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

where 7 comes from (2) and '^f = (7^f,.., 7 J) where 7^f is the s^th element of vector — (I_r(_s)SfI_r(_s))~¹ I_r(_s)s(p^f)∙

Remark that out of equilibrium retail prices can be obtained from observed equilibrium retail
prices, retail margins at equilibrium and out of equilibrium retail margins using : p^fr(s) = 7^fr(s) —
(p_s — w_s — c_s)+ p_s where 7^fr(s) = p^fr(s) — w_s — c_s is the out of equilibrium retail margin∙ Moreover,
pʃ can be deduced from the differentiation of the retailer’s first order conditions with respect to
wholesale prices∙ These first order conditions are, for all r = 1,.., R and VJ ∈ S_r,

Sj (p^fr )+   ∑    ' — ^s — Cs) '    .'   =0

sCS_r ∖G_fr                   ^Spj

which gives for r = 1,.., R, J ∈ S_r and s = 1,.., J¹

₁      ∂ss (7^fr(j)) _{+ v} Ssi(p-^fr(j)) ⅜f^ω

^[sGSr} dpf^r(j     ^S dpf^r(j)    ∂ws

¹J              ^l^ ^sr      ^{j j}

Defining Spf the matrix (J x J) of the second derivatives of the market shares with respect to
retail prices whose element (s, I) is

we can write equation (16) in matrix form to obtain

FW [sf + irS^f_p' + (spfIr7^fr∣...∣sp;I_r7^fr)] I_r — IrSp (ι_r — I_r) = 0

where 7^fr = p^fr — w — c∙

Denoting Mf_r the matrix ^S~ + I_rSf' + (S^pfI_r7^fr∣...∣S^p_f^jI_r7^fr)] we can solve this system of

equations and get the following expression for J²W

_

Equation (15) shows that one can express the manufacturer’s price-cost margins vector as depen-
ding on the demand function and the structure of the industry by replacing the expression for

F^f

^ W ∙

17

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