Remark that with RPM, the retail buyer power does not change the retail equilibrium prices
(but only the fixed fees in the contracts).
Indeed, with RPM, the previous expression of the manufacturer profit can be written
J
∏f = ^ ((ps - μs - cs)ss(p) + ^ (ps - ws - Cs)ss(p) - ^(ps - Ws - cs')ss(pfr(a'>')
sGGf s(^Gf s=l
wherethepart ɪ) (ps - μs - cs)ss(p) + ɪ) (ps - ws - cs)ss(p) is the expression of the profit when
s∈ Gf s(^Gf
there is no incentive constraint and thus the buyer power is fixed exogenously and - ∑J=l(ps -
ws - cs)ss(pfr(s'>') = - ∑ (ps - ws - cs)s,s(pfr(s)) (because Ss(pfr(s'>) = 0 if s ∈ Gj) is the part
s$Gf
corresponding to the "endogenous" rent that the manufacturer has to leave to the retailer.
It is clear from this expression that the "endogenous rent" that the manufacturer leaves to the
retailer is not affected by the retail prices on its own products decided using RPM because the
vector pfris- corresponds to the vector of prices when firm f products are not sold by retailer r
and thus is not affected by retails prices of firm f products.
Now, we can use the first order conditions of the maximization of profit of f with respect to
retail prices pj ∈ Gj using the simpler expression of profit with no endogenous buyer power since
first order conditions are equivalent (as in Bonnet and Dubois, 2010) :
0 = Sj (p) + ∑ (ps - ws - cs) + ∑ (ws - μs) ;'^s(p'
s = l L pj -l s<EGf pj
As Rey and Vergé (2004) argue, a continuum of equilibria exist in this general case with RPM,
with one equilibrium corresponding to each possible value of the vector of wholesale prices w.
As we can re-write the retail margins (p - w - c) as the difference between total margins
(p - μ - c) and wholesale margins (w - μ), the previous J - J first order conditions can be written
in a matrix form as
If Sp(p + Γ) + If s(p) - If Sp(I - Ij)Γ = 0 (12)
where Γ = (ws - μs)s=l j is the full vector of wholesale margins and 7 + Γ the vector of total
margins.
14
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