## Contract Theory

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### 資料紹介

Contract Theory
Let us set up the problem as follows. The set of supplier’s type;Q = fq0;q1g, where 0< q0 < q1. Here supplier
of typeq0 is superior, and supplier of typeq1 is inferior. And assume that the true type of supplier is only known to
supplier himself, that is, it is a private information of supplier. Maker does not know whichq0 orq1 supplier is but
he knows the probability, which are known to both maker and supplier.
Prob(q = q0) = p;Prob(q = q1) = 1 p
Timing is as follows, whereci(x) = qix(i = 0;1)is the cost for production, andx2 X = [0;¯x]is the quality ofb().
Maker
w: payment!
b(x): production
Supplier withci(x)
The utility functions of both are given by,
b(x) w :for maker;
Ui(x;w) = w ci(x) :for supplier:
(1)
Then let us change the timing into the one including contract, where
m:M ! X R,d(m)2 X is quality and
r(m)2 R is price.
Maker
m2 M
Supplier
+
Maker
!
m(m) = (d(m);r(m))
Instruct quality and price
Supplier
The problem for worker are
(i) What kind of report does he demand to supplier? That is, how does he chooseM ?
(ii) What kind of instruction does he do, depending on reportm? That is, how does he choose
m() = (d();r())?
So that maker considers(M;m); mechanism design.
1.1 Symmetric information case as Bench mark
If type of supplierq is not private information of supplier, and if it is known to both supplier and maker, then it
is obvious that the desirable quality and price depend on the type of supplier. Let us denote t

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Contract Theory
Let us set up the problem as follows. The set of supplier’s type;Q = fq0;q1g, where 0< q0 < q1. Here supplier
of typeq0 is superior, and supplier of typeq1 is inferior. And assume that the true type of supplier is only known to
supplier himself, that is, it is a private information of supplier. Maker does not know whichq0 orq1 supplier is but
he knows the probability, which are known to both maker and supplier.
Prob(q = q0) = p;Prob(q = q1) = 1 p
Timing is as ..

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