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

7 Appendix

7.1 Detailed resolution of system of equations

Generically we have systems of equations to be solved of the form

ʃ Af(7 + Γ) + Bf =0
ɪ for f = 1,..,G

where Af and Bf are some given matrices.

Solving this system amounts to solve the following minimization problem

min £ [Af (7 + Γ) ₊ Bf ]' [Af (7 + Γ) ₊ Bf ]

^7+Г f=1

leads to the first order conditions

(G ∖ G

£Af Af (7 + Γ) - £Af Bf =0

f = 1 ) f =1

that allow to find the following expression for its solution

(∖ -1
G ∖ G

Σ ^Af ^Af Σ ^Af ^Bf

f =1 f f = 1

7.2 Detailed proof of the manufacturers profit expression under two-
part tariff’s

We use the theoretical results due to Rey and Vergé (2004) applied to our context with F firms

and R retailers. The participation constraint (9) being binding, we have for all r ɪ) [(p_s — w_s —
siS_r

c_s)s_s(p) — F_s] = ∏ which implies that

ɪɪ F_s = ∑(ps — w_s — c_s)s_s(p) — ∏^rs∈S_r s∈S_r

and thus

∑F_j +∑ Fj = £ Fj = £ £ Fs

J∈^g∕ j∈Gf ^j=¹,.,J r-!,._iRs∈S_r

= £ £⁽ps^— Ws^—^cs'^)ss⁽p)^— £ ∏^,

£ ⁽pj^—^wj^—^cj^)sj⁽p⁾^— £ ∏^,j=1,.,J r=1,.,R

r=1,.,Rs∈Sr r=1,.,R

so that

£ ^f3 = £ ⁽pj^—^w3 ^—^cj^)sj ⁽p^{) —} £ ^f3 ^— £ ∏^,

j∈Gf j = 1,..,J j∈Gf r=1,.,R