$title Parts Supply Problem w/ Efficient Type w/o Asymmetric Information (PS2_F_eff,SEQ=360)
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Hideo Hashimoto, Kojun Hamada, and Nobuhiro Hosoe, "A Numerical Approach
to the Contract Theory: the Case of Adverse Selection", GRIPS Discussion
Paper 11-27, National Graduate Institute for Policy Studies, Tokyo, Japan,
March 2012.
Keywords: nonlinear programming, contract theory, principal-agent problem,
adverse selection, parts supply problem
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option limCol = 0, limRow = 0;
Set i 'type of supplier' / eff /;
Parameter theta(i) 'efficiency' / eff 0.2 /;
* Definition of Primal/Dual Variables
Positive Variable
x(i) "quality"
b(i) "maker's revenue"
c(i) "cost";
Variable Util "maker's utility";
Equation
obj "maker's utility function"
rev(i) "maker's revenue function"
pc(i) "participation constraint";
obj.. Util =e= sum(i, (b(i) - c(i)));
rev(i).. b(i) =e= x(i)**(0.5);
pc(i).. c(i) - theta(i)*x(i) =e= 0;
* Setting Lower Bounds on Variables to Avoid Division by Zero
x.lo(i) = 0.0001;
Model FB1 / all /;
solve FB1 maximizing Util using nlp;
Parameter
db(i) 'derivative of b'
w(i) 'price';
db(i) = 0.5*x.l(i)**(-0.5);
w(i) = c.l(i);
display x.l, b.l, c.l, util.l, db, w;