FEP - Working Papers - Universidade do Porto

Analogously, if i ∈ IR, we have that:
piL =
n − k + 1
2n − 2k + 1 t and piR =
n − k
2n − 2k + 1 t.
The expressions for the marginal consumers, ˜xL and ˜xR, follow immediately:
˜xL =
n − k + 1
2n − 2k + 1
and ˜xR = k
2k + 1 .
It is straightforward to see that ˜xL > ˜xR. Contradiction.
Proof of Proposition 3.
By Lemma 2, there is no equilibrium with prices in S2. Therefore, we must seek prices
satisfying the condition � n
i=1 |piL − piR| ≤ t.
As obtained in (8), the first-order conditions of the n shops at the malls imply that:
Therefore:
PL = n
n + 1 (t + PR) and PR = n
n + 1 (t + PL) .
Using (7), we obtain the individual prices:
PL = PR = nt.
piL = piR = t, ∀i ∈ I.
To complete the proof, we must verify that these local maxima are global maxima. We
need to check if each shop chooses the price t, when the remainders charge t for their
products. Without loss of generality, we consider the shop that sells good iR.
Substituting pjL = pjR = t, ∀j �= i and piL = t in the demand for good iR, given in (10),
we obtain:
qiR =
⎧
⎨
⎩
1 − piR
2t , piR ∈ [0, 2t]
0, piR ∈]2t, +∞[
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