⎪⎧ ⎡ 2t ( L − b − a )( b − a ) ⎤
⎨⎢
⎥ + L − b + a⎬
2 ⎪⎩ ⎣
3t ( L − b − a )
⎪⎭
⎦
⇒
xˆ h =
(32)
⎤
1 ⎡ 2 (b − a )
+ L − b + a⎥
⎢
2⎣
3
⎦
⇒
xˆ h =
1
( 3L − b + a )
6
And:
L − xˆ h = L −
1
1
(3L − b + a ) = (3L + b − a )
6
6
(33)
Note again that if a = b (i.e. the firms are located the same distance from either end of the
line) then p Ah = pBh = tL and xˆ h = L − xˆ h = L 2 . Firm A’s profits are given by:
π A = p Ah x h =
t
b−a⎞
(3L − b + a )( L − b − a ) ⎛⎜ L −
⎟
6
3 ⎠
⎝
(34)
Note that:
∂π A t ⎡
b−a⎞
b − a ⎞ ⎛ 3L − b + a ⎞
⎤
⎛
⎛
= ⎢( L − b − a ) ⎜ L −
⎟ − ( 3L − b + a ) ⎜ L −
⎟+⎜
⎟ ( L − b − a )⎥
∂a 6 ⎣
3 ⎠
3 ⎠ ⎝
3
⎝
⎝
⎠
⎦
⇒
∂π A t ⎡
b−a⎞
b−a⎞ ⎛
b−a⎞
⎤
⎛
⎛
= ⎢( L − b − a ) ⎜ L −
⎟ − ( 3L − b + a ) ⎜ L −
⎟+⎜L−
⎟ ( L − b − a )⎥
∂a 6 ⎣
3 ⎠
3 ⎠ ⎝
3 ⎠
⎝
⎝
⎦
⇒
∂π A t ⎛
b−a⎞
= ⎜L−
⎟ ( L − b − a − 3L + b − a + L − b − a )
∂a 6 ⎝
3 ⎠
⇒
∂π A t ⎛
b−a⎞
= ⎜L−
⎟ ( − L − b − 3a )
∂a 6 ⎝
3 ⎠
⇒
∂π A
t⎛
b−a⎞
= − ⎜L−
⎟ ( L + b + 3a ) < 0
∂a
6⎝
3 ⎠
17
(35)
Industrial Organisation (ES30044)
Topic Four: Horizontal Product Differentiation
It is apparent that Firm A maximises its profit by making a as small as possible, taking b as
given and assuming that p A and pB are determined non-cooperatively to maximise profit,
once locations are chosen. Thus, to maximise profits Firm A (Firm B) will want to locate as
far to the left (right) of the market as possible. With quadratic transportation costs, therefore,
the non-cooperative equilibrium locations are a = b = 0 such that p Ah = pBh = tL2 and
xˆ h = L − xˆ h = L 2 such that each firm supplies half of the market. With quadratic
transportation costs, the principle of minimum differentiation is replaced by the principle of
maximum differentiation: to maximise profits, duopolists locate as far apart as possible.
18