ps3_s_mn.gms : Parts Supply Problem w/ 3 Types w/ Monotonicity Constraint

**Description**

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, monotonicity

**References**

- Hashimoto, H, Hamada, K, and Hosoe, N, A Numerical Approachto the Contract Theory: The Case of Adverse Selection. GRIPS Discussion Papers, National Graduate Institute for Policy Studies, 2012.
- Itoh, H, A Course in Contract Theory. Yuhikaku, Tokyo, 2003.

**Small Model of Type :** NLP

**Category :** GAMS Model library

**Main file :** ps3_s_mn.gms

```
$title Parts Supply Problem w/ 3 Types w/ Monotonicity Constraint (PS3_S_MN,SEQ=366)
$onText
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, monotonicity
$offText
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"
mn(i) "monotonicity 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);
mn(i).. x(i) =g= x(i+1);
* Setting Lower Bounds on Variables to Avoid Division by Zero
x.lo(i) = 0.0001;
Model SB4 / all /;
solve SB4 maximizing Util using nlp;
* The Case w/ alternative p(i)
p("0") = 0.30;
p("1") = 0.10;
p("2") = 0.60;
solve SB4 maximizing Util using nlp;
* The Case w/ alternative theta(i)
* Assumning the original p(i)
p("0") = 0.20;
p("1") = 0.50;
p("2") = 0.30;
* Assumning alternative theta(i)
theta("0") = 0.10;
theta("1") = 0.30;
theta("2") = 0.31;
solve SB4 maximizing Util using nlp;
```