ps10_s_mn.gms : Parts Supply Problem w/ 10 Types w/ Random p(i)

**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. <a href="http://r-center.grips.ac.jp/DiscussionPapersDetails/247/#">http://r-center.grips.ac.jp/DiscussionPapersDetails/247/#</a>

**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 :** ps10_s_mn.gms

$Title Parts Supply Problem w/ 10 Types w/ Random p(i) (PS10_S_MN,SEQ=369) * Hideo Hashimoto, Kojun Hamada, and Nobuhiro Hosoe, "A Numerical Approach * to the Contract Theory: the Case of Adverse Selection", GRIPS Discussion * Paper 11-27, National Graduate Institute for Policy Studies, Tokyo, Japan, * March 2012. * * http://r-center.grips.ac.jp/DiscussionPapersDetails/247/# Option limcol=0,limrow=0,solprint=off; * Definition of Set Set i type of supplier /0*9/ t no. of Monte-Carlo draws /1*1000/; Alias (i,j); * Definition of Parameters Parameter theta(i) efficiency pt(i,t) probability of type p(i) probability of type; theta(i) =ord(i)/card(i); * Generating probability loop(t,pt(i,t) =uniform(0,1);); pt(i,t) =pt(i,t)/sum(j,pt(j,t)); *pt(i,"1")=1/card(i); Parameter F(i,t) "cumulative probability (Itho p. 42)" noMHRC0(i,t) "no MHRC combination between i and i-1" * (MHRC: monotone hazard rate condition) noMHRC(t) ">=1: no MHRC case"; F(i,t) =sum(j$(ord(j) le ord(i)), pt(j,t)); noMHRC0(i,t)$(ord(i) lt card(i)) =1$(F(i,t)/pt(i+1,t) lt F(i-1,t)/pt(i,t)); noMHRC(t)$(sum(i, noMHRC0(i,t)) ge 1)=1; Scalar ru reservation utility /0/; * Definition of Primal/Dual Variables Positive Variable x(i) quality b(i) maker's revenue w(i) price; Variable Util maker's utility; Equation obj maker's utility function rev(i) maker's revenue function pc(i) participation constraint licd(i) incentive compatibility constraint licu(i) incentive compatibility constraint ic(i,j) global incentive compatibility constraint mn(i) monotonicity constraint; * Specification of Equations obj.. Util =e= sum(i, p(i)*(b(i)-w(i))); rev(i)..b(i) =e= x(i)**(0.5); pc(i).. w(i)-theta(i)*x(i) =g= ru; licd(i)..w(i)-theta(i)*x(i) =g= w(i+1)-theta(i)*x(i+1); licu(i)..w(i)-theta(i)*x(i) =g= w(i-1)-theta(i)*x(i-1); ic(i,j)..w(i)-theta(i)*x(i) =g= w(j)-theta(i)*x(j); mn(i).. x(i) =g= x(i+1); * Setting Lower Bounds on Variables to Avoid Division by Zero x.lo(i)=0.0001; * Defining and Solving the Model Model SB_lic /obj,rev,pc,licd/; Model SB_lic2 /obj,rev,pc,licd,mn/; * Options to solve models quickly SB_lic.solvelink=5; SB_lic2.solvelink=5; Parameter Util_lic(t) "Util solved w/o MN" Util_lic2(t) "Util solved w/ MN" Util_gap(t) "Gap between these two Util" x_lic(i,t) "x solved in w/o MN" x_lic2(i,t) "x solved in w/ MN" MN_lic(t) "Monotonicity of x solved w/o MN" MN_lic2(t) "Monotonicity of x solved w/ MN" loop(t, p(i)=pt(i,t); * Solving the model w/o MN Solve SB_lic maximizing Util using NLP; Util_lic(t)=util.l; x_lic(i,t)=x.l(i); MN_lic(t)=sum(i, 1$(round(x.l(i),10) lt round(x.l(i+1),10))); * Solving the model w/ MN Solve SB_lic2 maximizing Util using NLP; Util_lic2(t)=util.l; x_lic2(i,t)=x.l(i); MN_lic2(t)=sum(i, 1$(round(x.l(i),10) lt round(x.l(i+1),10))); ); Util_gap(t)=1$(round(Util_lic(t),10) ne round(Util_Lic2(t),10)); * Computing probability that MHRC and MN holds. Parameter p_noMHRC no MHRC case [%] p_noMN_lic no MN case [%] p_Util_gap no Util-equality case [%] ; p_noMHRC =sum(t$(noMHRC(t) gt 0), 1) /card(t)*100; p_noMN_lic =sum(t$(MN_lic(t) gt 0), 1) /card(t)*100; p_Util_gap =sum(t$(Util_gap(t) gt 0), 1) /card(t)*100; Display p_noMHRC,p_noMN_LIC,p_Util_gap; * End of Model * Generating CSV file for summary File sol /solution_lic.csv/; put sol; sol.pc=5; sol.pw=32767; put ""; loop(i, put "pt(i,t)";); put "" "" "" ""; loop(i, put "x: w/o MN";); loop(i, put "x: w/ MN";); put /; put ""; loop(i, put i.tl;); put ">=1: no MHRC" "Util: w/o MN" "Util: w/ MN" "Util_gap: =1: not equal"; loop(i, put i.tl;); loop(i, put i.tl;); put "MN_lic: >=1: no MN" "MN_lic2: >=1: no MN"/; loop(t, put t.tl; loop(i, put pt(i,t):10:5;); put noMHRC(t) Util_lic(t):20:10 Util_Lic2(t):20:10 Util_gap(t); loop(i, put X_lic(i,t);); loop(i, put X_lic2(i,t);); put MN_lic(t) MN_lic2(t)/;); put /; * End of CSV file generation