$title Parts Supply Problem (PARTSSUPPLY,SEQ=404) $onText This model is based on the ps2_f_s.358 .. ps10_s_mn.396 models by Hideo Hashimoto, Kojun Hamada, and Nobuhiro Hosoe. Using the following options, these models can be run: ps2_f : default ps2_f_eff : --nsupplier=1 ps2_f_inf : --nsupplier=1 --alttheta=1 ps2_f_s : --useic=1 ps2_s : --useic=1 ps3_f : --nsupplier=3 ps3_s : --nsupplier=3 --uselicd=1 ps3_s_gic : --nsupplier=3 --useic=1 ps3_s_mn 1st solve: --nsupplier=3 --uselicd=1 2nd solve: --nsupplier=3 --uselicd=1 --altpi=1 3rd solve: --nsupplier=3 --uselicd=1 --alttheta=1 ps3_s_scp 1st solve: --nsupplier=3 --alttheta=2 --modweight=1 --useic=1 2nd solve: --nsupplier=3 --alttheta=2 --modweight=1 --uselicd=1 --uselicu=1 ps5_s_mn : --nsupplier=5 --uselicd=1 --nsamples=1000 ps10_s : --nsupplier=10 --uselicd=1 ps10_s_mn : --nsupplier=10 --uselicd=1 --nsamples=1000 Alternatively, the corresponding original model files can be found in the GAMS model library. Keywords: nonlinear programming, contract theory, principal-agent problem, adverse selection, parts supply problem $offText $if not set nsupplier $set nsupplier 2 $if not set modweight $set modweight 0 $if not set useic $set useic 0 $if not set uselicd $set uselicd 0 $if not set uselicu $set uselicu 0 $if not set usemn $set usemn 0 $if not set altpi $set altpi 0 $if not set alttheta $set alttheta 0 $if not set nsamples $set nsamples 1 Set i 'type of supplier' / 1*%nsupplier% / t 'Monte-Carlo draws' / 1*%nsamples% /; Alias (i,j); Parameter theta(i) 'efficiency' pt(i,t) 'probability of type' p(i) 'probability of type for currently evaluated scenario' icweight(i) 'weight in ic constraints'; Scalar ru 'reservation utility' / 0 /; * Data $ifThen %nsupplier% == 1 $if %alttheta% == 0 Parameter theta(i) / 1 0.2 /; $if %alttheta% == 1 Parameter theta(i) / 1 0.3 /; Parameter p(i) / 1 1 /; $elseIf %nsupplier% == 2 Parameter theta(i) / 1 0.2, 2 0.3 / p(i) / 1 0.2, 2 0.8 /; $elseIf %nsupplier% == 3 $if %alttheta% == 0 Parameter theta(i) / 1 0.1, 2 0.2, 3 0.3 /; $if %alttheta% == 1 Parameter theta(i) / 1 0.1, 2 0.3, 3 0.31 /; $if %alttheta% == 2 Parameter theta(i) / 1 0.1, 2 0.4, 3 0.9 /; $if %altpi% == 0 Parameter p(i) / 1 0.2, 2 0.5, 3 0.3 /; $if %altpi% == 1 Parameter p(i) / 1 0.3, 2 0.1, 3 0.6 /; $else theta(i) = ord(i)/card(i); p(i) = 1/card(i); $endIf loop(t, pt(i,t) = uniform(0,1)); pt(i,t) = pt(i,t)/sum(j, pt(j,t)); $if %nsamples% == 1 pt(i,t) = p(i); 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" mn(i) "monotonicity constraint"; obj.. Util =e= sum(i, p(i)*(b(i) - w(i))); rev(i).. b(i) =e= sqrt(x(i)); pc(i).. $if %modweight% == 0 w(i) - theta(i) *x(i) =g= ru; $if %modweight% == 1 w(i) - icweight(i)*x(i) =g= ru + theta(i); ic(i,j).. w(i) - icweight(i)*x(i) =g= w(j) - icweight(i)*x(j); licd(i)$(ord(i) < card(i)).. w(i) - icweight(i)*x(i) =g= w(i+1) - icweight(i)*x(i+1); licu(i)$(ord(i) > 1).. w(i) - icweight(i)*x(i) =g= w(i-1) - icweight(i)*x(i-1); mn(i)$(ord(i) < card(i)).. x(i) =g= x(i+1); * Setting Lower Bounds on Variables to Avoid Division by Zero x.lo(i) = 0.0001; Model m 'parts supply model w/o monotonicity' / all - mn $if %useic% == 0 -ic $if %uselicd% == 0 -licd $if %uselicu% == 0 -licu /; Model m_mn 'parts supply model w/ monotonicity' / m + mn /; * Parameters to store some solution values Parameter Util_lic(t) 'util solved w/o MN' Util_lic2(t) 'util solved w/ MN' x_lic(i,t) 'x solved w/o MN' x_lic2(i,t) 'x solved w/ MN'; * Solving the Model option limRow = 0, limCol = 0; loop(t, p(i) = pt(i,t); icweight(i) = theta(i)$(not %modweight%) + (1 - theta(i) + sqr(theta(i)))$(%modweight%); solve m maximizing Util using nlp; Util_lic(t) = util.l; x_lic(i,t) = x.l(i); solve m_mn maximizing Util using nlp; Util_lic2(t) = util.l; x_lic2(i,t) = x.l(i); option solPrint = off; ); $if %nsamples% == 1 $exit * Evaluation and display results as in ps5_s_mn Parameter MN_lic(t) 'monotonicity of x solved w/o MN' MN_lic2(t) 'monotonicity of x solved w/ MN' Util_gap(t) 'gap between Util_lic and Util_lic2' F(i,t) 'cumulative probability (Itho p. 42)' noMHRC0(i,t) 'no MHRC combination between i and i-1 (MHRC: monotone hazard rate condition)' noMHRC(t) '>=1: no MHRC case' p_noMHRC 'no MHRC case [%]' p_noMN_lic 'no MN case [%]' p_Util_gap 'no util-equality case [%]'; MN_lic(t) = sum(i, 1$(round(x_lic (i,t),10) < round(x_lic (i+1,t),10))); MN_lic2(t) = sum(i, 1$(round(x_lic2(i,t),10) < round(x_lic2(i+1,t),10))); Util_gap(t) = 1$(round(Util_lic(t),10) <> round(Util_Lic2(t),10)); F(i,t) = sum(j$(ord(j) <= ord(i)), pt(j,t)); noMHRC0(i,t)$(ord(i) < card(i)) = 1$(F(i,t)/pt(i+1,t) < F(i-1,t)/pt(i,t)); noMHRC(t)$(sum(i, noMHRC0(i,t)) >= 1) = 1; * Computing probability that MHRC and MN holds. p_noMHRC = sum(t$(noMHRC(t) > 0), 1)/card(t)*100; p_noMN_lic = sum(t$(MN_lic(t) > 0), 1)/card(t)*100; p_Util_gap = sum(t$(Util_gap(t) > 0), 1)/card(t)*100; display p_noMHRC, p_noMN_LIC, p_Util_gap; * 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 /;