(2)
Max CEiM =Ε∑
Qik i k=1
ï ï
m n m
- b2∑Q - b3∑∑Q - qkPk
k=1 j=1 k=1
j ≠i J J
Taking the first-order necessary condition for a maximum in (2) yields:
m nm
(3) pk =b1Ε(sk)-2b2∑qik -b3∑∑qjk
k=1 j=1 k=1
j≠i
Aggregation of (3) across the demands for grower k from the processors yields:
(4) Pk = biE( sk ) - 2 b2 + b3 (n 1) ∑ qk
n k=1
The grower k´s problem for the derived demand (4) is to choose his effort in quality and
quantity to maximize his certainty equivalent CEkM :
(5) Max CEM = E\ qkPk - c xkek V pσ2k
xk, ek V 2 J 2 k
Upon expanding the above expression, the following is obtained:
M- ' M Mi'''-' \ ! r! ∖ 2b2 + b3 (n 1) ï C 2 ρ 2-ι2 2 2
(6) MaxCEk = qj b1E(sk)--2---31----∑∑qk I--χkek - —qkb1ekσs
xk, ek V n k=1 J 2 2
Taking into account that qk = xk and maximizing (6) with respect to xk and ek , a
system of two equations with two unknowns is obtained:
(7a)
∂ CEm
dx k
(2b2 +b3(n-1))\ m ï c 2 2 2 2
= b1 ek---I Σ xk + xk I --ek - Pxkb1 ekσs= 0
n V k =1 J 2
∂CEm
(7b) —:---= xkb1 - cxkek - Pxkb1 ekσs = 0
s
∂ek
Since processors and growers face common equations, without loss of generality, in
what follows we omit the subscript i and k in the variables. Analyzing equation (7a), we
see an inverse relation between quantity and quality.