with respect to all retail prices, i.e.
/
Sp ≡
∖
∂sγ
dpi
∂sj
dpi
.
dsi
∂p J
.
ds j
dp j
In vector notation, the first order condition (1) implies that the vector 7 of retailer r’s margins
(rows corresponding to products not sold by r are set to zero), i.e. the retail price p minus the
wholesale price w minus the marginal cost of distribution c, is1
7 ≡ p — w — c = — (IrSpIr) 1 Irs(p)
(2)
Remark that for private labels, this price-cost margin is in fact the total price-cost margin p — μ — c
which amounts to replace the wholesale price w by the marginal cost of production μ in this
formula.
Concerning the manufacturers’ behavior, we assume they maximize profit choosing the whole-
sale prices Wj of their own products and given the retailers’ response (1). The profit of manufacturer
f is given by
∏f = ∑j∙eσ (wj— μj )sj (p(w))
where μj∙ is the manufacturer’s (constant) marginal cost of production of product j. Assuming the
existence of a pure-strategy Bertrand-Nash equilibrium in wholesale prices between manufacturers,
the first order conditions are
sj+ Σ Σ (ws— μs)jsS^p. = 0’ foranjeGf∙ (3)
s<EGf 1=1,..,J pι j
We denote If the ownership matrix of manufacturer f that is diagonal and whose jth element is
equal to one if j is produced by the manufacturer f and zero otherwise.
We introduce Pw the matrix (J x J) of retail prices responses to wholesale prices, containing
1Abusing notations, we consider the generalized inverse when noting the inverse of non invertible matrices, which
means that for example
2 0 ’_ Γ 1/2 0
Iool I 0 0 ∣.
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