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

the fixed fees such that all the constraints (10) will be binding, we have Vr = 1,.., R

∑ [(p_s - w_s - c,)s,(p) - F_s]=   ∑ [(pf - w_s - c_s)s_s(p^fr) - F_s]

s∈S_r                                     s∈S_r ∖G_fr

In general, if constraints (10) are satisfied, the constraints (9) will be satisfied. The binding

constraints (10) imply that the sum of fixed fees paid for the products of f sold through r is

V Fs = V [⁽Ps ^{- w}s ^{- c}s'⁾Ss⁽p^{) - (}p^f_s^{r -} Ws ^{- c}s)Ss⁽p^{fr )}]
s∈Gf_r       s∈S_r

because s_s(p^fr ) = 0 when s ∈ Gf_r.

Using this expression, one can rewrite the profit of the manufacturer f as

∏^f = Σ2 ^[(ws ^- Ms^)ss⁽p⁾+ ^Fs^] =    ⁽W ^- G)^ss ⁽P^)F E_{r = 1} ∑ ^Fs

s∈Gf                               s∈Gf                               s∈Gf_r

=   Σ ^(ws ^- Ms^)ss⁽p^{) +} Σ_r=₁ Σ [⁽ps ^{- w}s ^{- c}s')Ss⁽p) ^{- (pf}_s^{r - w}s ^{- c}s)^ss⁽P^{f )}]

s∈Gf                               s∈S_r

because ^r=∖Gf_l- = Gf (and Gfr ∩ Gfr^f = 0). The manufacturer’s profit is then

J

∏^f = ^ (Ws - Ms)^ss(p) + ^ (pPs - Ws - cs')Ss(p) - (p{^ri'^sf - Ws - c.. ) Ss ( p^frs )       (11)

s∈Gf                     s = 1

where r(s) denotes the retailer of product s (∈ {1,.., J}).

We will also consider a simpler case where constraints (10) do not exist because it is assumed
that if one offer is rejected then all offers must be rejected as in Bonnet and Dubois (2010). Then,
the outside opportunities depend on a fixed exogenous reservation utility and we will say that the
buyer power of retailer is exogenous.

3.2.1 With Resale Price Maintenance

Let’s consider the case where manufacturers use resale price maintenance (RPM) in their
contracts with retailers. Then, manufacturers can choose retail prices while the wholesale prices
have no direct effect on profit. In this case, the vectors of prices p>^fr are such that p^fr = p⅛ if
i ∈ Gf_r and the profit (11) of manufacturer f can then be written as⁵

∏^f = ∑_s∈_g ^(ws ^- Vs^)ss⁽p^{) +} ^_s=1 ⁽ps ^{- w}s ^{- c}s] [^ss⁽p^{) - s}s⁽p^fr(s))]

⁵Because also s_s(p^fr^^s^) = 0, p^fr = ÷∞ for s ∈ Gf_r and by convention s_s(p^fr(^s))p^fr = 0.

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