This PDF is a selection from an out-of-print volume from the National Bureau of Economic Research

Volume Title: Annals of Economic and Social Measurement, Volume 6, number 5

Volume Author/Editor: NBER

Volume Publisher: NBER

Volume URL: http://www.nber.org/books/aesm77-5

Publication Date: December 1977

Chapter Title: An Optimization Approach in Multiple Target Problems Using Inequality Constraints: The Case Against Weighted Criterion Functions

Chapter Author: Gunther Bock V. Wulfingen, Peter Pauly

Chapter URL: http://www.nber.org/chapters/c10547

Chapter pages in book: (p. 613 - 630)

ffl/j/t (I /0fl oil tilleh( (i/ /()'.5

AN OP11MI7AIION APPROA(ii IN MtJ1FIpI l\jjF FPROBiI\1S iSlN(; INFQUAIIIY ('(INS1RAINIS Fill.('At. A(iAINSF 'FlG I lilt) (i il1RR)N IVN( lR)NS

Iv (IflNIlu:R I()(K V. 'rYtil iIN(1N \NI) PIFIR y

/ lie cant ti: hunts! sait/irtito ii iii Oflitiflit jiuli u/'tl?nl:slh,un u :nt a ttuh le/ i nitr,r,all :quat/falit I rueful,, /11114 110,! t ('lillil.'t'i/ /0! I; S i trnte/Iual iillut/equas toni ar/j!r !ltlr5 1 1,1alitrilal,, e (i/tF0th ii i.c ifs e/u;n ti. re/Itt' senhat lilt' tulure pro/i/en, En a ri'orous Inequalulranu'tt tirA / 1,' eo'n/'ulaluinai rra/,_'ai,w, I /Ot 'oi/ itu on oiie!Fu.,hEni/ aquethI' 0/ O/UifllJ:gj-lieu z.ys toi/ eialua:,,,n at duo! aria/nt's / lit' c,/,,'orj,/,,,, for File sit/it!,, ill it/ I/Ic l?ii/'/ht/51.i' prvsHs'in it ituilvuil anti 1/sr st/tale jsrotrdure i/lu s ImIt',/ sill, a 'illintro al e iniple' lotnt/it ale II' ,neuh,sds fir i/ri lit u and a se/ulnes some nwdi/itaons an,i e len 'opts 44/ ih,'I'ttsl( n/ta are positlecl out

INikol ) ( I l( )N

I)uring the last few years the discussion on optimal control ot econotiliL'ssstems has primarily focused on the developnien t of adequate optim iia-tion techniques. t/n lii recently, however, coinpa ra b!' less ellint hasbeen devoted to the aspects ot an adequate lonna I represent at ion ofthe objectives o eCOilOinic policy. Ihe standard quadratic criterion l'unc-tion, originally advocated by Si fllOfl, Theil and I lolL,' has been appliedalmost uniformly, only sOnlctinlesaceornpanjcrj by a dissociative phrase.This paper sets forth an alternative approach applying inequality con-straints instead of quadratic or otherwise nenalitcd deviations, In ch. I westart with a short critical discussion ol ihe conventional approach. t'hcbasic ideas of our lorniulation are to he found in ch. 2: while ch. 3 dealswith the problem oi its computational implementation, the approach isapplied to a medium-sued nonlinear economic model in eh 4. the follow-ing ch. 5 being devoted to the discussion of some modifications and e-tensions of' the basic concept. The final ch. 6 summariies what we thinkto he the mu am advantages oft he proposal.

'Fiim ('oNvl:NlIoNAI. API'kOAeiI

Consider the standard macroeconomic model formulation

(I) /(x,v ii, U 0

are ,rutetncd Its tao ,IIonnhous retirees for alu.ilsIi crltleisn, of ,i etirlicr dr.stiof thi5 paper.

(f Simon ( l')5(,j, 1 hell 'iS?, '/65), I loll (I '162).

6 13

I

vhere x is the vector of conten)poraneotls endugenous varjftesX IS a veCtor of lagged cnuogenous variables

U is the vector of contefli poraneous instrumentsU is a vector of lageed inst rumen s

is a vector of exogenous variables, hich. in he cac Ut i

stochastic systems. contains the error tcrn as 't elI

/ is a vector of nonlinear, interdependent. aiitl iiflpliitI (Iefjneclun Ct ion S.

In general the optimization of models of this k intl proceeds alone the

following lines:Take the preferred values of' targets and instruments aiid put thenitogether in a vector = . Note that the term "values'' in this con-text is meant iii the most comprehensive sense; it includes e e.

ratios an(l changes of viiriahks as well 2

Specify a criterion function sw ted sa , to stahi li/c the CCI)flO!fljCpath around some a priori track and/or to rc(1 nec the period-to.period fluctuations in the econo ni ic path

(2) ( - z )' .1 (:: -

Hence the coefficients of the weighting mat ri.x are depending onthe interpretation of the respective elements of and intended.first, to describe the politicians' preferences. Second, to account forpenalty term efiects and, third, to represent certain smoothnessrequirements on the solution paths of targets and instruments.lirially, to derive the optimal policy m mimi/c the (expected) sel-fare loss, i.e.

(3) nn (z - Ytl( -

s.t. f(.) = 0

This approach is widespread in application primarily due to its opera-

tional convcnience. One ol' the most frequent justifications for using thequadratic formula has been that this is probably (he simplest form hich

allows for decreasing marginal rates ol suhstitution: in the absence of in-equality constraints this is in general necessary for the existence of finitesolutions (cf. Friedman (1975), p. 3). The standard approach has, Iìo-ever, been subject to growing criticism: some of the must signilicaiit itenisshall be discussed briefly.

21or a thurotig}i dIcusNinn ol i tu1LcrprlalioI1 ri !ciiur,ti ( tii' I'"). I

(iarba'Jc (I)75), cli. 5, tlothrook (973 cli i 974, I97s or Nriiri \.rIiiIri l',tI.h(1975).

3C1. TIieit iI9(), pp. 3 '. Oierc rrriic Idrtion,it reicrerice i. tiicn to rcLiic&! rn

epics in staitsiics and cngtncering.

(i 4

a) First, it is to he noted that the theoretically rather flexible ap-proach has lost SOflie ot Its attractiveneSs in application since ire-qitcittly a diagonal Weighting matrix is used.4h) lurthermore, it Seems almost unlikely that [lie 1)OliCymaker'spreferences lit into such an artificially limiting framework as thequadra tic function. In this Context the most Serious problem

seems to he the implied symmetric reaction It is well known thatsminetric tunetions incorporate a Potential to bias policy be-havior ii the n umerical values of the z * -elements are not chosenappropriately (ci. Palash (1977), Shupp (1977)). This can be over-come by the use of truncated or exponential criterion functions(cI'. Palash (1977)). Apart from this, however, as Friedman (1975,p. I l3) points out, ". . .often policy makers see certain variablesmore as Constraints, iii the sense of bearing an implicit preferenceloss only For val tics outside of a particular range. Tb is requiresan asymmetrical and possibly piecewise formulatjon

c) Regardless of the degree of sophisticatedness of the functionalkrin of the criterion function a numerical weighting is indis-pensible. As numerous examples indicate, however, the optimi-iation results in general turn out to be rather sensitive as to thechoice of the coefficients in the weighting matrix.6 Hence a mean-ingful application of this approach requires a precise knowledgeof the coellicients' numerical values. For obvious reasons it seemslitrly improbable that the policymaker is able to specify in anappropriate numerical form his preferences concerning the rela-tive importance of concurrent targets and, even more tedious, inaddition to that to fix the weights of the cross products betweentargets and instruments.7' The coincidence of both these factsseems to give the entire approach a somewhat arbitrary touch.l'here have been some proposals to integrate the iterative processof optimization and preference revelation into a unified ap-

the other hand the further restrictive assumption of positive definiteness oF Aofin to he found in earlier stud cs is no more signi fica iii, sii)ce itowadevs models gei1eriIIyare neither COO VCX 1101 ci)iicaVe.

icr. Friedman ( l972 975, pp. 183 96), lair (1974), Poirier (1976).0An:ifvr teal deriva tiuni of consequences of a nhisspecificut ion of A are to he found in

I heil ( 1968, chs. 2, 3 t 5), /eIInr/Gciscl 968), /eltner (1974) or IlalIcti (977).1Even if the model builder tries to figure out the policymakers' preferences by an

adequate procedure it is to ask too much to expect a consistent preference scheme on anPrim basis. l'or some of thcse a priori procedures to derive a functional representa-

tion of the politicians preferences ci. e.g. Johansen (1974) Rra (1974, 975). Additionalin form a tii)i) COI1CC In tog the pen a I /at tO fl of instrument variations can, however, undercertain circumlistarices he drawti according to Gordon (1976).

1 he problem of relative weighting is generally aggravated by the fact that theVii ma hfcs u,e to he unnurm a li,ed tie rice OflC ca ni not di sell in in a Ic between that part of t he'seightiig scheme serving fur iti)riiiiiii/atiofl arid that one L'spressing preference ordering.

615

L

I

I'

proach.9 As far as the application ot these algorithm5 to latnonlinear systems is concerned there seems, hoscver t he fluconvincing evidence up to nov.d) An integrated welfare loss function of the standard type ohvi)fl5f

lacks the facility to discriminate eflIcicntl between targets and re-strictioris. Although restrictions can he made etkctive SifllpIv ban arbitrarily high wetghting term, this causes numerical proh!efl)in the case of more than that particular argument in the Criterionfunction since the value of the criterion function is dominatedbpenalty terms,° If there are more than one "restriction" of thattype taken into account the weights may in eliect cancel out. Fur-

thermore, it seems possible that the policyrnaker is not indifferentconcerning the order of activation of certain instruments pOssiblydue to decentralized or hierarchical decision processes in thestandard approach this has to he expressed within the framesorkof' the general weighting scheme too.e) Evaluating the optimization results by means of just a Single wel-fare index may be insufficient under various aspects: first, there isno obvious economic interpretation of the results; second, this int-plies among other things that the performance of different runs(employing different z4-values) cannot be evaluated in terms ofultimate targets; third, due to the fact that a lot of sinlulations isnecessary to evaluate the local properties of a particular solution

a systematic sensitivity analysis turns out to he somewhat tedious.As far as the first three items are concerned Livesey's (l973a. p. II)conclusion seems inevitable: "Hoping to come up with the unique social

welfare function is a fruitless task. For this reason it would be desirable tokeep the welfare function as simple as possible and to incorporate policyobjectives. . . in the model as inequality constraints." This is the waywhich is to he pursued in the present paper. A consequent applicatiort ofnonlinear programming (NIP)- techniques will. as can be shownthroughout the presentation..... in addition surmount at least in pait theshortcomings of the traditional approach indicated under d) and e) above

2. AN ALTERNATIVE APPROA('IIIn the previous chapter we tried to bring out some of the main prob-lems associated with the application of a weighted loss function in eco-

9Cf. Rustem/veIupjll.ij/yç101 (1977) and earlier work b Lelenv/Cochrane (I73).followed b) Wallenlus/Wallenius/vartia (1976) or Donckels (1977) All these approachesare, however, developed within the standard quadratic framework.'°Cf. Luenberger (1969), p. 302. The signifjcan or this point depends upon the relj-live magnitude of the weighting terms as compared with the numerical preciseness of thecomputer. For an alternative pe'taliztng scheme handling limited discretion cf. Garhade(1977).

nomic policy optlmiiation. In what follows We outline the features of ouralternative approach. In order to keep to the essentials and to clarify ourposition we Sttrt With the description of a rather rigorousVCFSIOfl of thebasic procedure. In ch. 5 below sonic of the stronger assumption will be

relaxed in order to point out some potential modificationsOur starting point is a vector z + too, although we make a different

use of the information contained therein. As has been outlined above,z is a rather heterogeneous mixture of targets of economic policy, ex-pressed in numerical values of certain endogenous variables (but of dif-fering importance for the policy-maker), and "preferred" values of in-struments and/or their respective paths. One should, however, recognizethat some of the elements of z have originally been incorporated intothe criterion function in order to approximate some more or less tech-nically determined restrictions on the time paths of those particular vari-ables. In a first step. we try to separate these elements from the entirez-vectot. This is done by direct formulation of inequality constraints onthe respective variables: these constraints are combined into a vector xand added as an integral part to the model, As Kornaj (1967, p. 398)points out this ". . system of constraints expresses thus the compellingforces of outward circumstances," which are to be distinguished from thewishes of economic policy entering the objective function to express thepreference of economic administration. For expository purposes we willassume that z contains only instruments recognizing that in short-termplanning because of the inertia of legislative processes this vector mightcontain more elements than in the long-term framework."

En view of the criticism raised in the previous chapter we furtherreformulate the whole bundle of remaining f-values in terms of suitablychosen inequality restrictions, the upper bounds of which are integratedinto a vector zr.'2 Finally, we require an initial ordering of these in-equalities according to their relative importance. Actually, it is not neces-sary to assign to each zr-value a cardinal preference number: all what isrequired is an ordering.0

Without having made it explicit up to now, the fundamental dif-ference between this approach and the traditional formulation can heexemplified figuring out the basically different interpretations of z undz. While the former is assigned to "desired levels." the latter is to indi-cate the maximum (or minimum) tolerable level of certain variables.

''For the moment we further cave out of consideration that there may be preferencesconcern,ng a sequential activation of instruments. In our application (ci. ch. 4) this (leais taken up again.

12Note that this general formulation may without any further complications implyupper and lower bounds on the range of instruments and/or endogenous variables. Theseparation of targets and instruments in : and Z2 actually is nonessential, ci ci,. 5.

3Esen if there is only a grouped ordering our approach -although somes+ hat morecornpl,cated is still applicable: ci. ch. 5.

617

Actually, as far as upper and lower hounds are COncerIle(j W{ coilciderthis to he a more adequate representation of the real policy PrOhle.furthermore, in most realistic situations we expect sonic orderij)g

f t;tr.gets to exist. If this s the case a formulation as outlined hov turns oato he somewhat more straightforward than to try to catch U) this rdcr;,..

Jwithin a reasonably complex weighting scheme; apart from that weCOO.

sider it to be intuitively more pausible to the policvniakcr. We are nowa position to identify the basic methodological ditlerences White thestandard approach can be characteriied as a minlml,ation problem underequality constraints the essence of this method is to construct lt\jhle soiUtiOflS of a mixed equality inequality system h' an iterative procedure

Before we now come to the exposition of our procedure, let us suiriup: z is a vector containing upper bounds on instruments, the respectivevariables are contained in the vector Z2 (the elements of :, form a subsetof a). is the vector of upper hounds on endogenous variables its ele-ments being ordered according to their importance lor the politician Leus assume that the system's endogenous variables are ordered such thatthe i-th order restriction refers to the variable x. Let : contain the tirst

endogenous variables, which are to be restricted. From what v,as es-plained above it follows that the politician wants to know a po}ic\ hichfulfills the following equality/inequality system:

(4) Z1

f(.) = 0K '

But, as it was pointed out--among others- by Livesey (1976): 'Theformulation of economic policy is . - an iterative procedure, with the out-come of one policy evaluation influencing the formulation of the nextplanning exercise.' So our aim is to give the policy-maker together with asolution of (4) as much information as possible for the evaluation of thederived policy under the aspects of the relative importance of his prel-erences and the implications of their modification.

With this model formulation we now come to the calculations. Thebasic idea of the solution method is, starting from a :-feasible point, inthe procedure stepwise addition of the restrictions in according to theirordereither to construct a (4)-feasible point or region or to shoc that(4) has no solution; in this case we compute the feasible value for thatparticular restriction that causes in feasibility.

In the simplest formulation the steps of computation are therefore:0) (Construction of a feasible starting vector) Choose any (reason-

able) zr-feasible set of instrument values and compute viaf(.) = 0

618

(5)

the respective endogenous variables Set the -Ieiitent Counter1: = 0.Set i: i + 1. If I > i the number of elements in , then stop.

2) Test, whether the i-th order restriction in is vin!ated by the cur-rent value of the respective variable x,. If not, go to I).

3) Solve the following probkni:

mm x

s.t.

j(.) 0

:, < z'where is the truncated vector containing only the firstI - I elements.'4 ie. those restrictions which have been dealt within earlier steps 2) or 3),1

4) Substitute

'40r cquivalentl: a vCCtor . whose elements + . are fixed at the highestecononi,callv still meaninc'ful values. Note that the minimum of .v is independent of theordering of !.

Noie that proceeding th:s ssav the restrictions are in 'any case luhilled whereas in theacighted critCrlon function approach it is generally not guaranteed that the solution is

"close" to the slesircd path: cf. e.g. 1.ivesey( i973 a,h:l974).'' Fvaluaiion of the multipliers in the present paper means that we make use of the well

knossn propert of the Laurangean multipliers, nameR to indicate the derivative of thecriicrioii function v. chanites of the right hand side of the respective inequality con-siraints; ef. e.g. Liienherger (1969). pp. 221 223, Peterson (1973). This allows for any easycalculation of local elasticities.

619

(6) = max (,x)where is the maximal permitted value (of the i-tb order restric-

Note that step 4) above is essential: we have either constructed a

tion) for x1, and xm the minimum achieved in (5). Go to step I).

'-feasible point or in the case where v is greater that the originalvalue shown that this i-th order restriction is incompatible with thepreceding ones of higher importance (including the vector zfl. For thesolution of problem (5) we make use of a Lagrangean approach (cf. thenext chapter for more details), especially to compute the multipliers for

I

binding restrictions.Going through steps I) to 4) finally leads to the information output of

the ultimate feasible solution as well as the respective results front inter-mediate steps. If an optimization step 3) was performed, this would in-clude the values of the multipliers'6 to indicate the relative importance of

the effective - and z-restricttons, which indicate th 1SitlVit' of th1results w.r.1. variations of the :*_values; this is the inlorma,t

neededfor policy analysis. From the well-k nown nianiloki evaluatiri

Possihilitjc,We only note the following: a high dual foi an instrument at the

houndstates on the one hand its effectiveness arid on the othcr hand the

necessiltto control it precisely: a high in tilt ip!ier For an endogenous variable

at thel)Otlfl(l stales the crucial importance oF a precise knowledge of the respeoti ye -va I tie. /

\Vtth these results at hand the policymaker has the follo'irgOption

either to accept the result as definite or, if he considers the inforritionaicontent of the multipliers to he insufficient, to respecily the vectors

ancj,possibly, z concerning values and ordering. In the latter Case We ouIdstart the computational procedure anew. hence the essence of this

methodis to perform an alternating sequence of computation and

evaluationSte

3. ri A!.cRlTlit

For the solution of (5) and the evaluation of alternative policies aoutlined above an algorithm is needed which is not restricted to ihe h'i-dling of quadratic criterion functions, which allows for inequality restric-tions on endogenous variables as well as on instruments and, fmnallv. com-putes the values of multipliers at least for the inequality constraints.

The literature on solution methods for this general NLP-prohlem ()is not very extensive, especially for the numerical treatment of medium.sized or large-scale problems. In the following we outline our algorithrn.a

A) Transform all inequality restrictions to equalities by introductionof quadratic slack variables:

7Note that in a stochastic svste m where ihe (estriettons get a stochastic character too,the mu I tipliers can he evai hated under th isaspeci. lieu risiica liv. the restriction then could heread as

+ or s- -

s here is a random term.5Reordering of course can onI he expeded to he eikctive in ihe case 01 a suiiiieiiinonlinearity of the muitiptiers.9iust to see shether the preferences arc feasible, obviousI. other (and simpler) pro-cedures siould do as seii. e.g. the changing 0) the target variable in l in each step siouldhe Uflflecessir', To choose hiisser the lossesi preference variable as a tixed target ssouMnot give as much inforniatton

concerning the leasihie relijon as does the changinti procedureadopted here.20This sequential proceduec is in the spirit of Kornai (191171 and 1ive,e t i97br cI fin9 10(1.

21 I-or a more detailed exposition cf. Hock v. WiiI(ingen/j'atils jljh pp I 0 Oneof the referees noted th ...the superiorit) of the nonlinear prograttini ing algoritltni to an'other existing algorithm is not at all clear.' Ax a general remark this is correct: hut one otthe a priori reasons for the development ot this algorithm s as its high rate of oh1sergence.ss hich in our moditicatiori (em plo i ng an idea of I aasonen (19W))) is os C i uad ratic. More-over the Jacobian ifl (lOi S ers easy to cons pute for ecoiloltict nc models

6 2()

thededties

LI ndSit

theec-

zi +

Z2 -f .

B) Transform the restricted OPtimization problem (5) to a non-restricted cxtrcmal pioblem b introduction of the Lagrangean.L = x, + M(z1 - .c

+ 41;(z. S ;:) + A'f(.) mmonnal C) Set up the whole set of necessary conditions for an extrernum of Lud, (L ilL IlL ilL L ilLuld ( )

ulu' es,' th2 01t11' SM,' ILJodon Note that in an case of nonlinearity in the system (4) the respec-tive derivative equation in (9) Contains a nonlinear mixture ofsystem variables and multipliers.

The respective derivative for a variable entering the systemLS only in linear form is, of course, a linear combination of mul-In- tipliers. The derivatives w.r.t. the slack variables simply slate thcell known condition that the slack or the multiplier must equalzero.D) Solve this system (9) simultaneously to get the optimal values of(5) instruments, endogenous variables and multipliers.m- The iinplenientatjon22 is characterized by the following nrnnrtv21 . The model fis coded in data form, not in a program or procedure

2'' and z are represented simply by indices and critical values ofthe respective variables. This makes model modifications orchanges a very easy task, and in either case we do not need totranslate any program anew.The system (9) uses analytical derivatives. They are created inter-nally by the program using the input for the model f and the vec-tors 2 and z. Thus we circumvent the rather time consuming

fl approach via a formula interpreter and code generator, whose out-put normally has to be translated before further processing.(9) is solved with the Newton-method. The iteration prescriptiond

23d iS

(10) = -where v is the vector of unknowns, which in our application con-tairis the whole set of variables x and u, the slack variables s, andthe multipliers A and M. Note that the derivatives of (9) in the

2Cf. ibid. for a more detailed discussion of the imple!nentational advantages.73Cf. any standard reference. e.g. OrLcga/Rheinholdt (1970).

621

Jacobian I contain the first and second derivatives of the Otigiflmodel (I).The derivatives needed iii the Jacobian J ale also CO1ttptitdanalytically and generated internally: 10 CaCII iteF(jWi / the Wholesystem (9) is evaluated siriiultaneouslv for I hC

cOftiPUtiltioii of thenew Jacobian of the last tera ted value ', F denotes the Vector efresiduals of all equations for i',

o I he main computational burden iii the approach IS associated withthe solution of I or equivalentl the solution 0 a large Scaleiiiiear eq uation System. In on r implemental ion W first eXtrjct fromthe original system (9) the upper and lower triangular part, theiitest the remaining interdependent system for bl0Ckdjag,I;il struc_ture2 and 1 it has one repeat the truing uli,atioil and

deconi_position procedure for each block and so on. l:Ventuallv

( O)applied to each indeconiposable block separately. employing ifnecessary sparse matrix cch ii iq ecs as familiar from

I_P-i iflpk_nientat Oils.

4. A Fx,sipii

The Function of the following chapter is to give a rough impressionof' the working of' the procedure. For expository purposes we have ap-plied our niethod to a medium-sized, nonlinear, and interdependenttheoretical model, It can he characterized as a modified Keynes_Wicksellmonetary growth model of an open econolnv with price 11exhifitv aridlabor market disequilihriun special emphasis has been Put on the intro-duction of stock and flow constraints as well as on the fulfilment of thgovernment budget constraint.17 It is a condensed version of a larger two-sector model, which has been described in detail elsewhere25 In order tofocus primarily on the application of the proposed algorithm and thespecific form of the criterion function, in the present paper a more com-prehensive discussion of the model has been sk ipped.2

Within the context of this model we try to solve the following prob-lem. assume there is an ordered vector of targets of' economic policy

he hajc dcis hase been ouilined e.g ii lJeIIernIan/Rarjck (1972)2S( e.g. Icher (i9721.SUISC% of nmdcts ot ihi kuid d Pra-(io i ( i97)- I he iUndiii,jJ !lnporiaIJce ol i t1cc ispcc1 in ii crenom Ic poiu niak sirccciiij hen pointed 0th e.g. b lirinson (I 97bL Iiuk s. UIthnCfl/J'k 977. pp 0 5. Iscu lliis coittleiised sersionup lo about i() Cqu.i(ions ippro\IhiiiicI till ol (hem ,irc C eii(,iji inierdepeiideni \iihregard to I he degree of diigerciiuiii a tsell 1 iii u iric coni ii) Frinen or.k ihmodel is er nLiLh in the spiro of recefl developed more cmii prehensise riiodels, ciNlu1 ( 97(i).

2A detailed 'ersioii of (he equ tori sstein is. of course, as,iilablc honi ihe iiithrsIII requesi

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here 1' = rate of inliatioriU unemploynierit rate1)IF = budget delicitB PA = balance of payments aceou nt

At the policymakers' disposal in this (lixed exchange rate)SvstCfll

there are three instruments: government expenditures (G ). the amount ofthe monetary authoritv's autoiionious open market transactions (Q,tf

)

and the exchange rate (it ): two of these three nstrumcnts. however areconstrained as well, namely

-- (; - 2514

= vitIi =- - I .O

---it. -- .9

The optimization results can be dra ri from Fig. I Since in our ap-plication there are upper and lower bounds on the sante varahle a slightmodilication of the general procedure is applied: if the violation of theupper (lower) restriction in step 2) makes an optiniitation (5) necessar

e account br the respective lower (upper) con strai ut at the same time:this is just to reduce the computational expense. Note further that 'seadopted the special case referred to in the exposition of SUCCeSSiVe activa-tion of policy instruments.

Starting ith the choice of a - feasible vector the solution pro-ceeds along the following steps:

Add: P < 2; no corn putationAdd: < 3: minimize U.

Ill) Add: 1)1 1 < 2; no computation.Add: l)EF I: minimize - DE F.Add: -- BI'A IS: minimize tWA.Add: B PA < 10: no computation

The final solution I has the undesirahie property that the mostimportant restriction is at its hound; we would assume that the policy-

°oie that, 'Inc the nstiumCnts hiae hecn set tree iecesSI\eh\ the \aIUC 0t 1 t,it5tariahle n1a. ri a sutrsequeni siep. still hail helo that aiueachteed PfeIUa. ii hen the

respective sariahk ssas chosen as a tarnet sari,ihle

(24

I)

1) [1'vith- I - 1)11

J.BPA '5L BP,\ I U

hiIC

ry

-e

a

0-

maker takes this final result as the basis to ask for further analysis of theloc:tl properties. Nevertheless it turns out in this exampte that the polw -

maker's preferences are at least feasible. It this were not the case we hadto ask the policyriiaker to respecily his preferences, e.g. by weakening therestrictions in . Then the procedure had to be applied anew.

5. Mo111ruAnoNs ANI) EXTENSIONS

lip to this point we have explicitly considered only one-period prob-

lems. in a niultiperiod approach there could be slight modifications ofthe formulation. E.g.: the z 4-value in z would be treated as restrictions

m on the average value of the respective variables:

1/T. :, <

where T is the long run planning horizon. If it is the policymaker's pref-erence not to allow some variables to deviate in a single period from theaverage value by more than a certain amount. we would add the followingrestrictions:

Z1, I I

where

b = diag(l + O.O1b,)

and b, is the maximal tolerable (percentual) deviation from the respec-tive average value.3'

if- -in the multiperiod case the politicians preference actually is tomaximize a certain variable (say capacity growth or per capita consump-tion) upon holding of the restrictions formulated above, this would causeno computational problem: the last Step of the procedure described inch. 3 would be the maximization of that particular variable under therestriction of (4). A discounting in the criterion function would be un-necessary because we can reformulate any kind of intcrtemporal pref-erence into an adequate inequality restriction.32

Of course concerning our main aim, the entire elimination of theweights from the approach. the linear criterion function in (5) is not essen-tial. Any other, e.g. a quadratic function containing only one single vari-able would do as well and he conceptually equivalent. In this case.

HThe problems arising in the rnultiperiod case concerriiflg the algorithm hase been

discussed in Bock v. Wtiltingen/Paul) (1977). p.6.a recent paper Ku/Atliafls/Varaia 1977) hae pointed out the crucial irflpO[-

lance of the numerical value of the discount rate for the stahilit) properties ol dy nanti

s) stems. Despite that the introduction of a discounting factor ss ould onI) be another ek-

ment of arbitrariness.

625

I

b

/ hO\s ever, the numerical evaluation of the in nIt ipliers s o uld ht'eto takeinto acCount the specific form of the criterion ftincho11 this would he thcase particul:Lrlv for piecewise defined functions

from the Cxposilion of the computationalProcedUre inl;uuld he evident that the assumption of : containi °iily iIisirLjnptand con taming only eridogenous variables is not essential either Withcontairii also instruments the approach would h identical II thrgare eI1(l&)gen()u: variables in the construct ion of a feasible

startingpoint . oiild have to he slightly modified, becauseit could iitvols'e stillsome Optimization steps as in (5).

One of the fundamental ingredients of the present procedure is a SStematic evaluation of the multipliers. Using an algorithni which does notcompute the values of the multipliers, changes of the criterion functioflvalue ith regard to variations of the restrictions on the instrtlnlentsmight as well he computed simply by sim ulation runs with modifiedvaluesof the instruments (as long as no restrictions on endogenj5

variables ireor become active) In contrast to that the corn putations of !Uultipljerss ft. restrictions on endogenous variables would deserve for repe1tedOptinittatjon runs for slightly altered *51!Ue5 of the respective elementsIn our example we gave the multipliers only for those steps in whichan optimi/atjon was necessary and trivially the duals arc Zero fornonbinding restrictions To gain a deeper insight into the implications ofthe given preference order, we could perform step 3) afler adding an ele-ment of, irrespective of whether the restriction was eflCctjve or not.Furthernmre, we- could in any case insert the following step:(3a) Set each element of c . where the restriction is not binding at avalue with a zero slack.33 Now compute the multipliers of therestrictions34 the information contained herein being of Specialimportance in the stochastic case (see ftn. 17).Finall let us look at the requirerne13 of strong ordering of all mestrictions. A m1aturtl weakening would be a grouped ordering in the sense thatsome *vIlues would

he considered of equal importance by the Politician.In this case "e 'sould have to add not only one hut several restrictions ina particular step of thecomputation procedure

If none of the group is violated by the solution of the last step, thereis rio further prohle and can proceed to the next group or singlerestriction1 Otherwise se first try with sonic auxiliary criterion function toconstruct a ne\s solution which fulfills those restrictiojic to he added. Ifthat turns out to be impossible we raise all restrictions of the groupe.g. i 62 2 = SCL * 1.62 - ), small hut rIoI),ero ()hsiijsl= 0 sout(t kad to degcnci-ac

34Butthc ness Sojuuon ot cotirse has not alt restrjetjo,15 hut- so to et theFflUitJpItr lntorrnit1on possibt (lCsCf\ Cs or more th onC addjtion,i OptjIt1uatii step

626

f)

simultaneotislY by the same rate to achieve feasibility35 Note that thisC states an infeasibtlity of the original vector z too.

We have tried to show that the present approach is a very flexibletool and allows for a variety of extensions and modifications- the funda

(S mental idea in any case is based on one single unweighted target variablelb in the course of solution and on the manifold application and evaluationrc of the multipliers.

C

ill 6. Sutsixtv

In the present paper we set Irth an alternative approach to the)t formulation of the criterion function in multiple target optimization prob-ii lems based on the use of one- or two-sided inequality constraints on tar

ts gets and instruments. The computational aspects of its implementationes have been discussed in detail. As far as the application to problems of

IC economic policy optimization is concerned, the main advantages of ourproposal can be summarized as follows:

d It allows to separate distinctly between targets of econon policand more or less technically determined restrictions.Most of the notorious problems involved in the specification ol theweighting matrix in the standard quadratic or otherwise penaliteddeviations approach can be avoided.The problem of sensitivity of the results w.r.t. variations in theweighting matrix is circumvented.The formulation of targets and restrictions requires a considerablylower degree of preference revelation. Actually, all we need is an

e ordering of the targets-- at least grouped and numerical values

d of upper and lower bounds for targets and instruments.As far as policy evaluation for different preferences, i.e. modiliedsets of * or -vaIties is concerned, we suppose our approach to bemore suitable than the resort to a single welfare index in the stan-dard approach.

n The local elasticities of the target values w.r.t. the restrictions uponother targets and/or instruments may be evaluated making use ofthe Lagrangean multipliers.If a global analysis is preferred, in this approach a systematic

0 variation of the bounds allows for a straightforward constructionof feasible regions of solution.

"Assuming irnplici.l some sort ol locul honiogeneit ol the politiciari preler-

ences. In spite of ihe grouping thk ritas he iuesiionahle. just as the tolio mg procedure: add

all restrictions of the group simultaneously and change the bounds successisely until all miii-tipliers of the group have the same value (which is practicable of course only in the caie of

sulticient nonlinearit : ii furthermore ititroduces an additional problem of choice. sitice the

duals are conditional on the respcctls c target satiable

627

Although ve make no stochastic optlrniiation in ur proceduthe stochastic nature ol a particular s stein can he taken

into account h an appropriate evaluation 0) I he Ci)nstra jilts' dual varja h ks.As f'ar as Our experience indicates these advcintages Outweigh thcomputational impediments which may he connected with the

tPPlicatjonof our method in a particular problem. Thus e arc not in aPoSition 10share the pessimism sometimes expressed Concerning the t0ffl1Uh1tjo

ofeconomic policy problems in terms and techniques of NI_P. As'stefl1aIiccomparison, however, must he left to future research

Lnit'ersj',1' o/ /JwnhurgCut o/ i'('OflOtflj'ç

R is H RI N ('iS

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Conferenceon Dynamic Modelling and Control of National tcononii Vienna, January 1977(forthcoming th the conference proceedings)Branson, Vi". II. (1976). The sinai rote of the governnienl budget and the balance of paymentsri the flloveriient from short -run to lo Fig-ru ii eq iii hi) urn. Quorit'rf,- Journal o;iioaiii.',, 90/ 976, pp. 345 367

J. ( I 974). Predtctive corttrot 0) a stchasiic model of ilic (5K eeOitomv simulatingpresent polic makin practice by !he P K L'overflirieni0/ /.co,,,,,,,w a,ts Sth'ialIfeciai,r,'ccfr,jl 3/I 974, pp. 239 255

Bray, J. (1975). Optimal control of it itsitsy econo,n with th I 5K is an esaniple JOUTSCIof the Rot'a! S1atiy/i,j( Scr. A, 35/I 97S pp. 3.49 366(hii (i. C. I 975): .4 "a/coa anl eoniyo/ of dt'awnjc (5000 iHi' iCvfi'?flc Nets York I 97SIknr,is, J. L, J. Mord (1977)' Qticisj-Ness ton methods niotivatiori arid theory S/4 .1/ Re-taos- 19/1977 Vol. I

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625

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at Ic

)Urg

flies

Ia inc

Oflo-

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1977.

cntt

Ii igoil!

Re-

and

tiC.

eels,

len-

sir-

darn

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2)

I

(,1

a

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