FEP - Working Papers - Universidade do Porto

7 Appendix
Proof of Lemma 1.
Assume, by way of contradiction, that the prices that maximize the profits of a department
store at L (given the prices at R) are such that p ∈ S2.
(i) We start by considering the possibility of an interior solution, assuming that 0 < ˜xL <
˜x < ˜xR < 1 and that piL �= piR, ∀i ∈ I.
In this case, the profit function is continuous with respect to piL, ∀i ∈ I, in a neigborhood
of p. This implies that the first-order conditions must be satisfied.
In the interior of any branch in S2, the profit of the department store is given by:
� � �
ΠL = ˜xL pjL + ˜xR pjL = ˜xL pjL + (˜xR − ˜xL) �
j∈IR
j∈IL
j∈I
Using the fact that ˜x − ˜xL = ˜xR − ˜x, we can write:
ΠL = ˜x �
�
�
pjL + (˜x − ˜xL) pjL − �
j∈I
The first-order conditions, for i ∈ IR, are:
∂ΠL
∂piL
= 0 ⇔ ˜xL = 1
t
j∈IL
j∈IR
pjL
�
�
pjL ⇔ 2 �
pjL = t + �
j∈IR
while, for i ∈ IL, the first-order conditions are:
∂ΠL
∂piL
= 0 ⇔ ˜xR = 1
t
j∈IR
j∈IL
.
j∈IR
�
pjL ⇔ 2 �
pjL = �
j∈IL
j∈IL
j∈IL
pjR,
pjR.
Using the two last equalities, we can rewrite the expression of ΠL as follows:
ΠL = ˜xPL − t
2 (˜x − ˜xL) .
If the department store had chosen prices such that p ∈ S1 with the same PL, its profit
21
pjL.