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

'll h ∕T^ ∖               TT /-∏ ∖     T∕∖      /-∏ ∖ I A       /-∏ W        1       J 1                J

with eJ_t(Γ_t) = Pj_t - Hj(Γ.) - f (λ_b^j^ (Γ.) + Λ_r(_j∙) (Γ.))pjt where the parameters
are solutions to the following minimization problem

min        ^2[Pjt ^{- H}j^(rt^{) - f(A}b(j) ⁺ Mj))Pj't]²

tΛfe,Λ_r >6=ι,,.,B,_r = ι,..,β jt

Writing the identification problem in this way allows to more simply find a lower bound on the
degree of underidentification. Indeed, it could be that S^h is not an empty set for many different
vectors of {λb, Λ_r }b_{=1 b r-1 r} and not only the one that minimize the criterion above. However,
in practice, this will not happen in our empirical application and we prefer to present here this
weaker result which is sufficient to explain our method.

Thus, the degree of underidentification of the supply model depends on card(S^h ). The vector
of margins is underidentified if card(S^h) > 1, and overidentified if S^h = 0. As remarked above,
the case of just-identification does not necessarily correspond to card(S^h) = 1,because S^h defined
as above is a lower bound of the "identification set".

In practice, we will see that the demand shape is such that we always get overidentification and
we will consider the solution

Γ* = -'I.       = arg min  T^!"^ ⅜(Γt)²                (23)

M_i=ι,..,τ    ^j-1,∙∙0

as the equilibrium solution.

4.2.4 Testing across different models

Then, in order to test between alternative models once we have estimated the demand model
and obtained the different price-cost margins estimates according to their expressions obtained in
section 3, we apply non nested tests à la Vuong (1989) exactly as in Bonnet and Dubois (2010).
The tests allow to draw some inference between any two alternative models for which we obtained
total marginal costs. The tests statistics are based on the difference between lack-of-fit criterion
of each cost equation that can be estimated for each model once price-cost margins are obtained.
Details on the specification of these cost equations will be given in the next section.

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