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 = (7f,.., 7 J) where 7f is the sth element of vector (Ir(s)SfIr(s))~1 Ir(s)s(pf)

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 :
pfr(s) = 7fr(s)
(ps ws cs)+ ps where 7fr(s) = pfr(s) ws cs is the out of equilibrium retail margin∙ Moreover,
p
ʃ 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 Sr,

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

sCSr Gfr                   Spj

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

^    ∂sj (pfrjfS) ∂pfrj

S-~f        Sffrjjl    ∂ws

IC[1,..,J }Gfr   °pl               s


1      ∂ss (7fr(j)) + v Ssi(p-fr(j)) ⅜fω

[sGSr} dpfr(j     ^S dpfr(j)    ∂ws

1J              l^ sr      j j

+ Σ

lCSrGfr


(pfr wι cι)    ∑

sG[1,..,J }\Gfr


a2sl(p'fr(j)) ap'fr(j)

∂pfj∂pfsrj dws


(16)


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

)     ∙ p

affr(f)affr(f) , ∙ ∙


/

Sp

J2f


a2s1 (pfr<f>)
a
pfr(f)apfr(f)

2s1 (pfrt-f~>)
∂pfrωdpfjrf


2sj (pfrt-^)   

apf,r<f>apfr<f>


2sj (ffr<f>)

∂pfr-f∂pjr-f /


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

FW [sf + irSfp' + (spfIr7fr...sp;Ir7fr)] Ir IrSp r Ir) = 0

where 7fr = pfr w c

Denoting Mfr the matrix ^S~ + IrSf' + (SpfIr7fr...SpfjIr7fr)] we can solve this system of

equations and get the following expression for J2W

_


f∑r-1 Ir Mf r Ir Sf (Ir


i)) (∑Π1


Ir Mf r Mfr Ir


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

Ff

^ W

17



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