ps3_s_scp.gms : Parts Supply Problem w/ 3 Types w/o and w/ SCP

**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

**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_scp.gms

```
$title Parts Supply Problem w/ 3 Types w/o & w/ SCP (PS3_S_SCP,SEQ=367)
$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
$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 0.4, 2 0.9 /
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"
ic(i,j) "incentive compatibility constraint"
licd(i) "incentive compatibility constraint"
licu(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)+(1 - theta(i) + sqr(theta(i)))*x(i)) =g= ru;
ic(i,j).. w(i) - (theta(i) + (1 - theta(i) + sqr(theta(i)))*x(i))
=g= w(j) - (theta(i) + (1 - theta(i) + sqr(theta(i)))*x(j));
licd(i).. w(i) - (theta(i) + (1 - theta(i) + sqr(theta(i)))*x(i))
=g= w(i+1)- (theta(i) + (1 - theta(i) + sqr(theta(i)))*x(i+1));
licu(i).. w(i) - (theta(i) + (1 - theta(i) + sqr(theta(i)))*x(i))
=g= w(i-1)- (theta(i) + (1 - theta(i) + sqr(theta(i)))*x(i-1));
* Setting Lower Bounds on Variables to Avoid Division by Zero
x.lo(i) = 0.0001;
Model
SB_gic_wo_SCP / obj, rev, pc, ic /
SB_lic_wo_SCP / obj, rev, pc, licd, licu /;
solve SB_gic_wo_SCP maximizing Util using nlp;
solve SB_lic_wo_SCP maximizing Util using nlp;
```