The optimization problem in equations (9)-(11) can be solved sequentially. First, the
optimal solutions to the grower’s decision on efforts in quantity and quality in equation
(10) are obtained:
Max CEiIkC
xik,ek
r (
n m
= Ε∑ ai + βiQik b1 Sk - b2 ∑ Qik
i i ik 1 k 2 ik
i=1 k=1
к к
Ï
nm
- b з ∑∑ Qjk
j=1 k=1
j ≠ i j
_£ = '
2 xe:
J
∑n ρk βi2Qi2kb12ek2σs2
i=1 2
Optimizing and substituting the values of Qik and E(Sk), we obtain:
.. ( , s >
IC m n m n
(12) ʌ = βi b1 ek - b 21 xk +∑ xik I-b3 ∑∑ xk - b3 ∑βjxjk — e= - Pkβikxikb1 skσs' = 0
i 1 k 2 ik ik 3 jk 3 j jk k k ik ik 1 k s
dxik к k=1 J j=1 k=1 j=1 2
к j≠i J j≠i
(13) d CEk =∑(β1xkb1 - cxikek - ρkβi2 x2kb2 ekσs2 )= 0
i ik 1 ik k k i ik 1 k s
Here, we should substitute equations (12) and (13) into equations (9) and (11) and
maximizing with respect to β, the optimal share for the grower would be obtained6.
Finally, substitutingx*, e*and β*into equation (11), we would obtain the optimal fixed
rent α* . But similarly than in the previous model, these equations can not be solved
analytically and a numerical simulation exercise is obtained instead.
Simulation and discussion
As we mentioned earlier, the mean-variance approach has been used to analyze various
contractual issues. However, it is usually very difficult to explicitly solve the first-order
conditions that define the values of the decision variables in the optimal contract.
Quantitative applications of mean-variance models require numerical simulations
(Robe, 2001).
Theoretically, growers respond to incentives in contracts obtaining greater levels of
quality than in the spot market. And in practice, some authors have obtained conclusive
support empirically (for example, Curtis and McCluskey 2003, Alexander, Goodhue