The Advantage of Cooperatives under Asymmetric Cost Information



In particular, di > c" => ¾(c<) ≤ <7l(cz∕), as claimed in the proposition. By
<7,(∙) monotonously decreasing, it follows also that
dqi(di)∕dc,i exists almost
everywhere (a.e.), cf. LafFont and Tirole(1993) p. 63.

Rewriting (2), we get that for all di and d- with di > 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' ^^ Ci - cj' - di - c"

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

d(¾(ci)-<⅞∙¾(cj))

-----J------- -QiW ≤ θ «•

This shows that the less efficient types earn less profit and it implies
fc? _ _

4^ ɑi ʧi(ɛi) d* J Qi[Ci) dci
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 + ci ∙ ςi(ci) + / qi(ci) d⅛ - ci qi(ci) >

Γci _

ki + ⅛∙qi(di) + J qi(ci) dci - ci ∙ qi(c,i) ∀i,ci,c'.

Reducing and rewriting, we get

,c<z                     ∕*< __                  Γci

/ qdci)dci> / ρi(c')dci+ / qi(ci)dci Vi,ci,di
Jci                    Jci                  Jcti

which holds because qi{.) 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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