The Advantage of Cooperatives under Asymmetric Cost Information

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) = ∑E_clj>-⅛(c) -s_i(c)) = ∑E_ci (p∙⅛(c_i) -β_j(c_i))
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 c_i = sup{c_i]¾(c_i) >
0}. By the first property in Proposition 1, g_i(c_i) > O for all c_i < ⅛ and
¾(c_i) = O for all c_i > Ci. Also, it follows from the monopsonist interest in
reducing payment that s_i(c_i) - c_i ∙ q_i(c_i) = O.² Using the second property in
Proposition 1, we therefore have

Si(c_i) = c_i ∙ q_i(c_i) + / q_i(c_i)dc_i Vc_i,i
Jci

Substituting this into the objective function and using partial integration,
we get

G^jo⅛,s) =52/ (p-⅛(Ci)-c_i∙q_i(c_i)- [ q_i(c_i) dcλ f_i(c_i) dc_i

*=ι^jc< ×                       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, c_i : O ≤ ¾(c_i) ≤ gf^z.

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. F_i(c_i)∕∕_i(c_i) is weakly increasing on [cf, cf] for

²By IR s_i(c_i) - c_i - ρ_i(c_i) > O. Now if s_i(c_i) - c_i ∙ g_i(c_i) = ε > O, we also have
s_i(c_i) — ɑt^, 9i(c⅛) ≥ ε Vc_i < 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 c_i ≤ c_i. This would not affect the IC constraints.

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