The Advantage of Cooperatives under Asymmetric Cost Information

third set of constraints are the budget balancing (BC) constraints. They say
that the farmers can not get more than what is earned at the market place.¹

The incentive compatibility constraints induce considerable structure on
the production and payment plans. For an arbitrary production and payment
scheme for fanner г, ¾(c) and s_l(c), let <j_t(c_t) and s_i(c_i) be the corresponding
conditional expected production and payment when the types of the other
farmers have been integrated out, i.e.

¾(c_j) := ¾ (¾(c_i,c-_j)) Vc_i
S_i(c_i) := ¾(s_j(c_i,c-_i)) Vc_i

We now have the following useful proposition.

Proposition 1 The production and payment schemes ρ(.) = (ς⅛(.))i∈j and
s(.) = (s_i(.))_ie/ are incentive compatible if and only if

• c^,_t > c" ≠> q_i(d_i) ≤ q_i(<ff) Vi,⅛(f_i, and

• s_i(c_i) = k_i + c_i ∙ q_i(c_i) + q_i(c_i) dc_i ∀t,c_i

Proof. Initially, we note that by independence, the conditional expectar
tion operator E£_. (.) does not depend on the specific value of the costs c⅛.
Therefore, the incentive compatibility constraints are equivalent to

¾(c_i) - c_i ■ q_i(c_i) ≥ ¾(⅛) - c_i ∙ ¾(ξ) ∀i, c_i, ξ.             (1)

To show the two properties in the proposition, we consider a given i and note
that (1) for arbitrary c^f_i and c" with d_i > ^implies

¾(c^,_i) - _c; ∙ ⅛(ξ) ≥ _s-_i(_c") - < ∙ ¾(<) = s-_i(<) - < ■ ¾(<) + « -₁⅛)¾(<)
Si(<) - < ∙ ¾(<⅞') ≥ ¾(⅛) - c" ∙ ¾(c^,_i) = 5i(c!) - C^,_i ■ ¾(c') + (ξ - e'')g,(eɔ

or equivalently

. <J'X ⅛⅛ - ⅛ ’ ft(⅛) - (*(<) - 4' ■ ¾W))      ∕√x /nʌ

-ft(ɑi ) ≤ --------------^⅛-⅛-------------- ~ ~¾(^ci)    V²)

¹We assume that the incentive problem is related to the cost types only. The planned
production levels can be implemented without additional incentive problems eg because
the chosen production levels are directly verifiable such that deviations can be avoided
with infinitely harsh punishment treats. We therefore do not need to let s∣(.) depend on
the actual production levels.

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