An optimal contract offered to grower i specifies a payment ri depending on observed
feed levels x, ri (x). Let ui(x)=U[ri (x)] denote utility payments and the inverse U-1[ui(x)]= ri(x)
denote equivalent income. Since U(.) is increasing and strictly concave, U-1 is increasing and
strictly convex. To derive the optimal utility payments, we characterize the incentive-efficient scheme
assuming that the integrator benefits by implementing effort eH for the growers. Hence , the
incentive-efficient scheme u'(x) solves the following problem:
Max py-∫xxH.....∫xxH U 1[ul(x)]χ(x/e = eH)dx 1....dxn
ui (x),l∈N L L
+ω [∫χH ....∫χH u (x)χ(x/e = eH)dx∖..dxn -c(e = eH)]
xL xL H H
subject to
∫xH ....∫XxLH U(x)X(x/e = eH)dx∖..dxn -c(el = eH) ≥ 0 ∀i (3’)
∫XXH ....∫XXH U (x)X(x /e = eH)dx 1....dxn - c(e = eH) ≥ ∀
∫xH ....∫xHul(x)X(x/e1 = eL,e-1 = eH)dx 1....dxn -c(e = eL)
where ω is welfare weight on the grower’s utility and E(xl /e) is the expected feed utilization
by grower i given effort e for all growers. Also where the constraints in (3) are lndlvldual
ratlonallty constralnts, and those in (4) are Nash lncentlve compatlblllty constralnts
3. Results
Since from conditional probability we know thatχ(x/e)= h(xi / e) g(x -i/ x‘ ,e) it can be shown
that the optimum incentive efficient scheme satisfies:
1 (5)
U '( r (x ) =-----------------------------;-----;-------------;------------------::::----------- ∀x, l
h h Γ1 h ( x1 / e = eL, e- = eH ) g ( xl / xl, el = eL, el = eH )
(λ + ω ) + μ [1 - —H/ —l----—-----—]
h(xl / e = eH )g(x l / xl, e = eH )
where λ and μ are multipliers for constraints (3) and (4).