1
The model: linear transport case
The basic setting is similar to that of Matsushima (2004). The only difference is that an
upstream firm incurs a transport cost τ s to supply a downstream firm at distance s from
the upstream firm. That is, the upstream firm’s transport cost is linear in distance. In the
model, we only consider endogenous locations of upstream firms.
There are two downstream firms, D1 and D2 , which produce the same physical product.
A linear city of length 1 lies on a line, and consumers are uniformly distributed with density
1 along this interval. Suppose that D1 (resp. D2 ) is located at point l1 ∈ [0, 1] (resp.
1 − l2 ∈ [0, 1]). Without loss of generality, it is sufficient to consider only the case in
which l1 ≤ 1 − l2 . A consumer living at y ∈ [0, 1] incurs a transport cost of t(l1 − y)2
(resp. t(1 − l2 − y)2 ) when purchasing a product from D1 (resp. D2 ). Consumers have unit
demands, i.e., each consumes one unit of the product.
Two upstream firms, UA and UB , supply inputs to two downstream firms. Suppose that
UA (resp. UB ) is located at hA ∈ [0, 1] (resp. 1 − hB ∈ [0, 1]). Upstream firms produce a
homogeneous output, which is the input to downstream firms. Upstream firms engage in
price competition for the business of downstream firms. To supply an s distant downstream
firm, an upstream firm incurs a transport cost τ s. We assume that τ ≤ t. τ can be interpreted
as the degree of difficulty, which is a reflection of the transport of downstream firms’ inputs
to their locations.
We analyze a four-stage game. In the first stage, downstream firms and upstream firms
simultaneously choose their locations. In the second stage, each upstream firm, Ui (i = A, B),
simultaneously chooses its wholesale prices, wij ∈ [0, ∞) (j = 1, 2), where j is the index of
the downstream firm, Dj (j = 1, 2). For instance, wA2 is UA ’s wholesale price for D2 . Each
upstream firm engages in price competition for the business of downstream firms. In the third
stage, observing the wholesale prices, each downstream firm chooses its supplier between UA
and UB and then sets its retail price pi ∈ [0, ∞) (i = 1, 2) simultaneously. In the fourth
stage, observing the retail prices, consumers select between sellers D1 and D2 .
1
2
Basic analysis: linear transport case
2.1
The third and fourth stages
In this paper, we only consider the case in which l1 ≤ 1 − l2 . For a consumer living at
x=
1 + l1 − l2
p2 − p1
+
,
2
2t(1 − l1 − l2 )
(1)
the full price (transport cost plus price) is the same at either of the two firms. The profit of
each downstream firm is given by
πd2
µ
¶
p2 − p1
1 + l1 − l2
+
,
2
2t(1 − l1 − l2 )
µ
¶
p1 − p2
1 − l1 + l2
≡ (p2 − w2 )
+
.
2
2t(1 − l1 − l2 )
πd1 ≡ (p1 − w1 )
(2)
(3)
In (2) and (3), w1 = min{wA1 , wB1 } and w2 = min{wA2 , wB2 }. These mean that D1 (D2 )
procures his or her input from the upstream firm (UA or UB ) that offers the lower wholesale
price. Inserting the first-order conditions of downstream firms, we have
πd1 =
πd2 =
2.2
((1 − l1 − l2 )(3 + l1 − l2 )t − w1 + w2 )2
,
18(1 − l1 − l2 )t
((1 − l1 − l2 )(3 − l1 + l2 )t + w1 − w2 )2
.
18(1 − l1 − l2 )t
(4)
(5)
The second stage
For each downstream firm, each upstream firm sets its price at the rival firm’s transport cost
if its cost is lower than the rival’s (Bertrand competition). The prices set by UA and UB are
as follows:
UA : wA1 = max{τ |l1 − hA |, τ |1 − hB − l1 |}, wA2 = max{τ |1 − l2 − hA |, τ |l2 − hB |}, (6)
UB : wB1 = max{τ |l1 − hA |, τ |1 − hB − l1 |}, wB2 = max{τ |1 − l2 − hA |, τ |l2 − hB |}. (7)
2.3
The first stage
In Figure 1-(1) and 1-(4), if UB moves rightward, the following two changes occur: (a) UB
efficiently supplies D2 without change of price wB2 (for τ |1 − l2 − hA | > τ |l2 − hB |); (b) wA1
increases (for τ |1 − hB − l1 | > τ |l1 − hA |). Then, D1 becomes less efficient and the quantity
2
supplied by D2 is higher. Both changes enlarge UB ’s profit. Therefore, UB has an incentive
to move rightward. In Figure 1-(2) and 1-(3), UA earns zero profit and has an incentive to
relocate.
(1) |l1 − hA | < |1 − hB − l1 |
UA
l1
hA
D1
0
(2) |l1 − hA | > |1 − hB − l1 |
UB
1 − hB
UA
1 − l2
1
0 hA
D2
(3) |l1 − hA | > |1 − hB − l1 |
UA UB
l1
0 hA
1 − hB D1
1 − l2
1 − l2
1
D2
(4) |l1 − hA | < |1 − hB − l1 |
UA
l1
1
0
D2
hA
D1
(5)
UA
UB
l1
1 − hB
D1
UB
1 − l2
1 − hB
D2
1
(6)
l1
0 hA
D1
UB
1 − l2
1 − hB1
D2
0
UA
l1
hA
D1
UB
1 − l2
1 − hB 1
D2
Figure 1: location patterns
Thus, we have only to consider cases (5) and (6). From the discussion above, we have the
following lemma:
Lemma 1 If a location pattern is an equilibrium outcome, the location pattern has to be that
in which (0 ≤)hA ≤ l1 and (0 ≤)hB ≤ l2 .
We now consider the location pattern in which (0 ≤)hA ≤ l1 and (0 ≤)hB ≤ l2 . From (1),
(4), (5), (6), and (7), we derive the four firms’ profits:
πd1 =
πd2 =
[(3 + l1 − l2 )(1 − l1 − l2 )t + (l1 − l2 − hA + hB )τ ]2
,
18t(1 − l1 − l2 )
[(3 − l1 + l2 )(1 − l1 − l2 )t − (l1 − l2 − hA + hB )τ ]2
,
18t(1 − l1 − l2 )
πuA ≡ (wA1 − τ |l1 − hA |)x
3
(8)
(9)
=
τ [1 − 2l1 + hA − hB ][(3 + l1 − l2 )(1 − l1 − l2 )t + (l1 − l2 − hA + hB )τ ]
, (10)
6t(1 − l1 − l2 )
πuB ≡ (wB2 − τ |l2 − hB |)(1 − x)
=
τ [1 − 2l2 − hA + hB ][(3 − l1 + l2 )(1 − l1 − l2 )t − (l1 − l2 − hA + hB )τ ]
. (11)
6t(1 − l1 − l2 )
The first-order conditions are as follows:
∂πd1
∂l1
= [(3 + l1 − l2 )(1 − l1 − l2 )t + (l1 − l2 − hA + hB )τ ]
×
∂πd2
∂l2
(2 − l1 − 3l2 − hA + hB )τ − (1 − l1 − l2 )(1 + 3l1 + l2 )t
,
18t(1 − l1 − l2 )2
= [(3 − l1 + l2 )(1 − l1 − l2 )t − (l1 − l2 − hA + hB )τ ]
(2 − 3l1 − l2 + hA − hB )τ − (1 − l1 − l2 )(1 + l1 + 3l2 )t
,
18t(1 − l1 − l2 )2
τ ((3 + l1 − l2 )(1 − l1 − l2 )t − (1 − 3l1 + l2 + 2hA − 2hB )τ )
,
6t(1 − l1 − l2 )
τ ((3 − l1 + l2 )(1 − l1 − l2 )t − (1 + l1 − 3l2 − 2hA + 2hB )τ )
.
6t(1 − l1 − l2 )
×
∂πuA
∂hA
∂πuB
∂hB
(12)
=
=
(13)
(14)
(15)
From (14) and (15), we have the following lemma:
Lemma 2 In an equilibrium outcome, the following location pattern must hold: hA = l1 and
hB = l2 .
Proof:
From (14) and (15), we have the reaction functions of the upstream firms:
hA = hB +
(3 + l1 − l2 )(1 − l1 − l2 )t + (1 − 3l1 + l2 )τ
,
τ
(14’)
hB = hA +
(3 − l1 + l2 )(1 − l1 − l2 )t + (1 + l1 − 3l2 )τ
.
τ
(15’)
¿From (14’) and (15’), we find that the reaction functions are parallel. We depict them:
hB
hB (hA )
hA (hB )
A
hA
0
B
4
where, A ≡
(3−l1 +l2 )(1−l1 −l2 )t+(1+l1 −3l2 )τ
τ
and B ≡ −
(3+l1 −l2 )(1−l1 −l2 )t+(1−3l1 +l2 )τ
.
τ
If A > B,
the Lemma holds.
A−B =
=
(3 − l1 + l2 )(1 − l1 − l2 )t + (1 + l1 − 3l2 )τ
τ
µ
¶
(3 + l1 − l2 )(1 − l1 − l2 )t + (1 − 3l1 + l2 )τ
− −
τ
6(1 − l1 − l2 )t − 2(1 − l1 − l2 )τ
> 0.
τ
Q.E.D.
One of the upstream firms locates at the same point as one of the downstream firms and
supplies its input to that downstream one, even though the upstream firm is independent
of the downstream firms. Each upstream firm produces a basic input for each downstream
firm, as if each of the upstream firms belonged to each adjacent downstream firm.
We now show the intuition behind Lemma 2. In Figure 1-(5), if UA moves rightward,
the following two changes occur: (a) UA efficiently supplies D1 without change of wA1 (for
τ |1 − hB − l1 | > τ |l1 − hA |). (b) wB2 decreases.1 That is, it enhances the quantity supplied
by D2 . The former (resp. latter) effect is positive (resp. negative) to UA . In (10), hA in the
first (resp. second) pair of brackets represents the former (resp. latter) effect. The former
(resp. latter) effect is the first (resp. second) order effect on UA ’s profit. Therefore, the
former effect dominates the latter one, and Lemma 2 holds.2
From Lemma 2 and the first-order conditions, we solve the following simultaneous equations:
∂πd2
∂πd1
= 0,
= 0, hA = l1 , hB = l2 .
∂l1
∂l2
(18)
We derive the following proposition:
Proposition 1 The following location pattern is an equilibrium outcome:
t
, and
2
√
t
(24 2 − 30)t
2τ − t
, if
<τ ≤
.
l1 = l2 = hA = hB =
4t
2
4
l1 = l2 = hA = hB = 0, if τ ≤
1
(19)
(20)
We implicitly assume that |l2 − hB | < |1 − l2 − hA |. If the assumption does not hold, UB earns zero profits
and has an incentive to move leftward.
2
As shown by Matsushima (2004), the latter effect is not dominated by the former, if the transport costs
of upstream firms are quadratic in distance.
5
Proof
Substituting hA = l1 and hB = l2 into (12) and (13), we have
(3 + l1 − l2 )((1 + 3l1 + l2 )t − 2τ )
,
18
(3 − l1 + l2 )((1 + l1 + 3l2 )t − 2τ )
.
= −
18
∂πd1
∂l1
∂πd2
∂l2
= −
(21)
(22)
Solving the following simultaneous equations: ∂πd1 /∂l1 = 0 and ∂πd2 /∂l2 = 0, we have:
µ
t − 2τ
t − 2τ
,−
(l1 , l2 ) = −
4t
4t
¶
,
µ
5t − τ
t+τ
,−
2t
2t
¶
,
µ
5t − τ t + τ
−
,
2t
2t
¶
.
(23)
For any τ ≤ t, the second and the third pairs violate the boundary condition, l1 ≥ 0 and
l2 ≥ 0.
If τ ≤ t/2, the first pair also violates the boundary condition, and the optimal location
may be l1 = hA = 0 and l2 = hB = 0. We now check whether the location pattern is an
equilibrium outcome when τ ≤ t/2. From Lemma 2, we find that neither upstream firm
has an incentive to change its location. We now show that neither downstream firm has
an incentive to change its location. By symmetry, it is sufficient to consider D1 ’s incentive.
Given the locations of the other three firms, there exist two patterns of its location.
1. 0 ≤ l1 ≤ 1/2: UA supplies D1 and UB supplies D2 .
In this case, πd1 is (8) and the first order condition is (12). Substituting l2 = hA =
hB = 0 into (12), we have
[(3 + l1 )(1 − l1 )t + l1 τ ][(7 + 5l1 )(1 − l1 )t − (2 − 3l1 )τ ]
∂πd1
=−
< 0.
∂l1
18t(1 − l1 )2
(24)
For any 0 ≤ l1 ≤ 1/2, l1 = 0 is optimal.
2. 1/2 ≤ l1 ≤ 1: UA supplies D1 and UB supplies D2 .
In this case, UB supplies D1 and D2 . The wholesale price for D1 is not wA1 but wB1 .
The wholesale price for D2 is wB2 . From (4), (6), and (7), πd1 is:
πd1 =
((3 + l1 − l2 )(1 − l1 − l2 )t + (1 − l1 − l2 )τ )2
.
18(1 − l1 − l2 )t
The first order condition is
∂πd1
∂l1
= −
(3t + l1 t − l2 t + τ )(t + 3l1 t + l2 t + τ )
< 0.
18t
We find that l1 = 1/2 is optimal for any 1/2 ≤ l1 ≤ 1.
6
(25)
From the result of the two patterns, we find that l1 = 0 is optimal.
If τ > t/2, the first pair is an interior solution. We now check whether the first pair in
(23) is an equilibrium outcome, that is, we check whether (26) is an equilibrium outcome:
µ
t − 2τ
t − 2τ
t − 2τ
t − 2τ
(l1 , l2 , hA , hB ) = −
,−
,−
,−
4t
4t
4t
4t
¶
.
(26)
From Lemma 1, we find that neither upstream firm has an incentive to change its location.
We now investigate the condition that neither downstream firm has an incentive to change its
location. By symmetry, it is sufficient for us to consider D1 ’s incentive. Given the locations
of the other three firms, there exist three patterns of location:
1. 0 ≤ l1 ≤ 1/2: UA supplies D1 and UB supplies D2 .
In this case, πd1 is (8) and the first order condition is (12). Substituting l2 = hA =
hB = − t−2τ
4t into (12), we have
∂πd1
(2τ − t − 4tl1 )(15t − 10τ − 12tl1 )(65t2 − 32tτ − 4τ 2 − 16t(2t − τ )l1 + 16t2 l12 )
=
.
∂l1
288(5t − 2τ − 4tl1 )2
(27)
We now consider the following three cases. If 0 ≤ l1 ≤ (2τ − t)/4t, the value is positive.
If (2τ − t)/4t ≤ l1 ≤ 5(3t − 2τ )/12t and 5(3t − 2τ )/12t < 1/2, or (2τ − t)/4t ≤ l1 ≤ 1/2
and 5(3t − 2τ )/12t > 1/2, the value is negative. If 5(3t − 2τ )/12t ≤ l1 ≤ 1/2 and
5(3t − 2τ )/12t < 1/2, the value is positive. Therefore, if πd1 in which l1 = (2τ − t)/4t
is larger than that in which l1 = 1/2, the optimal location is l1 = (2τ − t)/4t for any
0 ≤ l1 ≤ 1/2.
πd1
µ
2τ − t
4t
¶
=
3t − 2τ
,
4
πd1
µ ¶
1
2
=
(3t − 2τ )(15t + 2τ )2
.
1152t2
(28)
We now compare πd1 ( 2τ4t−t ) with πd1 ( 12 ):
µ
¶
µ ¶
(3t − 2τ )(15t + 2τ )2
3t − 2τ
−
4
1152t2
2
(3t − 2τ )(63t − 60tτ − 4τ 2 )
=
.
(29)
1152t2
√
If 63t2 − 60tτ − 4τ 2 > 0, that is, if τ < (24 2 − 30)t/4(∼ 0.984t), the optimal location
πd1
2τ − t
4t
− πd1
1
2
=
is l1 = (2τ − t)/4t for any 0 ≤ l1 ≤ 1/2.
7
2. 1/2 ≤ l1 ≤ 1 − (2τ − t)/(4t): UA does not supply but UB supplies D1 and D2 .
In this case, UB supplies D1 and D2 . The wholesale price for D1 is not wA1 but wB1 .
The wholesale price for D2 is wB2 . From (4), (6), and (7), πd1 is:
πd1 =
((3 + l1 − l2 )(1 − l1 − l2 )t + (1 − l1 − l2 )τ )2
.
18(1 − l1 − l2 )t
(30)
The first order condition is
∂πd1
∂l1
= −
(3t + l1 t − l2 t + τ )(t + 3l1 t + l2 t + τ )
< 0.
18t
We find that l1 = 1/2 is optimal for any 1/2 ≤ l1 ≤ 1 − (2τ − t)/(4t).
3. 1 − (2τ − t)/(4t) ≤ l1 ≤ 1: UA does not supply but UB supplies D1 and D2 .
In this case, πd1 and πd2 are
µ
πd2
µ
p2 − p1
1 + l1 − l2
+
2
2t(1 − l1 − l2 )
µ
¶
p2 − p1
1 + l1 − l2
+
= (p2 − wB2 )
.
2
2t(1 − l1 − l2 )
πd1 = (p1 − wB1 ) 1 −
¶¶
,
(31)
(32)
Each firm’s pricing in the third stage is
p1 =
p2 =
(l1 + l2 − 1)(3 − l1 + l2 )t + 2wB1 + wB2
,
3
(l1 + l2 − 1)(3 + l1 − l2 )t + wB1 + 2wB2
.
3
(33)
(34)
πd1 and πd2 are
πd1 =
=
πd2 =
=
((l1 + l2 − 1)(3 − l1 + l2 )t − wB1 + wB2 )2
18(l1 + l2 − 1)t
((l1 + l2 − 1)(3 − l1 + l2 )t − (l1 + l2 − 1)τ )2
,
18(l1 + l2 − 1)t
((l1 + l2 − 1)(3 + l1 − l2 )t + wB1 − wB2 )2
18(l1 + l2 − 1)t
((l1 + l2 − 1)(3 + l1 − l2 )t + (l1 + l2 − 1)τ )2
.
18(l1 + l2 − 1)t
(35)
(36)
The first order condition is
∂πd1
∂l1
=
((5 − 3l1 − l2 )t − τ )((3 − l1 + l2 )t − τ )
> 0.
18
8
(37)
For any 1 − (2τ − t)/(4t) ≤ l1 ≤ 1, l1 = 1 is optimal. The profit is
πd1 =
(2τ − t)(7t − 2τ )
.
1152t2
(38)
This is smaller than πd1 ( 2τ4t−t ) in (28).
From the results of the three patterns, we find that l1 = (2τ − t)/4t is optimal if τ <
√
Q.E.D.
(24 2 − 30)t/4(∼ 0.984t).
9