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

All margins are then identified.

General case : A less restrictive identification method may consist in adding restrictions on the
vectors of marginal costs and margins. Actually, the total marginal cost Cj_t of product j being the
sum of the marginal cost of production μ_j∙_t and of distribution Cj_t, we will consider the following
assumption to get identification of retail and wholesale margins in two-part tariffs models :

Identification assumption :

Cjt = μjt + Cjt = f (λ_b(j) + Λ_r(j∙))pj∙_t for all j = 1,.., J and t = 1,.., T           (22)

where δ(j) denotes the brand of product j, r(j') the retailer of product j, and f (.) is a function to
be specified.

This assumption means that total marginal cost Cj_t is a positive share of retail price pj_t
which is non time varying, brand and retailer specific. It introduces some restrictions between the
J x T unknown marginal costs Cj_t and the (B + R) x T unknown parameters ʌb and Λ_r (where
B + R< J = Bx R and B is the number of brands and R the number of retailers). In practice, we
impose f (æ) = _1+e*_p(^) which proved to be the preferred specification among several tested ones
in terms of tractability of empirical estimation.

Then, this identification assumption implies that

Pjt - Hj (Γt) = f (ʌb(j) + Λr(j∙))pjt for all j = 1,.., J and t = 1,.., T

which reduces the degree of underidentification since it adds J x T restrictions and only (B +
R) additional unknown parameters. The true degree of underidentification will depend on the
properties of the non linear function H(.).

The identification of margins will thus depend on the set S^h of vectors of wholesale margins
Γ = (Γ₁,.., Γ_t) solutions to the identification restrictions (22). This set can be described as the set
S^h defined by

S^h = {Γ ∈ R^jt I eht(Γ)=0, ∀j, ∀t}

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