The Advantage of Cooperatives under Asymmetric Cost Information

In particular, d_i > c" => ¾(c<) ≤ <7_l(c^z∕), as claimed in the proposition. By
<7,(∙) monotonously decreasing, it follows also that dq_i(d_i)∕dc^,_i exists almost
everywhere (a.e.), cf. LafFont and Tirole(1993) p. 63.

Rewriting (2), we get that for all d_i and d- with d_i > d-

-(ξ - <)⅛(<) < SiW - SiW) + < ■ ¾(<) - ⅛ ∙ QiW ≤-Wi - <)¾(ξ)
or equivalently

<⅞(¾(4) - QiW)} _< ⅝(cj) - ¾(<⅜') _< <(¾(g^,.) - ¾(⅛))
Cj - ɑi' ^^ C_i - cj' - d_i - c"

We see therefore that since ⅛(c^z_l)∕dc^z_l exists (a.e.), so does dsɪ(eɔ/dej. Fur-
thermore, going to the limit (d_i → eɔ in (2), we get that

d(¾(c_i)-<⅞∙¾(c_j))

-----J------- -QiW ≤ θ «•

This shows that the less efficient types earn less profit and it implies
f^c? _ _

4^ ɑi ʧi(ɛi) d* J Qi[Ci) dc_i
which is the last property in the proposition.

We shall now show that the two properties in Proposition 1 implies in-
centive compatibility. Inserting the expression for ¾(.) into the incentive
compatibility constraint (1) we get

A⅛ + c_i ∙ ς_i(ci) + / q_i(c_i) d⅛ - c_i ■ q_i(c_i) >

Γ^ci _

k_i + ⅛∙q_i(d_i) + J q_i(c_i) dc_i - c_i ∙ q_i(c^,_i) ∀i,c_i,c'.

Reducing and rewriting, we get

∕^,c<^z                     ∕*< __                  Γ^ci

/ qdc_i)dc_i> / ρ_i(c')dc_i+ / q_i(c_i)dc_i Vi,c_i,d_i
Jc_i                    Jc_i                  Jc^t_i

which holds because q_i{.) is weakly decreasing. ■

According to Proposition 1, the less efficient types produce less. Also,
the expected payment is - up to an integration constant - determined en-
tirely from the production scheme. Proposition 1 makes it easy to analyze
alternative organizations as we shall see below.

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