FEP - Working Papers - Universidade do Porto

In (the candidate) equilibrium, p∗ iR , we have PR − PL = n−1t.
As a result,
n+2
�
j∈I\{i}
(pjR − pjL) + 3
n + 2 t − piL =
n − 1
n + 2 t ⇔ sL =
n − 4
n + 2 t + sR + piL. (12)
As the value of sL does not depend on piR, it has the same value in D2 and in D3. Then,
substituting (12) in (11), we find that p ∗∗
iR is outside D2 when:
sR
2
Then, if sR + piL ≥ √ 18t,
p∗∗
n+2 iR
piL
+
2 ≥ −t + 2sR
n − 4
+ 2piL +
n + 2 t ⇔ sR + piL ≤ 4
n + 2 t.
= sR
2
equilibrium, p ∗ iR . We conclude that p∗ iR
+ piL
2 is in D2 and, therefore, upsets our candidate
∀i ∈ I, sR + piL ≤
is an equilibrium if and only if:
√ 18
n + 2 t.
An equivalent, but more elegant, condition is obtained by using (12):
�
� �
�
�piL − 3
n + 2 t
�
�
�
� ≤ n + 6√2 − 7
t < t.
n + 2
i∈I
The equilibrium demand and profits follow immediately. �
Proof of Proposition 2.
By Lemma 1, the best response prices are such that � n
i=1 |piL − piR| ≤ t.
The first-order conditions for the profit-maximization problems of the department stores
imply that:
yielding:
PL = PR
2
t
+
2 and PR = PL t
+
2 2 ,
PL = PR = t.
The equilibrium demand and profits follow immediately. �
Proof of Lemma 2.
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