The previous equations stand for the pricing of brands owned by manufacturers who retail their
products through a downstream intermediary. Private labels (store brands) pricing obviously does
not follow the same pricing equilibrium. However the retailers’ profits coming from private labels
are implicitly taken into account in the incentive and participation constraints of retailers when
manufacturers make take-it-or-leave-it offers. Taking into account the possibility of endogenous
entry and exit of private label products by retailers is out of the scope of this paper.
Thus, in the case of private label products, retailers (who are also "manufacturers") choose
retail prices and bear the marginal cost of production and distribution, solving :
max V (ps - μs - cs)ss{p) - V (ps - ws - cs)ss(p)
{pi }-^0 δ—'s∈Sr s—<s∈Sr ∖Sr
k^2 оj о t ɔr
where Sr is the set of private label products of retailer r. Thus, for private label products, additional
equations are obtained from the first order conditions of the profit maximization of retailers that
both produce and retail these products. The first order conditions give
Σ ⅛s - Ms - Cs) ds" (p>> + Sj (p) + ^ (ps - Ws - Cs) ds" (p>> = 0 for all j ∈ Sr
*ppj *ppj
s∈Sr s∈Sr ∖Sr
which can be written
∑ (p - Ms - cs) dSs(p) + Sj (p) - ∑ (ws - Ms) dSs(p) = 0 for a∏ j ∈ Sr
s^ ∂pj s^ ∂pj
<jΓ<√ j -J ,-rv∖p5 j -J
s∈sr s∈ Sr ∖Sr
In matrix notation, these first order conditions are : for r = 1,..,R
(IrSpIr)(7 + r)+ IrS(p) - IrSpIrΓ = 0 (13)
where Ir is the ownership matrix of private label products by retailer r.
We thus obtain a system of equations with (12) and (13) where 7 + Γ and Γ are unknown,
which is the following :
ʃʃfSp(7 + Γ) + If s(p)~- If Sp(I - If )Γ = 0 for f = 1,.., F
t (IrSpIr)(7 + Γ) + Irs(p) - IrSpIrΓ = 0 for r = 1,.., R
After solving the system (see appendix 7.1), we obtain the expression for the total price-cost margin
of all products as a function of demand parameters, of the structure of the industry and the vector
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