The investor owned (IO) monopsonist’s contract design problem can there-
fore be formulated as the general problem (P) with the following more spe-
cific objective
n n
G'o(q,s) = ∑Eclj>-⅛(c) -si(c)) = ∑Eci (p∙⅛(ci) -βj(ci))
t=l t=≈l
We see that the processor’s objective - as the constraints - depends only on
average production and payments. Also, the objective - as the constraints -
are effectively separable in farmer specific problems.
Using Proposition 1, we shall now characterize the solution to this prob-
lem. Assume that (s(.), g(.)) is a feasible solution and let ci = sup{ci]¾(ci) >
0}. By the first property in Proposition 1, gi(ci) > O for all ci < ⅛ and
¾(ci) = O for all ci > Ci. Also, it follows from the monopsonist interest in
reducing payment that si(ci) - ci ∙ qi(ci) = O.2 Using the second property in
Proposition 1, we therefore have
Si(ci) = ci ∙ qi(ci) + / qi(ci)dci Vci,i
Jci
Substituting this into the objective function and using partial integration,
we get
Gjo⅛,s) =52/ (p-⅛(Ci)-ci∙qi(ci)- [ qi(ci) dcλ fi(ci) dci
*=ιjc< × Λ⅛ /
≈ Σ / ” ci)¾(ci)Λ(ci) ~ <*⅛
i=l *'ci'
This objective must be maximized subject to the constraints that production
levels are weakly decreasing, i.e. Vt,eJ,cJ' : cj > c" => ¾(c⅛) ≤ ⅞(c⅛'), and
that they do not exceed the capacities, i.e. Vi, ci : O ≤ ¾(ci) ≤ gfz.
This is easy, however, if we inwoke a bit of regularity on the cost distri-
butions. Specifically, we will assume that the cost distributions have weakly
increasing hazard rate, i.e. Fi(ci)∕∕i(ci) is weakly increasing on [cf, cf] for
2By IR si(ci) - ci - ρi(ci) > O. Now if si(ci) - ci ∙ gi(ci) = ε > O, we also have
si(ci) — ɑt, 9i(c⅛) ≥ ε Vci < Ci since the producer’s expected profit is decreasing in the
cost type, cf the proof of Proposition 1. In this case, the contract could be improved by
reducing payments with ε for all ci ≤ ci. This would not affect the IC constraints.
10