The Advantage of Cooperatives under Asymmetric Cost Information

3 Central Planner’s Solution

We first characterize the set of production and payment plans that are socially
optimal. We shall talk about this as a central planner’s solution. What
defines the central planner (CP) is his objective - his aim is to maximize the
market value minus the production costs of all farmers, i.e. the integrated
profit from production and processing

n                         n

G^cp(q, s) = ∑ E_c ((p - c_i) ∙ _φ(c)) = 52 ¾_l((p - c_i) ■ φ(c_i))
i=l                          i=l

The central planner is assumed to have no more information about the costs
of any farmer than does the other farmers or an investor-owned processor.
Therefore the general design problem from Section 2 is still relevant.

We see that the central planner’s objective - as the constraints - depends
only on average production and payments. Also, the objective as the con-
straints are effectively separable in n farmer specific problems.

To maximize the net benefits from production and processing, the central
planner would like to implement the following production plan

Note that for the average production to be either the minimal or the maximal,
0 or çf ,the specific production level for all possible cost values must be either
0 or ⅛^z,i.e. gf^p(c) ~ ρf^p(c_i) ∀i,c. This plan is the first best plan, i.e. the
optimal production plan with perfect cost information.

To show that this ideal solution is actually feasible, we must specify the
payment plan that make the production-payment plan satisfy the IR, IC and
BB constraints. However, this is easy. Using Proposition 1, we know that to
be incentive compatible, the expected payment must satisfy

sf^j5(<⅛) = ⅛ +1⅛ ■ 9f^p(t⅛) + [ qf^p(ε_i)dci

Jc_i

( k_i+p q^  if c_i <p<$

~ < fc_i ⅛ c_l^c7 ∙     if C_i < cf ≤ p      Vi

I ⅛_i            otherwise

To be individually rational, we furthermore need fc_i ≥ 0.

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