Non Linear Contracting and Endogenous Buyer Power between Manufacturers and Retailers: Empirical Evidence on Food Retailing in France

margins received by the retailer). The profit function of firm f is equal to

∏^f = ∑_s∈_g, K^{Ws -} ^s)^s≡(P) + ^f≡]∙                            ⁽⁸⁾

Allowing retailers to enjoy some endogenous buyer power, we consider that retailers may be able to
refuse some contracts proposed by manufacturers while accepting other two-part tariffs contracts.
Contract offers are simultaneous but the incentive constraints of the retailers are such that contracts
offered by a manufacturer to a retailer must provide to the retailer a profit at least as large as
the profit that the retailer would obtain when refusing the proposed contract but accepting all
other offers. Moreover, it must be also that the retailers profits are at least larger than some fixed
reservation utility level denoted ∏^r for retailer r.

Thus, the manufacturers set the two-part tariffs contracts parameters (wholesale prices and
fixed fees) in order to maximize profits as in (8) subject to the following retailers’ participation
constraints

∏^r ≥ ∏^r,                                         (9)

and incentive constraints

∏^r ≥ Σ_f,     [ɑ^ - ■ - ^c≡)^s≡(^^r) - ^f≡]                    (¹°)

s s S'-.S

^l^ ∖ ʃ r'

for all r = 1, ∙∙,R, where ∏^r is the retailer’s profit (7) when accepting all the offers, where ∏^r is
the retailer r reservation utility, where Gf_r is the set of products owned by firm f and distributed
by retailer r, and p^fr = (p{^r, ∙∙,p^f_j^r) is the vector of retail prices when the products of Gf_r do not
exist (by convention we will have p^fr = +∞ if i ∈ Gf_r ).

When the retailer r refuses the offers of the manufacturer f, he can accept all other offers and
sell all products not manufactured by f, whose set is denoted S_r∖Gf_r. The market share s_s(p^fr)
of each product of the set S_r ∖Gf_r corresponds to the market share of product s when all products
in Gf_r are absent.

Then, following Rey and Vergé (2°°4) arguments, since the manufacturers can always adjust

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