The name is absent

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 = bi^E( 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 CE_k^M :

(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) MaxCE_k = qj b1^E(sk)--²---³¹----∑∑qk I^--^χ_ke_k ^- —qkb1ek^σs

x^k, e^k              V                    ⁿ         k=1 J ^{2          2}

Taking into account that q_k = x_k and maximizing (6) with respect to x_k and e_k , a
system of two equations with two unknowns is obtained:

(2b₂ +b₃(n-1))\ ^m ï c _{2         2 2 2}

= b1 ek---I Σ xk ⁺ xk I --ek - Pxkb1 ek^σs= ⁰

n        V k =1         J 2

∂CE^m

(7b) —:---= x_kb₁ - cx_ke_k - Px_kb₁ e_kσ_s = 0

s

∂e_k

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.

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