A Numerical Approach to the Contract Theory: the Case of Adverse Selection

Transcription

1 GRIPS Dscusson Paper A Numercal Approach to the Contract Theory: the Case of Adverse Selecton Hdeo Hashmoto Kojun Hamada Nobuhro Hosoe March 2012 Natonal Graduate Insttute for Polcy Studes Roppong, Mnato-ku, Tokyo, Japan

2 GRIPS Polcy Research Center Dscusson Paper : A Numercal Approach to the Contract Theory: the Case of Adverse Selecton Hdeo Hashmoto *, Kojun Hamada, Nobuhro Hosoe Abstract: By buldng and solvng numercal models of the parts supply problems (an example of the adverse selecton problems), and analyzng varous ssues of the contract theory, we demonstrate the benefts of the numercal approach. Frst, ths approach facltates the understandng of the contract theory by begnners, who fnd t dffcult to comprehend the theoretcal and general models. Second, ths approach could extend the analyss areas beyond those of the theoretcal models, whch are lmted by the smplfyng assumptons mposed n order to make ther analyss possble. The expanson of the number of the suppler types s one example. Keywords: Numercal approach; prncpal-agent problem; adverse selecton; numercal and computatonal model; Spence-Mrrlees sngle crossng property; monotoncty * Professor Emertus of Osaka Unversty Faculty of Economcs, Ngata Unversty Correspondng author. Natonal Graduate Insttute for Polcy Studes, Roppong, Mnato, Tokyo E-mal: nhosoe@grps.ac.jp. Page 1

3 GRIPS Polcy Research Center Dscusson Paper : A Numercal Approach In ths paper, we buld and solve varous numercal models of the parts supply problems as an example of the adverse selecton problems for the followng two purposes. Frst, the paper ams to facltate begnners of the contract theory to understand t wth the numercal models. Snce many earler studes have developed the contract theory models n a theoretcal and general manner, the begnners often encounter dffcultes of understandng them. Second, we demonstrate some advantages of the numercal approach over that theoretcal approach. For example, we can easly assume a larger number of the suppler s types than two or three, whch s often assumed n theoretcal studes. Furthermore, we can extend the analyss area that s lmted by some smplfyng assumptons used n those theoretcal studes. In addton by applyng the Monte Carlo smulaton method, we can nvestgate the lkelhood that those assumptons hold. To understand the numercal models fully, the readers must know the computer programs, whch wll be explaned n detals n the earler part of ths paper. To analyze the parts supply problems, we formulate the models as non-lnear programmng problems. Then, by specfyng functonal forms and feedng some numbers nto parameters, we buld and solve the numercal computer models. In ths numercal approach, we obtan not only the optmal values of the prmal varables, such as the parts qualty and the prce to be pad to the suppler, but also the optmal values of the Lagrange multplers of the mposed constrants. The nformaton rent, whch s the most mportant varable n problems wth asymmetrc nformaton, can be derved drectly from the solved values of some Lagrange multplers. Furthermore, we can easly conduct comparatve statcs by changng the values of some parameters. To buld and solve the numercal models, we use GAMS (General Algebrac Page 2

4 GRIPS Polcy Research Center Dscusson Paper : Modelng System) as computaton software. 1 Whle n ths paper we explan the essental GAMS syntax, refer to Hosoe et al. (2010, Chapter 3) for more detaled explanaton. The readers can realze the contract theory well, by playng around wth the models presented n ths paper,.e., changng the values of some parameters n our sample programs wth use of the tral verson of GAMS. On one hand, of course, we must be careful n dervng any general conclusons from numercal models whch are based on specfc functonal forms and assumed parameters. In addton, t s not an easy job to master a new computer language to deal wth numercal models freely. On the other hand, our approach to develop numercal models, whch depct the essence of the theoretcal models, wll afford the readers ease n comprehendng the contract theory. The authors hope the readers fnd the heurstc usefulness n ths paper. Ths paper orgnates from our prevous paper by Hashmoto et al. (2011) wrtten n Japanese where varous numercal models of the parts supply problems are developed, based on the theoretcal models by Itoh (2003). We also use the analytcal frameworks demonstrated by Itoh (2003) when we develop our numercal examples. 2. Adverse Selecton: the Parts Supply Problem The parts supply problems are the basc problems of adverse selecton. A maker (a prncpal, called wth a female pronoun), makng a unt of a fnal product, purchases ts parts from a suppler (an agent, called wth a male pronoun) for her producton. There are several techncal types (effcent or neffcent, etc.) of the suppler. Only the suppler knows about hs type (.e., prvate nformaton). Under these crcumstances, the maker s supposed 1 GAMS s commercal software; however, ts tral verson can be downloaded from the webste of GAMS Development Corporaton ( and used wthout charge. The numercal models presented n ths paper are so small that they can be solved wth the tral verson. Page 3

5 GRIPS Polcy Research Center Dscusson Paper : to offer a contract that maxmzes her utlty. The contract that the maker offers s the mechansm to determne the qualty of the parts x to be produced by the -th type suppler and the prce w to be pad to hm, n such a way as to maxmze the maker s utlty, based on the suppler s report about hs type. If, hypothetcally, the maker knew the suppler s type (symmetrc nformaton), there were no possblty that the suppler would report hs type falsely; thus, the frst-best would be realzed. However, n realty, the maker does not know the suppler s type (asymmetrc nformaton) and, thus, must offer a contract n such a way as to make the suppler obtan no extra gan, even f he should report hs type falsely. Such a contract s the second-best optmal n the sense that t mnmzes the neffcency resulted from asymmetrc nformaton. As the maker must be compromsed wth the suppler explotng hs asymmetrc nformaton, her second-best utlty level s lower than the frst-best one. The suppler s type n our models s a dscrete varable. In Secton 2.1, we consder the cases wth two suppler types (effcent and neffcent). We solve the frst-best equlbrum by assumng no asymmetrc nformaton as the benchmark case. Next, we present the second-best model, where the (global) ncentve compatblty condtons are needed to prevent the suppler from reportng hs type falsely. Subsequently, we consder the cases wth three suppler types n Sectons 2.2 and 2.3. When the number of types ncreases from two to N n general, the ncentve compatblty condtons become complcated. Because the number of possble combnatons that the -th type suppler reports hs type truly or falsely as type j ncreases, that of ncentve compatblty condtons to prevent hs false reportng ncreases rapdly. Addtonal assumptons the Spence-Mrrlees sngle crossng property (SCP) and monotoncty (MN) are ntroduced to smplfy these condtons. (The SCP and monotoncty wll be explaned n detals n Secton ) These smplfyng assumptons make the analyss of the theoretcal contract models possble. In Secton 2.2, we examne the smplest models wth these two assumptons. In Secton 2.3, we extend the models n two drectons. Frst, we ncrease the number of Page 4

6 GRIPS Polcy Research Center Dscusson Paper : suppler types. It can be done easly wthout complcated programmng technques. Second, we deal wth the models where these assumptons of the SCP and MN do not hold. Ths also can be done wthout causng techncal dffcultes. Fnally, n Secton 2.4, applyng the Monte Carlo smulaton method, we estmate the lkelhood that such assumptons as the SCP and MN hold, and dscuss the mportance of these assumptons n a contract theory analyss. 2.1 The 2-type Models We consder two suppler types (effcent and neffcent). In the former part of ths subsecton, we develop the frst-best models, where the maker knows the suppler s type,.e., hs margnal cost of the parts producton. For benefts of those unfamlar to GAMS computer models, we frst present the model only wth the effcent suppler; then, the one only wth the neffcent suppler separately. Next, we combne these two models nto one model that ncludes both types. In the latter part, we present the second-best model, where the maker does not know the suppler s type. Then, we develop one model that ncludes both the frst-best and the second-best. Fnally, we compare the second-best solutons wth the frst-best solutons The Frst-best Model wth Only One Suppler Type The frst-best model (or, the benchmark model) of the parts supply problem s for the maker to offer a contract that determnes the parts qualty x to be produced by the suppler and the prce w to be pad to hm, knowng the suppler s margnal cost, theta(). In the frst, we buld a model of the case that the maker knows the suppler s effcent, and n the second a model of the case that the maker knows the suppler s neffcent. These models are bult separately. The models are to smply maxmze the maker s utlty under non-negatvty constrants on the decson varable x. Page 5

7 GRIPS Polcy Research Center Dscusson Paper : The Fst-Best Model wth the Effcent Suppler (PS2_F_eff.gms) 2 Let us go through the nput fle of the frst-best model wth the effcent suppler (Lst 2.1). The frst-best model when the suppler type s known to be effcent s expressed as the followng maker's utlty maxmzaton problem: subject to b x c x max Utl (2.1) 0. 5 b x x (2.2) x x 0 c (2.3) Lst 2.1: The Frst-best Model wth the Effcent Suppler (PS2_F_eff.gms) 7 * Defnton of Set 8 Set type of suppler /eff/; 9 * Defnton of Parameters 10 Parameter 11 theta() effcency /eff 0.2/; 12 * Defnton of Prmal/Dual Varables 13 Postve Varable 14 x() qualty 15 b() maker's revenue 16 c() cost; 17 Varable 18 Utl maker's utlty; 19 Equaton 20 obj maker's utlty functon 21 rev() maker's revenue functon 22 pc() partcpaton constrant; 23 * Specfcaton of Equatons 24 obj.. Utl =e= sum(, (b()-c())); 25 rev()..b() =e= x()**(0.5); 26 pc().. c()-theta()*x() =e= 0; 27 2 Those n the parentheses after the secton ttles show GAMS nput fle names shown n the followng part. Page 6

8 GRIPS Polcy Research Center Dscusson Paper : * Settng Lower Bounds on Varables to Avod Dvson by Zero 29 x.lo()=0.0001;; * Defnng and Solvng the Model 32 Model FB1 /all/; 33 Solve FB1 maxmzng Utl usng NLP; Parameter 36 db() dervatve of b 37 w() prce 38 ; 39 db() =0.5*x.l()**(-0.5); 40 w() =c.l(); Dsplay x.l, b.l, c.l, utl.l, db, w; 43 * End of Model In Lne 8, the suppler s type s expressed rght after the Set drectve as an ndex (.e., the suffx n ordnary algebrac equatons), such as =eff (the effcent suppler) and =nf (the neffcent suppler). As ths model has only the effcent suppler, the ndex can be omtted. However, the ndex s kept because the same model s used later as the model wth an neffcent suppler by replacng eff wth nf. The effcency of each suppler type, denoted by hs margnal cost of the parts theta() ( n the mathematcal model) s shown n Lne 11. In Lnes 13 18, the decson (endogenous) varables are declared. The decson varables are the qualty of suppler s parts x(), the maker s revenue receved from suppler s parts b(), and suppler s cost c(). And, as these varables must be non-negatve, they are declared wth the Postve Varable drectve. 3 The other endogenous varable s the value of the objectve functon, whose name s defned as Utl. As the sgn of the value of Utl s not known before solvng the model, t must be declared wth the Varable drectve so that ts value can be ether postve or negatve. Lnes show the names of the objectve functon (.e., the maker s utlty functon) and the 3 In GAMS, Postve Varable means non-negatve varable; thus, t can be zero. Page 7

9 GRIPS Polcy Research Center Dscusson Paper : constrants or the equatons (.e., revenue functon and cost functon). In GAMS programs, names must be gven to all equatons (ncludng the objectve functon). 4 The names of the objectve functon (2.1), revenue functon (2.2), and cost functon (2.3) are named as obj, rev(), and pc()n the program, respectvely. Lnes defne the equatons (ncludng the objectve functon), whch consttute the maxmzaton problem presented above. Lne 24 shows the objectve functon. 5 Lne 25 expresses the maker s revenue b, whch s assumed to be a smple concave 0. 5 functon b. Lne 26 shows the suppler s cost x x c as a lnear functon of x wth the constant margnal cost. In GAMS, =e= means strct equalty, * (unless t appears n the frst column n a lne) s multplcaton, and ** means power. In Lne 29, an arbtrary very small postve value s assgned as the lower bound of x() so as to avod computatonal problems (e.g., dvson by zero). If a soluton matches ths lower bound, we must reconsder the lower bound, because ths soluton may be affected by ths artfcally-set lower bound. Lne 32 gves the model name FB1 to the model consstng of all the equatons ncludng the objectve functon. Lne 33 s a statement to solve the model FB1 maxmzng Utl by usng a non-lnear programmng problem (NLP) solver. The lnes after Lne 34 are added for further analyses. The symbol db() means the value of the frst-order dervatve of b() wth respect to x() evaluated at the soluton of x(). Although the contract must nclude not only the parts qualty x() but also the 4 Whle we often use mechancal equaton names, such as Equaton 1, Equaton 2, etc. n mathematcal expressons, we can freely make names that suggest the meanngs of the equatons n GAMS. 5 The objectve functon s the weghted sum of the dfference between revenues and costs for all. The sum n algebrac equatons s expressed as sum(, ) n GAMS. The summaton, however, does not carry any sgnfcance here because we consder only one suppler type for. Page 8

10 GRIPS Polcy Research Center Dscusson Paper : prce w() to be pad to the t -th type suppler, the latter s not to be shown n the maxmzaton problem specfed above. Thus, the value of w() s computed n Lne 40 by beng smply equated to c(), followng the take-t-or-leave-t offer assumpton the maker has the full barganng power at the tme of the contract negotaton n thee frst-best model. (When the solved values are used for computaton,.l s attached to the varables, such as x.l(). Note that l n the suffx s not a numeral 1 but a Roman letter el. To use the solved values of the Lagrange multplers of constrants,.m s attached as, say, pc.m(). Whle the equalty symbol = =e= s usedd for equatons wthn the maxmzaton problem, the ordnary equalty symbol = = s used forr equatons outsde thee maxmzaton problem. The Dsplay statement n Lne 42 s to prnt the solutons ncludng those of the Lagrange multplers and the computed values of the equatons specfed n the lnes after Lne 34. Fnally, lnes startng wth * (astersk) are smply memos, whch do not affect maxmzaton or any computaton. These memos not only would facltate other people to understand the model contents but also could help the modelerss recall therr thought process. (As tme passes, the modelers themselves often cannot recall what w they have wrtten..) It s strongly recommended to wrte as muchh detaled memos as possble. When you make a computer program stated above by usng a software called GAMS IDE, clck the on the menuu bar; then, GAMS computes the models wrtten n the nput fle and generates the output fle. 6 The output fle s shown n Lst 2.2. Note that the numbers n the furthermost left column are put only for explanaton and are the same as n the nput fle. Those numbers should not be put n the nput fle, when you wrte t. Lst 2.2: Output Fle for the Frst-best Model for the Effcent Suppler S (PS2_F_eff.lst) 6 As for GAMS and GAMS IDE, refer to Hosoe et al. (2010, Chapter 3 and Appendx C), Brook et al. (2010), and McCarl (2009). Page 9

11 GRIPS Polcy Research Center Dscusson Paper : * Defnton of Set 3 8 Set type of suppler /eff/; 4 9 * Defnton of Parameters 5 10 Parameter 6 11 theta() effcency /eff 0.2/; 7 12 * Defnton of Prmal/Dual Varables 8 13 Postve Varable 9 14 x() qualty 11 S O L V E S U M M A R Y MODEL FB1 OBJECTIVE Utl 14 TYPE NLP DIRECTION MAXIMIZE 15 SOLVER CONOPT FROM LINE **** SOLVER STATUS 1 NORMAL COMPLETION 18 **** MODEL STATUS 2 LOCALLY OPTIMAL 19 **** OBJECTIVE VALUE ** Optmal soluton. Reduced gradent less than tolerance. 23 LOWER LEVEL UPPER MARGINAL EQU obj obj total surplus functon EQU rev maker's revenue functon LOWER LEVEL UPPER MARGINAL eff EQU pc partcpaton constrant LOWER LEVEL UPPER MARGINAL eff VAR x qualty LOWER LEVEL UPPER MARGINAL eff E INF EPS VAR b maker's revenue LOWER LEVEL UPPER MARGINAL Page 10

12 GRIPS Polcy Research Center Dscusson Paper : eff INF VAR c cost LOWER LEVEL UPPER MARGINAL eff INF LOWER LEVEL UPPER MARGINAL VAR Utl -INF INF VARIABLE x.l qualty eff VARIABLE b.l maker's revenue eff VARIABLE c.l cost eff VARIABLE Utl.L = total surplus PARAMETER db dervatve of b eff PARAMETER w prce eff In the frst part of the output fle, the codes orgnally put n the nput fle appear wth ther lne numbers (.e., echo prnt). Make sure that **optmal soluton appears n the SOLVE SUMMARY part after ths echo prnt as shown n Lne 21 of Lst 2.2. (If not, whatever results are meanngless as solutons.) Then, there are two types of solutons; EQU Page 11

13 GRIPS Polcy Research Center Dscusson Paper : (manly, expressng the Lagrange multplers of the constrants) and VAR (manly, the solved values of the decson or endogenous varables). The EQU block comes frst. In the EQU block, the Lagrange multplers are shown n the column under MARGINAL. 7 For example, the Lagrange multpler of constrant rev() s (Incdentally, n any maxmzaton problem, the Lagrange multpler of the objectve functon s always unty.) The followng VAR block contans the solutons of the decson or endogenous varables. We can see that the maker s margnal revenue db() evaluated at the solved parts qualty x() s equal to the suppler s margnal cost theta(). As the maker explots her barganng power n her take-t-or-leave-t offer to the suppler and does not allow any gans captured by hm, the prce w() s as low as the suppler s cost. Thus, ths problem to maxmze the maker s utlty can be also consdered as a maxmzaton problem of the total surplus (Table 2.1). Table 2.1: Numercal Soluton of the Frst-best Model for the Effcent Suppler Varable Varable Names n the GAMS Solved Values Program Parts Qualty x() Maker s Revenues b() Maker s Costs c() db dx db() Prce w() Maker s Utlty (or Total Surplus) Utl When the constraned maxmzaton s expressed as follows: max f x subject to x 0 g ther Lagrange multplers are postve (Varan 1992, Chapter 27). As the maxmzaton problems n ths paper are not necessarly expressed n such a form for the convenence of nterpretaton, most Lagrange multplers end up wth negatve values. Ths effects no substance. Page 12

14 GRIPS Polcy Research Center Dscusson Paper : The Frst-best Model wth the Ineffcent Suppler (PS2_F_nf.gms) The frst-best model wth the neffcent suppler can be bult by makng the followng two changes n the orgnal nput fle of PS2_F_eff.gms: n Lne 8, replace eff appearng as the element of set by nf n Lne 11, put the margnal cost of the neffcent suppler for theta() The solutons of the maxmzaton problem and the values computed from the solutons are n Table 2.2. As ponted out n Secton 2.1.2, the maker s margnal revenue db() evaluated at the solved parts qualty x() s equal to the suppler s margnal cost theta(), and ths utlty maxmzaton concdes wth the maxmzaton of the total surplus. Table 2.2: Numercal Soluton of the Frst-best Model for an Ineffcent Suppler Varable Varable Names n the GAMS Solved Values Program Parts Qualty x() Maker s Revenues b() Maker s Costs c() db dx db() Prce w() Maker s Utlty (or Total Surplus) Utl The Frst-Best Model wth Both Suppler Types (PS2_F.gms) Now that we deal wth a model that ncludes both suppler types, we ntroduce an ex-ante probablty p whch ndcates the -th type exsts. We assume that the maker knows ths probablty (common knowledge). The maxmzaton problem of the frst-best model s wrtten n the followng: subject to max Utl p b x w (2.1 ) Page 13

15 GRIPS Polcy Research Center Dscusson Paper : b x x (2.2) w x ru (2.3 ) In comparson wth the maxmzaton problem consstng of (1.1)-(1.3), there are two changes made. The frst change s made n the objectve functon (2.1 ). Now, the maker s utlty weghted wth the probablty s maxmzed. The second change s made n (2.3 ). In the prevous model, by assumng the take-t-or-leave-t offer on an a pror bass, no surplus s gven to the suppler; thus, c s solved frst. Then, c s equated to w. In the present model, we explctly use w as a decson varable. Then, we ntroduce the partcpaton constrant that the suppler s (net) utlty,.e., hs recept from the maker w() mnus the producton cost theta()*x(), must be greater than or equal to the reservaton utlty ru the mnmum utlty wth whch the suppler partcpates n the contract. If ths condton s not satsfed, the suppler does not accept the contract offered by the maker. The explanaton on the nput fle of the computer model n the followng wll be centered on the dfferences from the prevous models (Lst 2.3). In Lne 8, both suppler types eff and nf are put as the elements of Set. In Lnes 11 16, the margnal costs theta() and the probablty p() of both suppler types are put. In Lne 17, the reservaton utlty ru s gven. Although ru s set to be zero n ths numercal example, t can be any number. In Lnes 20 25, the decson varables are defned. Dfferently from the prevous models, w() s defned as a decson varable n ths model. In Lnes 26 29, the names of the objectve functons and the equatons are shown. Lnes contan the equatons of the maxmzaton problem. Lne 32 shows the objectve functon. The maker s utlty s the sum of net revenues (.e., the proft margn between b() and w()) weghted wth the probablty. Lne 33 s the revenue functon, defned as 0.5 b x as n the prevous models. Lne 34 represents the partcpaton constrant. The nequalty symbol s expressed as =g= n GAMS programs. Lne 40 defnes the model that conssts of all the equatons, and Page 14

16 GRIPS Polcy Research Center Dscusson Paper : Lne 41 s a drectve to solve the model by maxmzng the maker s utlty Utl. Lst 2.3: The Integrated Frst-best Model for Two Suppler Types (PS2_F.gms) 7 * Defnton of Set 8 Set type of suppler /eff, nf/; 9 10 * Defnton of Parameters 11 Parameter 12 theta() effcency /eff nf 0.3/ 14 prob() probablty of type 15 /eff nf 0.8/; 17 Scalar ru reservaton utlty /0/; * Defnton of Prmal/Dual Varables 20 Postve Varable 21 x() qualty 22 b() maker's revenue 23 w() prce; 24 Varable 25 Utl total surplus; 26 Equaton 27 obj suppler's proft functon 28 rev() maker's revenue functon 29 pc() partcpaton constrant; * Specfcaton of Equatons 32 obj.. Utl =e= sum(, prob()*(b()-w())); 33 rev()..b() =e= x()**(0.5); 34 pc().. w()-theta()*x() =e= 0; * Settng Lower Bounds on Varables to Avod Dvson by Zero 37 x.lo()= ; * Defnng and Solvng the Model 40 model FB1 /all/; 41 solve FB1 maxmzng Utl usng NLP; * End of Model The Lagrange multplers of the partcpaton constrants pc() are bndng for both suppler types (Lnes , of Lst 2.4). Ths means that the maker, who knows the Page 15

17 GRIPS Polcy Research Center Dscusson Paper : suppler type, offers hm the prce, dependng upon hs type, wth whch hs utlty s ndfferent from the reservaton level ru. The solved values of the parts qualty x(), the maker s revenue b(), and the maker s utlty Utl are shown n the VAR block. Lst 2.4: Output Fle of the Integrated Frst-best Model (PS2_F.lst) 74 S O L V E S U M M A R Y MODEL FB1 OBJECTIVE Utl 77 TYPE NLP DIRECTION MAXIMIZE 78 SOLVER CONOPT FROM LINE **** SOLVER STATUS 1 NORMAL COMPLETION 81 **** MODEL STATUS 2 LOCALLY OPTIMAL 82 **** OBJECTIVE VALUE EQU obj obj suppler's proft functon EQU rev maker's revenue functon LOWER LEVEL UPPER MARGINAL eff nf EQU pc partcpaton constrant LOWER LEVEL UPPER MARGINAL eff nf VAR x qualty LOWER LEVEL UPPER MARGINAL eff E INF E nf E INF EPS VAR b maker's revenue LOWER LEVEL UPPER MARGINAL eff INF. Page 16

18 GRIPS Polcy Research Center Dscusson Paper : nf INF VAR w prce LOWER LEVEL UPPER MARGINAL eff INF. 154 nf INF LOWER LEVEL UPPER MARGINAL VAR Utl -INF INF Utl total surplus The solutons of ths model, shown n Table 2.3, concde wth those of the models (Tables 2.1 and 2.2), whch are constructed for each suppler type separately. Table 2.3: Numercal Solutons of the Integrated Frst-best Model Varables Varable Names n Suppler s Type Solved Values GAMS Program Parts Qualty x() effcent (eff) neffcent (nf) Maker s Revenue b() effcent (eff) neffcent (nf) Prce w() effcent (eff) neffcent (nf) Maker s Utlty (or Total Surplus) Utl The Second-Best Model wth Both Suppler Types (PS2_S.gms) The second-best model of the parts supply problem s constructed n such a way that the maker offers a contract regardng the parts qualty x and the prce w to be pad to the suppler wthout knowng the suppler s type. The effcent suppler may report falsely the margnal cost hgher than hs real one to obtan the extra gan by explotng the nformaton asymmetry. To avod such a stuaton, the maker takes account of the ncentve compatblty condton that any suppler can obtan no extra gan even though he reports hs type falsely. Page 17

19 GRIPS Polcy Research Center Dscusson Paper : The second-best model can be constructed by addng the ncentve compatblty condton (2.4) to the frst-best model wth (2.1 ), (2.2), and (2.3 ) shown n Secton Whle the frst-best models can be bult for the effcent suppler and the neffcent suppler ether separately or jontly, the second-best model must nclude both supplers n one model. Ths s because the parts qualty x() and the prce w() of both supplers must be ncluded n the ncentve compatblty condton. The second-best model s as follows: subject to max Utl p b x w (2.1 ) 0. 5 b x x (2.2) w x ru j (2.3 ) w x w x j (2.4) j The left-hand sde of (2.4) s the -th type suppler s utlty ganed by reportng hs type type truly, and the rght-hand sde s hs utlty ganed by reportng hs type falsely type j. By offerng a contract satsfyng the condton that the value of the rght-hand sde cannot exceed that of the left-hand sde, the maker does not gve any type of supplers opportuntes to earn extra even though they reports ther types falsely. The drectve Alas n Lne 9 means that and j are used nterchangeably. In Lne 30, the names of the ncentve compatblty condtons c(, j) are gven; n Lne 36 the ncentve compatblty condtons are specfed. These constrants can be rewrtten n the followng four equatons: w eff x w x (2.4a) eff eff nf eff nf w eff x w x (2.4b) eff eff eff eff eff w nf x w x (2.4c) nf nf eff nf eff w nf x w x (2.4d) nf nf nf nf nf Page 18

20 GRIPS Polcy Research Center Dscusson Paper : The new model system ncludes the above four equatons, compared wth two equatons of the orgnal system (2.4). 8 Equatons (2.4b) and (2.4d) trvally holds wth strct equalty and, thus, are redundant n lght of the orgnal constrant (2.4). Consequently, there s no dfference between the system stated above and the orgnal system of (2.4). Lst 2.5: Input Fle of the Second-best Model (PS2_S.gms) 7 * Defnton of Set 8 Set type of suppler /eff, nf/; 9 Alas (,j); 10 * Defnton of Parameters 11 Parameter 12 theta() effcency /eff nf 0.3/ 14 prob() probablty of type 15 /eff nf 0.8/; 17 Scalar ru reservaton utlty /0/; * Defnton of Prmal/Dual Varables 20 Postve Varable 21 x() qualty 22 b() maker's revenue 23 w() prce; 24 Varable 25 Utl maker's utlty; 26 Equaton 27 obj total surplus functon 28 rev() maker's revenue functon 29 pc() partcpaton constrant 30 c(,j) ncentve compatblty constrant; * Specfcaton of Equatons 33 obj.. Utl =e= sum(, prob()*(b()-w())); 34 rev()..b() =e= x()**(0.5); 35 pc().. w()-theta()*x() =g= ru; 8 If you feel uneasy wth ths redundancy, you can rewrte Lne 36 as: c(,j)$(ord() ne ord(j))..w()-theta()*x() =g= w(j)-theta()*x(j); where $( ) means a dummy varable expressng a condton that the relaton nsde the parenthess holds. And, ord() means the order of ndex n the defned set, and ne means not equal. Page 19

21 GRIPS Polcy Research Center Dscusson Paper : c(,j)..w()-theta()*x() =g= w(j)-theta()*x(j); * Settng Lower Bounds on Varables to Avod Dvson by Zero 39 x.lo()= ; * Defnng and Solvng the Model 42 model SB1 /all/; 43 solve SB1 maxmzng Utl usng NLP; * End of Model 2.6). The nput fle s shown n Lst 2.5: ts soluton s prnted n the output fle (Lst Lst 2.6: Output Fle of the Second-best Model (PS2_S.lst) EQU rev maker's revenue functon LOWER LEVEL UPPER MARGINAL eff nf EQU pc partcpaton constrant LOWER LEVEL UPPER MARGINAL eff INF. 135 nf.. +INF EQU c ncentve compatblty constrant LOWER LEVEL UPPER MARGINAL eff.nf.. +INF nf.eff INF EPS VAR x qualty LOWER LEVEL UPPER MARGINAL eff E INF E nf E INF VAR b maker's revenue Page 20

22 GRIPS Polcy Research Center Dscusson Paper : LOWER LEVEL UPPER MARGINAL eff INF. 156 nf INF VAR w prce LOWER LEVEL UPPER MARGINAL eff INF. 163 nf INF LOWER LEVEL UPPER MARGINAL VAR Utl -INF INF The Frst-Best and the Second-Best Integrated Models (PS2_F&S.gms) In the prevous sectons, we bult and solved the frst-best and second-best models ndvdually. Now we present a program to solve both models as one model. Note that the dfference between both the frst-best and second-best models s found only n the use of the ncentve compatblty condtons. The codes up to Lne 39 n the new model (Lst 2.7) are dentcal to those n the second-best model (Lst 2.5). Lne 42 defnes the frst-best model as FB1, whch conssts of three equatons, obj, rev(), and pc(), and Lne 43 drects the computer to solve the model FB1 by maxmzng Utl. Smlarly, Lne 45 defnes the second-best model as SB1, whch conssts of four equatons, obj, rev(), pc(), and c(,j). The equaton c(,j) s the ncentve compatblty condtons. Lne 46 drects to solve the model SB1 by maxmzng Utl. There are two techncal ponts n the programmng. Frst, n the prevous models, when we defne the model, we nclude all the equatons descrbed n the precedng lnes; so we express the model contents as: Model model-name /all/; Alternatvely, we can express more explctly lke: Page 21

23 GRIPS Polcy Research Center Dscusson Paper : Model model-name /obj, rev, pc/; In other words, nsde of / /, we exactly place the equaton names used for the model. Note that the ndces such as () are not used n / /. Lst 2.7: The Frst-best and the Second-best Integrated Model (PS2_F&S.gms) 7 * Defnton of Set 8 Set type of suppler /eff, nf/; 9 Alas (,j); 10 * Defnton of Parameters 11 Parameter 12 theta() effcency /eff nf 0.3/ 14 p() probablty of type 15 /eff nf 0.8/; 17 Scalar ru reservaton utlty /0/; * Defnton of Prmal/Dual Varables 20 Postve Varable 21 x() qualty 22 b() maker's revenue 23 w() prce; 24 Varable 25 Utl maker's utlty; 26 Equaton 27 obj maker s utlty functon 28 rev() maker's revenue functon 29 pc() partcpaton constrant 30 c(,j) ncentve compatblty constrant; * Specfcaton of Equatons 33 obj.. Utl =e= sum(, p()*(b()-w())); 34 rev()..b() =e= x()**(0.5); 35 pc().. w()-theta()*x() =g= ru; 36 c(,j)..w()-theta()*x() =g= w(j)-theta()*x(j); * Settng Lower Bounds on Varables to Avod Dvson by Zero 39 x.lo()=0.0001;; * Defnng and Solvng the Model 42 Model FB1 /obj,rev,pc/; 43 Solve FB1 maxmzng Utl usng NLP; Model SB1 /obj,rev,pc,c/; Page 22

24 GRIPS Polcy Research Center Dscusson Paper : Solve SB1 maxmzng Utl usng NLP; * End of Model Second, as we now solve multple models n one computer program usng the Solve drectve for several tmes, the same number of SOLVE SUMMARY s appear n the output fle. At the top of each of SOLVE SUMMARY, the model name such as FB1 appears n Lne 81 and SB1 n Lne 197 (Lst 2.8). You wll fnd the output fles of ths new model encompassng the solutons of the frst-best and the second-best models. Lst 2.8: Output Fle of the Frst-best and the Second-best Integrated Model (PS2_F&S.lst) 79 S O L V E S U M M A R Y MODEL FB1 OBJECTIVE Utl 82 TYPE NLP DIRECTION MAXIMIZE 83 SOLVER CONOPT FROM LINE **** SOLVER STATUS 1 NORMAL COMPLETION 86 **** MODEL STATUS 2 LOCALLY OPTIMAL 87 **** OBJECTIVE VALUE S O L V E S U M M A R Y MODEL SB1 OBJECTIVE Utl 198 TYPE NLP DIRECTION MAXIMIZE 199 SOLVER CONOPT FROM LINE **** SOLVER STATUS 1 NORMAL COMPLETION 202 **** MODEL STATUS 2 LOCALLY OPTIMAL 203 **** OBJECTIVE VALUE Comparson of the Solutons between the Frst-best and the Second-best Models The comparson of the solutons between the frst-best and the second-best models are summarzed n Table 2.4. The ncentve compatblty condton s bndng only for the effcent suppler (.e., the Lagrange multpler for the effcent suppler s non-zero). In other Page 23

25 GRIPS Polcy Research Center Dscusson Paper : words, the effcent suppler s utlty generated when he reports hs type truly s ndfferent to the one obtaned when he reports hs type falsely. In contrast, snce the neffcent suppler has no ncentve to report hs type falsely, the ncentve compatblty condton s not bndng. The comparson s further summarzed below. Table 2.4: Varables and Constrants Numercal Solutons of the Frst-best Model and the Second-best Model Varable Names Suppler Type Second-best Frst-best n the GAMS Model Soluton * Model Soluton Program Parts Qualty x() Effcent(eff) Ineffcent(nf) Maker s b() Effcent(eff) Revenues Ineffcent(nf) Prce w() Effcent(eff) Ineffcent(nf) Maker s Utlty Utl Suppler s pc.l() Effcent(eff) Utlty (=Info. -pc.lo() Ineffcent(nf) Rent) Lagrange Mult. of the Incentve ceff( eff, nf ) Effcent pretends neffcent. cnf( nf, eff ) Comp. Const. Partcpaton Constrant Ineffcent pretends effcent. pc.m() Effcent(eff) Ineffcent(nf) Note: Solved values n the frst-best model are all dentcal to those shown n Table 2.3. Frst, the maker offers to the effcent suppler a prce w( eff ) whch s hgher than the one that would be offered n the frst-best case. Ths s because the maker wants to dampen the effcent suppler s motve to obtan an extra gan by reportng hs type falsely. As a result, a slack,.e., the nformaton rent, s generated n the partcpaton constrant of the effcent suppler. The value of ths slack s 0.237, whch corresponds to the dfference between the LEVEL value pc.l( eff ) and the LOWER value pc.lo( eff )of the Page 24

26 GRIPS Polcy Research Center Dscusson Paper : pc( eff ). 9 Because the reservaton utlty ru s assumed to be zero n the present model, the nformaton rent matches the effcent suppler s utlty. Even though the effcent suppler enjoys a hgher prce, hs parts qualty remans as the same level n the frst-best case. Hs hgher prce results solely from hs nformaton rent. Second, whle n the second-best case the effcent suppler earns the nformaton rent, the maker does not allow the neffcent suppler to earn any nformaton rent. The maker offers the neffcent suppler a prce w( nf ), whch s lower than the one n the frst-best case, n order not to make the effcent suppler s nformaton rent too lberal. As a result, the neffcent suppler ends up wth the lower qualty. Thrd, the maker s utlty decreases partly because of the nformaton rent (0.237), whch the maker must pay to the effcent suppler and partly because of the loss resulted from the degraded parts qualty (0.411) made by the neffcent suppler. 2.2 The 3-Type Models Outlne of the 3-Type Models In ths secton, we present the models dstngushng three suppler types (0, 1, 2; the smaller number ndcates the hgher effcency). The key dfference of the N-type ( N 3 ) models from the 2-type models les n the complexty regardng the ncentve compatblty condtons of the second-best models. The 2-type models need ncentve compatblty condtons to prevent a false-reportng only for two cases. The one s the case that the effcent suppler falsely reports neffcent, whle the other s the case that the neffcent suppler falsely reports effcent. In general as we must concern all the combnatons of each suppler aganst all the other supplers, N 1N ncentve 9 As for the meanngs on the values under LOWER, LEVEL, and UPPER n the EQU, refer to McCarl (2009, ). If Opton solslack=1; s put at any place before the SOLVE statement n the program; then, SLACK appears nstead of LEVEL n the output fle. Page 25

27 GRIPS Polcy Research Center Dscusson Paper : compatblty condtons are needed n the N-type models. A 3-type model requres sx ncentve compatblty condtons; that maybe fne. But, a 10-type model requres 90 condtons! The ncentve compatblty condtons as a round robn are called the global (ncentve compatblty) condtons. To cope wth the complexty regardng a large number of the global condtons (often n the theoretcal models), we substtute the two local (ncentve compatblty) condtons for the global condton by ntroducng an assumpton, the Spence-Mrrlees sngle crossng property (SCP). The local condtons are to smply prevent the suppler from falsely reportng hs type as one type lower or hgher than hs real type. Furthermore, we add another assumpton of the monotoncty (MN) of the parts qualty x wth respect to the suppler type ndex ndcatng effcency. That s, as the suppler s effcency ncreases, hs parts qualty mproves. Ths assumpton allows us to make the 3-type models wth only one local condton. The followng subsectons go as follows. In Secton 2.2.2, we buld the frst-best model, whch has nothng to do wth the SCP and the monotoncty assumptons, because the ncentve compatblty condtons are not needed. In Secton 2.2.3, on the bass of the frst-best model, we develop the second-best model wth the local condtons, assumng that both the SCP and the monotoncty assumpton hold. In Secton 2.2.4, we buld the second-best model, by replacng the local condtons wth the global condtons, and show the solutons of ths model concde wth those of the model wth the local condtons of developed n Secton In Secton 2.3, we deal wth the models where nether the SCP nor the monotoncty assumpton holds The Frst-best Model (PS3_F.gms) The 3-type frst-best model can be made by extendng the 2-type model dscussed n Secton wth the followng amendments. In Lne 8 of Lst 2.9, three types are defned. (The most effcent type s defned as 0.) In Lnes 12 14, the margnal costs theta() Page 26

28 GRIPS Polcy Research Center Dscusson Paper : are put, and n Lnes the ex-ante probablty s gven for each suppler type. In Lne 42, the model name FB2 replaces the prevous name FB1. Lst 2.9: The 3-type Frst-best Model (PS3_F.gms) 7 * Defnton of Set 8 Set type of suppler /0,1,2/; 9 Alas (,j); 10 * Defnton of Parameters 11 Parameter 12 theta() effcency / / 15 p() probablty of type 16 / /; 19 Scalar ru reservaton utlty /0/; * Defnton of Prmal/Dual Varables 22 Postve Varable 23 x() qualty 24 b() maker's revenue 25 w() prce; 26 Varable 27 Utl maker's utlty; 28 Equaton 29 obj maker s utlty functon 30 rev() maker's revenue functon 31 pc() partcpaton constrant * Specfcaton of Equatons 34 obj.. Utl =e= sum(, p()*(b()-w())); 35 rev()..b() =e= x()**(0.5); 36 pc().. w()-theta()*x() =g= ru; * Settng Lower Bounds on Varables to Avod Dvson by Zero 39 x.lo()=0.0001;; * Defnng and Solvng the Model 42 Model FB2 /all/; 43 Solve FB2 maxmzng Utl usng NLP; * End of Model Page 27

29 GRIPS Polcy Research Center Dscusson Paper : The Second-best Model wth the Local Incentve Compatblty Condtons (PS3_S.gms) The second-best model must have the ncentve compatblty condton, whch s the only dfference from the frst-best model. As stated before, the complexty of the second-best model s centered on the ncentve compatblty condton. Before movng nto the second-best model, we explan the relatonshps between the global and local ncentve condtons n a matrx format (Table 2.5). In that table all the types that the -th type suppler can report truly and falsely are shown as dark and lght grey areas. Table 2.5: Global and Local Incentve Compatblty Condtons Reported Suppler Type N 1 True Suppler Type 0 U U U U U U 1 U 0 U 1 U U 1 1 U 1 1 N U N 1 N 1 0 U N 1 For the sake of the explanaton below, we defne the -th suppler s utlty generated when he reports hs type as the j -th suppler as U. The global ncentve compatblty condtons to dscourage hm from false reportng can be wrtten as follows: U U Ths s equvalent to: U U U j U 0 U 1 U 2 j j Page 28

30 GRIPS Polcy Research Center Dscusson Paper : U U U U U 1 U (Note that ths s a trval one.) U 1 U N 1 One can see that the number of the global condtons ncreases rapdly, as the number of types ncreases. There are two methods to cope wth ths dffculty. The frst method s to ntroduce the SCP assumpton and to substtute the local condtons for the global ones. If the SCP assumpton holds, only two condtons are needed for each suppler among these many. The one condton, denoted as LICD, s to dscourage a suppler from reportng hs type as one type lower than hs real type. The other condton, denoted as LICU, s to dscourage hm from reportng one type hgher: U U U 1 U 1 (LICU) (LICD) These condtons are mposed for the cases shown n the dark grey areas n Table 2.5. The second method to cope wth the dffculty s to add the condton of the monotoncty (MN) regardng the parts qualty x as: x 0 x N If the monotoncty condton s added, ether LICD or LICU s suffcent for the local condtons. The commonly used approach to the second method s to buld and solve the model wthout mposng MN, and ascertan that the solutons do satsfy MN. If the solutons satsfy MN, the omsson of MN n the model s justfed. Conversely, f MN s not satsfed n the solutons, one must rewrte the model n such a way as to add ether MN or LICD (or Page 29

31 GRIPS Polcy Research Center Dscusson Paper : LICU) explctly. (More detaled explanaton wll be gven n Secton ) We follow ths approach here. In our assumed suppler s utlty U w ux, and ux x,, the frst dervatve of u wth respect to x,.e., u x s a decreasng functon wth respect to ; therefore, the SCP s satsfed. Accordngly, we can smplfy the model by replacng the global condtons wth the local condtons. In the followng, we develop the model wth LICD but wthout MN as:. subject to max Utl p b x w (2.1 ) 0. 5 b x x (2.2) w x ru w x w x 1 1 (2.3 ) (2.5) In the model (PS3_S.gms), LICD (2.5) appears n Lne 38. The suppler who s by one type less effcent than the -th type suppler can be wrtten as 1, as ntuton tells you. The partcpaton constrant s set for all the supplers as n the frst-best model to compute the nformaton rent earned by each suppler, whch s ndcated as the solved slacks of the constrants (Lne 37 of Lst 2.10). As ths constrant s not bndng for other than the suppler 2 (.e., the most neffcent suppler), none of these extra constrants does not dstort the solutons at all. Lst 2.10: The 3-type Second-best Model (PS3_S.gms) 7 * Defnton of Set 8 Set type of suppler /0,1,2/; 9 Alas (,j); 10 * Defnton of Parameters 11 Parameter 12 Theta() effcency /0 0.1 Page 30

32 GRIPS Polcy Research Center Dscusson Paper : / 15 p() probablty of type 16 / /; 19 Scalar ru reservaton utlty /0/; * Defnton of Prmal/Dual Varables 22 Postve Varable 23 x() qualty 24 b() maker's revenue 25 w() prce; 26 Varable 27 Utl maker's utlty; 28 Equaton 29 obj maker s utlty functon 30 rev() maker's revenue functon 31 pc() partcpaton constrant 32 lcd() ncentve compatblty constrant; * Specfcaton of Equatons 35 obj.. Utl =e= sum(, p()*(b()-w())); 36 rev()..b() =e= x()**(0.5); 37 pc().. w()-theta()*x() =g= ru; 38 lcd()..w()-theta()*x() =g= w(+1)-theta()*x(+1); * Settng Lower Bounds on Varables to Avod Dvson by Zero 41 x.lo()=0.0001;; * Defnng and Solvng the Model 44 Model SB3 /all/; 45 Solve SB3 maxmzng Utl usng NLP; * End of Model The soluton of the model s shown n ts output fle shown n Lst Lst 2.11: Output Fle of the 3-type Second-best Model (PS3_S.lst) EQU rev maker's revenue functon LOWER LEVEL UPPER MARGINAL Page 31

33 GRIPS Polcy Research Center Dscusson Paper : EQU pc partcpaton constrant LOWER LEVEL UPPER MARGINAL INF INF INF EQU lcd ncentve compatblty constrant LOWER LEVEL UPPER MARGINAL INF INF INF VAR x qualty LOWER LEVEL UPPER MARGINAL E INF E E INF EPS E INF EPS VAR b maker's revenue LOWER LEVEL UPPER MARGINAL INF INF INF VAR w prce LOWER LEVEL UPPER MARGINAL INF INF INF LOWER LEVEL UPPER MARGINAL VAR Utl -INF INF. Page 32

34 GRIPS Polcy Research Center Dscusson Paper : Comparson of the Frst-best and the Second-best Solutons The comparson of the frst-best (PS3_F.lst) and second-best solutons (PS3_S.lst) s shown n Table 2.6. Table 2.6: Solutons of the 3-type Frst-best and Second-best Models Varable Varable Suppler Second-best Frst-best Name n the Type Soluton Soluton Program Gap Parts Qualty x() Maker s Revenue b() Prce w() Maker s Utlty Utl Suppler s Utlty (=Info. Rent) pc.l() -pc.lo() Lag. Mult. of Incentve Comp. Const. Lag. Mult. of Partcpaton Constrant lcd.m() pc.m() Compared wth the frst-best solutons, the parts qualty x() n the second-best solutons s the same only for the most effcent suppler; the qualty s lower for all the other supplers. Ths corresponds to the fact that the prce w() net of the nformaton rent remans the same for the most effcent suppler ncreases, whle that of all the others decreases. In order to prevent the (not-the-least effcent) supplers from reportng the less effcent supplers types, the maker must make a lberal offer to these supplers n the Page 33

35 GRIPS Polcy Research Center Dscusson Paper : second-best case than the one n the frst-best case. The maker need not be too lberal; ths lberalty s determned at such a level that these supplers can receve no extra gan even though they report ther types falsely. The ncentve compatblty condtons are bndng only for 0 and 1. Those condtons are not bndng for the least effcent suppler 2, because he has no less-effcent suppler to pretend for an extra gan. Therefore, hs partcpaton constrant s bndng; he can obtan just as much as hs reservaton utlty level. The nformaton rents ganed by the more effcent supplers ( 0 and 1 ) are shown as the slacks n ther respectve partcpaton constrans (Lnes 137 and 138 of Lst 2.11). As the reservaton utlty s set at zero, ther rents mmedately ndcate ther utlty levels. The prce of the ntermedately effcent suppler ( 1 ) becomes lower than that n the frst-best case, but he obtans strctly postve utlty generated by hs nformaton rents offsettng the losses from the lower prce. Needless to say, the utlty of the most effcent suppler ( 0 ) ncreases. The maker s utlty decreases for two reasons. The frst s the loss resulted from the degraded parts qualty suppled by those but the most effcent one. The second s the nformaton rents exploted by those but the least effcent one. The solutons show the monotoncty of the parts qualty x. In other words, the more effcent the suppler s, the hgher hs parts qualty s. Thus, the omsson of MN n the models can be justfed The Second-best Model wth the Global Condtons (PS3_S_GIC.gms) In ths subsecton, we develop the second-best model by replacng the local condtons n the prevous model wth the global condtons, and analyze the solutons. Lne 38 of Lst 2.12 shows the condtons of (2.4). Note that the constrant c carres two suffxes: and j, and, s nserted between and j. The ncentve compatblty constrant c(,j) prevents the -th type suppler from ganng extra by reportng hs type as j. Page 34

36 GRIPS Polcy Research Center Dscusson Paper : Ths ncludes the case of j, where he reports hs own type truly. Because the constrant obvously holds wth strct equalty n the case of dstort the soluton at all. j, ths redundant constrant does not Lst 2.12: The 3-type Second-best Model wth Global Condtons (PS3_S_GIC.gms) 28 Equaton 29 obj maker s utlty functon 30 rev() maker's revenue functon 31 pc() partcpaton constrant 32 c(,j) ncentve compatblty constrant; * Specfcaton of Equatons 35 obj.. Utl =e= sum(, p()*(b()-w())); 36 rev()..b() =e= x()**(0.5); 37 pc().. w()-theta()*x() =g= ru; 38 c(,j)..w()-theta()*x() =g= w(j)-theta()*x(j); * Settng Lower Bounds on Varables to Avod Dvson by Zero The solutons of the second-best model wth the global condtons are equal to those of the second-best model wth the local condtons, because the present model satsfes the SCP (Lst 2.13). 10 Among the sx constrants of the (global) condtons, only two constrants those do not let the type 0 suppler to report hs type as type 1 (Lne 145) and the type 1 suppler to report hs type as type 2 (Lne 148) are bndng. (Type 2 does not have any less-effcent type to pretend.) In other words, only the constrants to prevent a suppler from reportng hs type as one-type less effcent are bndng. The other constrants are not bndng; thus, they can be omtted. If we omt these non-bndng constrants, the model s 10 The solutons of the Lagrange multplers are reported n a dfferent format between these two output fles, just because the one-dmensonal constrant of lcd() s replaced by the two-dmensonal one of c(,j) (Lst 2.13). However, the bndng equatons do not change n essence. Page 35