manufacturer branded products and J — J' are store brands (also called private labels). We denote
by Sr the set of products sold by retailer r and by Gf the set of products produced by firm f.
3.1 Linear Pricing
As in Rey and Vergé (2004), Villas-Boas (2007), Bonnet and Dubois (2010), among others, we
consider the game where manufacturers set wholesale prices first, and retailers follow by setting
the retail prices. We obtain the usual double marginalization result. For private labels, prices are
chosen by the retailer who bears both retailing and production costs. Using backward induction,
we consider the retailer’s problem who wants to maximize its profit denoted ∏r for retailer r and
equal to
∏r = ∑j∙es,. (Pj — wj — cj )sj (P)
where pj is the retail price of product j sold by retailer r, Wj is the wholesale price paid by retailer
r for product j, Cj is the retailer’s (constant) marginal cost of distribution for product j, Sj (p) is
the market share of product j, p is the vector of retail prices of all products.
Remark that we normalized the profit by the population size which amounts to define profits
as per household profit. Since we will take into account an outside good option denoted good 0,
this normalization is equivalent as if we had used the total demand of each good instead of market
shares.
Assuming that a pure-strategy Bertrand-Nash equilibrium in prices exists and that equilibrium
prices are strictly positive, the price of any product j sold by retailer r must satisfy the first-order
condition
∑ι WS«
s.8r (PS — WS — Cs) OPj =0’
for all j ∈ Sr.
(1)
Now, we define Ir as the ownership matrix (size (J x J)) of the retailer r that is diagonal and
whose jth element is equal to 1 if the retailer r sells product j and zero otherwise. Let Sp be the
market shares response matrix to retailer prices, containing the first derivatives of all market shares
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