$title Parts Supply Problem w/ 3 Types w/ Asymmetric Information (PS3_S,SEQ=364)
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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.
http://www.grips.ac.jp/r-center/en/discussion_papers/11-27/
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' / 0, 1, 2 /;
Alias (i,j);
Parameter
theta(i) 'efficiency' / 0 0.1, 1 0.2, 2 0.3 /
p(i) 'probability of type' / 0 0.2, 1 0.5, 2 0.3 /;
Scalar ru 'reservation utility' / 0 /;
* Definition of Primal/Dual Variables
Positive Variable
x(i) "quality"
b(i) "maker's revenue"
w(i) "price";
Variable Util "maker's utility";
Equation
obj "maker's utility function"
rev(i) "maker's revenue function"
pc(i) "participation constraint"
licd(i) "incentive compatibility constraint";
obj.. Util =e= sum(i, p(i)*(b(i) - w(i)));
rev(i).. b(i) =e= x(i)**(0.5);
pc(i).. w(i) - theta(i)*x(i) =g= ru;
licd(i).. w(i) - theta(i)*x(i) =g= w(i+1) - theta(i)*x(i+1);
* Setting Lower Bounds on Variables to Avoid Division by Zero
x.lo(i) = 0.0001;
Model SB3 / all /;
solve SB3 maximizing Util using nlp;