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

with respect to all retail prices, i.e.

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 = — (I_rS_pI_r) ¹ I_rs(p)

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 μ_j∙ 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⁾j^sS^^p. = ⁰’  ^foranjeGf∙               ⁽³⁾

s<EG_f 1=1,..,J               ^{pι     j}

We denote If the ownership matrix of manufacturer f that is diagonal and whose j^th element is

equal to one if j is produced by the manufacturer f and zero otherwise.

We introduce P_w 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

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