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) [(ps — ws —
siSr
cs)ss(p) — Fs] = ∏ which implies that
ɪɪ Fs = ∑(ps — ws — cs)ss(p) — ∏r
s∈Sr s∈Sr
and thus
∑Fj +∑ Fj = £ Fj = £ £ Fs
J∈g∕ j∈Gf j=1,.,J r-!,.iRs∈Sr
= £ £(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
39
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