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
Cjt of product j being the
sum of the marginal cost of production
μjt and of distribution Cjt, 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 b(j) + Λr(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 Cjt is a positive share of retail price pjt
which is non time varying, brand and retailer specific. It introduces some restrictions between the
J x T unknown marginal costs Cjt and the (B + R) x T unknown parameters ʌb and Λr (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 - Hjt) = 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 Sh of vectors of wholesale margins
Γ = (Γ
1,.., Γt) solutions to the identification restrictions (22). This set can be described as the set
Sh defined by

Sh = {Γ Rjt I eht(Γ)=0, j, t}

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