The assumption that types are independent is necessary to prove Propo-
sition 1. Specifically, (1) presumes independence. If the types are correlated,
a social planner could undermine the informational advantage of the agents
by comparing their messages. By paying most when an agent’s message is
likely given the messages of the other agents, the planner could reduce the
payment to the agents. With perfectly correlated types, it would suffice to
pay the true costs in all cases. Hence, with correlated costs, the cooperative
solution is but one possibility to get the first best outcome. Indeed, with
perfectly correlated costs, an IO processor would also generate the first best
outcome.
One way to relax the independence assumption without changing our
main results is to work with a refined set of IR constraints
si(c) - ci ■ qi(c) ≥0 ∀c,t
i.e. by assuming that the farmers must never end up with a negative cash-
flow. The stronger IR constraints can be interpreted as limited liability
constraints, safety first constraints, or as the usual participation constraints
coupled with extreme risk aversion (prohibiting negative cash flows). Using
the stronger IR constraints, and assuming that the joint distribution of types
has support C = ×t∈∕Ci, we get basically the same propositions as above
for the central planner and the cooperative - but we get it without using
Proposition 1. This is not difficult to prove: Assume that p < ejʃ. Tb be
socially optimal, we need farmer i to produce as long as c⅛ ≤ p. The c⅛ = p
type of farmer i must therefore be paid at least p per unit and since all the
more efficient types (by the full support condition) can imitate this type, they
must all be paid at least p per unit. The budget balancing constraint now
gives that they must be paid exactly p. Of course, there may still be room for
some zero mean lotteries. An IO processor will still ration production since
otherwise he will earn zero profit. Hence, in this case the socially optimal
outcome is accomplished by a cooperative and - modula some zero mean
lotteries - by a cooperative only.
The assumed constancy of the average net revenue product p is an as-
sumption that the processor has no market power and that there are no scale
economies in the processing (or less realistically, that these effects even out).
Our conclusions are sensitive to this assumption. Relaxing it may destroy
the cooperative’s ability to give the first best production levels. Truly, first
best production level may not be possible under any arrangement when p
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