Qualitative reasoning on economic modelsChristian Steinmann

Wirtschafts- and Sozialwissenschaftliche Fakultat, Universitat RostockD-18051 Rostock, Germany

e-mail: christianOwiwi .uni-rostock.de

AbstractQualitative analysis has a long tradition in eco-nomic modeling . The application of mathemat-ical methods, either quantitative or qualitative,to analyse an economic model stipulates strongassumptions. These assumptions restrict theanalysis on a special case of the more general,verbally stated model. Furthermore the applica-tion of these methods is successful in analysingsmall models only . In this paper it is dis-cussed, whether a qualitative formulation of aneconomic model and its qualitative analysis viaqualitative reasoning will result in statementsdescribing the behavior of the model. Doingthis, economists can avoid the pitfalls of havingto state the model as a concrete system of dif-ferential equations . However it is not expectedto achieve different results. Beyond this, maybemore complex models can be analysed qualita-tively .

Introduction

Whole seen the real economic process is ex-tremely complex and difficult to survey. This pa-per is not the right place to analyse the complex-ity itself, to aid further discussion I only want togive a short glimpse into the problem, first of all .Each household and each member of ahouseholddecides how to act in the economy. These deci-sions are the foundation of the business recessionand revival, of stability and growth of an econ-omy. They are influenced by many _economicvariables . On the other hand the decisions them-selves will change the value of these variables .The interaction between economic variables anddecisions of economic subjects, in combinationwith the vast number of economic subjects andgoods, prevents the description and analysis ofthe real process in detail . The elementary op-erations in economic acting are producing, sell-ing, and buying of goods. In principle each kind

of good has one market and one price. In ad-dition, firms try to distinguish their productsfrom those of their competitors . Hence, goodsof the same kind differ in quality, service, color,or equipment. Therefore on every goodsmarketthere are goods marked with differing featuresand dealt with different prices . There are hardlyany homogeneous goods. Adjustment of pricesand quantities leads to an equilibrium in pro-duction, supply, and demand of goods. It wouldbe hard to reason why this adjustment shouldtake place in the same way, and with the samespeed on every market. The adjustment of pricesand quantities in one market will be influencedby other external variables than those influenc-ing the process in another market . The sum ofall adjustments in every single market for goodsor production factors forms the dynamics of theentire economy. Hence, it is impossible to giveexact prediction of the real economic progress,since it is impossible to describe the real econ-omy in a model which can be analysed. By theway, it is not enough to look at the economy ofa single country, an economy has to be investi-gated worldwide, apart from the problem thatthe real functional relations between economicvariables and decisions of economic agents areunknown.

The crucial question concerning this mi-crobased view on the economic process iswhether suitable assumptions can be found todescribe the behavior of the economic agents .The economist has to deal with the samequestion like sociologists or, phsychologists asTESCH (18) asks : "Is the human/social world,just like the physical nature, governed by natural'laws'? Or is human behavior something we cannot predict, something onto whosc inner .work-ings we can only shed more or less light ,in . oureffort to understand." On the other hand, also ineconomics the whole thing is more than the sum

of its parts. The physicist can not predict thebehavior of a single electron in interaction withothers . Despite of the fact that every electronbehaves according known laws, a real system istoo complex to be analysed . But the behavior ofthe whole system, for example in thermodynam-ics, can be predicted . Obviously the economicprocess does not behave like a chaotic system .Without complete knowledge it can be assumedthat the system behaves according certain lawsalthough the behavior of every single economicagent is not completly known.

A central task of economics is to identify thebasic laws of the underlying economic process.For this purpose the goal of economic researchis not to represent reality completely . Insteadonly small cuts are considered . Doing this, un-realistic assumptions have to be accepted to re-strict the model on such a small cut. However,the main parts of reality have to be selected, andother parts are excluded with the help of theseassumptions. The result is a large gap betweenthe real world and the model. GOODWIN statesin his article "A growth cycle" in a similar con-text : "Any . . . economist should ask: why anal-yse an unreal, idealized system? The answer isthat to show the logic and plausibility of a typeof behavior and of its analysis, it is essential toget it clearly and simply stated." Accordingly anunrealistic, idealistic model is inspected in orderto determine certain aspects. In a more realisticmodel these aspects would otherwise be super-imposed by other effects. Additionaly, it wouldbe impossible to solve by analytical methods.This problem defeats most attempts to verifyeconomic models by econometric methods em-pirically. Every model, and especially every eco-nomic model, has to deal with a big number ofvariables as exogeneous constants. An econo-metric analysis of such a model has to filter outexogeneous shocks which happens in reality be-cause these variables are not constant .

The problem in this paper is the analysis ofan economic model. The economist states as-sumptions to build a model like Solow's stablefull employment model, described in the nextsection. His further task is to prove the dy-namic behavior of the economic process defined

by that model. Will this model lead for exam-ple to stable full employment, to stable underemployment, or will the process be unstable?

Qualitative methods in mathematicsEconomic modeling and model analysis are notimaginable without the application of qualita-tive methods . Proceeding from a verbal modelformulation the resulting growth cycle will bedescribed verbally, too. It is stated and arguedin linguistic terms like high, :low, increasing, ordecreasing. On the other hand, the formal de-scription of dynamic economic systems in formof differential equations is based on the work ofWalras, see (13) for example. The formal nota-tion in differential equations ensures an unequiv-ocal model description and allows analysis withmathematical methods. However, these differ-ential equation systems are investigated quali-tatively. "By qualitative analysis we mean theanalysis of the general properties of the model,abstracted from the single numerical case", saysGANDOLFO (6). According to Gandolfo qualita-tive Analysis divides in i) fixing the equilibriumpoint, ii) comparative static analysis, iii) equi-librium analysis and iv) analysis of parametervalue sensitivity. Statements concerning the dy-namic behavior of a model are derived withoutquantifying the parameters . To apply qualita-tive methods of mathematical analysis it is nec-essary to accept some restricting assumptions asstated above. Especially linear coherences areoften assumed . When some distinct relations areexpressed by linear differential equations, theseformal notation is describing a limited specialcase of the more general model which was ex-pressed and specified in verbal terms. The re-sults of the qualitative analysis of the differen-tial equation system are used to analyse and toexplain the behavior of the verbal model. How-ever, this paper concentrates on Gandolfos partiii) equilibrium analysis neglecting the others,mostly.For example, the fundamental equation of

Solow's stable full employment growth model is :

I = n-sf(l)

(1)

The growth rate of labor intensity 1 equals thegrowth rate of labor supply n less the growth

rate of the capital stock sf(l) .

The marginalpropensity to save s is assumed constant in therange [0,1] .

f(l) denotes the neoclassical pro-duction function in its intensive form expressedby means of labor intensity. The original pro-duction function is noted as F(K, L), produc-tion is a function of labor and capital input . Thefollowing assumptions conerning the productionfunction are made:

or

f, >0 and f"<0

FK>0, FL >0,

FKK < 0,

FLL < 0

and

FKL = FLK > 0.

Le. the slope of the function is strictly posi-tive and decreasing. This notation describes theshape of the function qualitatively. A concreteinstance of the function, like f(l) = 0.50, isnot given. The Solow model is analysable by aqualitative equilibrium analysis . After an exoge-neous shock has happened, the process developstowards the (new) full employment equilibrium.Under the stated assumptions, the equilibriumof the model is globally asymptotically stable,see (4) . To conclude this from the model, it isnot necessary to impute a concrete shape of theproduction function . Such an assumption hasonly to be made for a numerical analysis whichbecomes necessary in a different, more complexmodel where no statements result when usingthe above mathematical methods .On the other hand, strong assumptions are al-

ready stated to enable the application of math-ematical analysis . At least the concreteform ofthe differential equations has to be assumed any-way. Normally, linear or very simple coherencesare stated . Savings in the Solow model are as-sumed as S = sY with s E [0,1] as the constantmarginal propensity to save. This is undoubtlya strong assumption, but it is presumed that themain topics of the model do not depend exten-sively on this assumption .

For, another example consider the cross-dualadjustment rule for price and production in asimple market clearing model. The adjustmentrule is denoted cross-dual because of the as-sumed crossing effects . Quantity affects the

price development, price affects the quantity de-velopment:

P a(Yd _ Y.)

Q(P - c)

Q

_ IjCY .

Again linear coherences are assumed. Excess de-mand yd - Y' affects the price with a constantadjustment speed a . An additional unit demandincreases the price p, independently of the pricelevel or the level of demand and supply: Accord-ingly, the excess of the price over the productioncosts per unit c affects the supply Y' with a con-stant adjustment speed (3 . The analysis of themodel by application of the Routh-Hurwitz the-orem yields a stable equilibrium, if the demandis decreasing in changes of the price, or the pro-duction costs per unit are increasing in changesof supply, the production quantity. Concerningthe Jacobi matrix

yd has to be negative, or cy . has to be positive .In the case of a price independent demand, pd =0, and fixed production costs, cy. = 0, analysis,with the additional application of an appropriateLiapunov function, yields neutral oscillation, seefor example (2) or (9).Without quantifying the parameters a and A

in the stated market clearing model, or s in theSolow model above, the chosen mathematicalnotation contains strong assumptions . In thisway, the intended behavior of the economic sys-tem is fixed beyond the scope of a verbally statedmodel. Maybe it should be stated that the priceis increasing with the excess demand, or thatsavings are increasing with income under the re-striction that savings have to be smaller thanincome . Formal modeling has to solve the prob-lem to find a model description, which fulfillsthe presuppositions for the application of quali-tative mathematics, imputing only such assump-tions which do not falsify the intended core ofthegeneral verbal model. It is easy to imagine thatthere are assumptions, possibly stated in `not-ing a system of differential equations, which will

yield effects that are characteristic for this spe-cial case only, and do not represent the generalbehavior of the model.Adding these points concerning economics I

have to go a bit further than for example SIMON

does in his preface to the book "Qualitative Sim-ulation Modelling and Analysis" (3): "Some-times, we know the shape of the equation thatgoverns the phenomena of interest, but we do notknow the numerical value of parameters - per-haps, at most, we know their signs." I think atleast in economic modeling not even the shape ofthe equations is known, virtually . The microeco-nomic behavior of economic agents can only bedescribed verbally, resulting in qualitative effectson economic variables, or describing macroeco-nomic coherences between economic variables.Generally, only the direction of the effects isknown. Further assumptions that fix the shapeof the equations are stated for the sake of ap-plicability of the mathematical methods. Whenthose assumptions have been stated, then thereare parameter values, not earlier .

Numerical simulation

In various fields of science and practice simula-tion is used to examine and visualize complexprocesses, or systems, through simplified mod-els. It is beyond the scope of this paper to statethe aim and purpose of modeling and simulationin general . I presume every well disposed readerof this paper has a sufficient, maybe intuitive,insight into the application of simulation meth-ods .Running asimulation on a computer, the algo-

rithm deals with numerical data . These are mo-ments of the distribution of duration or interar-rival time, or other features of objects in discretesimulation, or values of level variables, parame-ters, and rates of adjustment in quasi continuoussimulation. . Of . what kinds are the aims of simu-lation?

In several situations the objective of simula-tion is parameter optimization . Here the aim isa numerical result . There are many examplesdescribing this kind of application . A typicalone is the control of a road crossing by a trafficlight system .

In other possible situations simulation is usedas an aid for decision . Someone has to chooseone of several alternatives and he is interestedto choose the best or at least an appropriate al-ternative . In this case the numerical result ismeaningless, and interesting only in compari-son to other results. Maybe there is addition-ally a point of interest whether the numericalvalue hits a special range. A typical examplehere is the arrangement of machines to'install aproduction process with some choosable alterna-tive arrangements . So, the target of simulationis to find, according to some given criteria, thebest possible alternative under a given system ofrestrictions . The numerical result of the simula-tion has to be classified, ordinally.

Finally there are other situations without anynumerical input data available. To run a simu-lation those variables have to be quantified, ar-tificially. Here a special case is modeled whilea general case is the point of interest . Usually,numerical results are meaningless in such situ-ations . Qualitative inputs are transformed intonumerical data, and the numerical results haveto be interpreted qualitatively . There are sev-eral applications in natural science, in physicsfor example, and in economics, especially . Theapplication of qualitative simulation is straightforward .Applying simulation, it is possible to analyse

more complex models which can not be analyseddirectly by mathematical methods. An interde-pendent economic model containing more thanthree differential equations usually can not beanalysed by qualitative methods of mathemati-cal analysis . Maybe it is possible to describe thebehavior of subsystems . Otherwise it is neces-sary to add additional assumptions for yieldingat least restricted results. For numerical sim-ulation of such a model again the model vari-ables have to be represented in numerical values .A concrete situation is considered as a specialcase of the more general, formally noted model.See (14) for example, for numerical methodsused with differential equations.As stated above, the numerical result yielded

by a simulation run is not the aim, but an in-termediate step . The result is to be interpreted

qualitatively . The main purpose in order to anal-yse the behavior of the more general model is togeneralize the result . Generalizing numerical re-sults always runs the risk of losing, if the resultsare specific only for this very limited special case .Inspecting a numerical model compared to a dif-ferential equation system is a stronger special-ization than considering a differential equationsystem compared to a verbal model .

Tpt(t)! qi(t)" yi(t)

Figure 1 : Numerical simulation

To figure a simple example I extend the differ-ential equation system stated in equation 2. Aneconomy containing two goods and two factorswith cross-dual adjustment has to be analysed .It is stated that households consume good 1 andfactor 1 . These are useful for the households .The utility function is of Cobb-Douglas type . Inthis way, supply of factor 2 is constant . Good 2and factor 2 are used together with good 1 andfactor 1 as inputs to produce the two goods. Theproduced quantities are yi and y2 . The prices ofgoods pi and p2 and of factors qi and q2 developin reaction on excess demand. The producedquantities are affected by the surplus of priceagainst costs, see (13) for more details on cross-dual adjustment . Figure 1 represents the resultsof a numerical simulation of this model, in reac-tion on an exogeneous shock reducing supply offactor 2 . The adjustment process takes a violentcyclic course to reach a new stable equilibriumpoint . Slight modifications of the model yieldin significant changes in adjustment (17) . If forexample households consume all goods and fac-

tors, or if there are firms with a different produc-tion structure, the oscillation is damped. Afteran equal exogeneous shock the new equilibriumpoint will be reached much faster .

The qualitative approachThe main problem in analysing a verbal statedmodel is the limited possibility to receive un-equivocal results by verbal argumentation . Itis not a simple task to consider all effects be-tween the economic variables completely . Addi-tionally, a verbal model representation is usuallyambiguous. For these reasons economists anal-yse a special case of the model depicted as dif-ferential equation system . Figure 2 shows theproceeding to analyse an economic model.

Figure 2 : Analysing an economic model

The verbally described model is considered byverbal argumentation only . A special case of thismodel depicted as a differential equation systemis inspected by qualitative methods of mathe-matical analysis with the aim to receive state-ments describing the behavior of the more gen-eral model. Finally a quantitative special case ofthe differential equation system is considered bynumerical simulation, if mathematical methodsmiss applicability. The arrows directing down-wards stand for the development of a specialcase, the other ones directing upwards expressthe attempt to generalize the results.The qualitative approach, analogous to quali-

tative physics, see (8), or qualitative process the-ory, see (5), extends the analysis of economicmodels . A new level of analysis is included .GRANTHAM and UNGAR state in their article"Qualitative physics" considering physical pro-

cesses : "The aim of qualitative reasoning is tocreate and use representations of the world whichare simplified so that irrelevant detail is ignoredwhile maintaining enough resolution to distin-guish and explain important features of behav-ior. "

This approach contains a formal notationwithout imputing the shape of functions or lin-ear coherences . The methods of qualitative sim-ulation and qualitative reasoning support thisapproach by making appropriate tools availiablefor analysis . Without these methods a formalnotation would be worthless at all . The corre-sponding proceeding of analysis can be seen infigure 3 .

differential equations system--yqualitative mathematicsI1

quantitative special case

Figure 3: Analysing an economic model

Qualitative Simulation

Laumerical simulation

To present a simple example followingKUIPERS (12) Dynamic Qualitative Simu-lation I will use the model describing cross-dualadjustment of price and supply, which had beenstated similarly, via differential equations in (2)on page 3. It has been analysed by mathemat-ical methods and numerical simulation in thecorresponding sections ahead. A negative effectof return per unit r = p- c on production Y' isassumed. c is the constant cost per unit output .Excess demand Ye =Yd-Y' has a positive effecton the price p and as a consequence on return,too. Furthermore demand yd is a negativefunction of price p. Qualitative functions are

denoted F in this paper.

Values of return and excess demand may bepositive, negative or zero . Price may be in therange (0, c), equal to c or larger than c. Zeroand-c are special points in the value range ofthese variables . They mark turning points in theadjustment process. Like c zero is a threshold,called landmark value in the qualitative context .Values can be increasing, decreasing or constant .The states of the variables return and excess de-mand are characterized by two values : First theactual range or landmark value respectively, andsecond the direction of movement (increasing T,decreasing 1, constant -, or indeterminate ?) .

Simulation begins in some start situation, forexample in a situation characterized by posi-tive return and negative excess demand. Equa-tions (3)-(7) give the corresponding values ofthe other variables and their directions of move-ment. Price p starts in the range (c, oo). Theother variables, Y', and yd, are bound to asingle qualitative value, the range (0, oo), dur-ing simulation . Y' and p are dynamically de-pended variables . Their direction of movementis given by the value of the variables r and Yeon the right side of equation (3) and (4): Pricewill increase with excess demand, supply will in-crease in reaction on positive return . Directionof movement of the other variables in (5)-(7) isgiven by the direction of movement rather thanvalue of the right side of these equations: De-mand is increasing and return is decreasing withdecreasing price. Direction of excess demandis indeterminate while supply and demand areboth increasing .The complete situation can be described by

the following tuples :

Y" _ {(0, 00) ; T}p =

{(c, 00); l}Yd = {(0, 00) ; T}

r = {(- co, 0) ;1}Ye = 1(-OO, o) ; ?}

Y' +-- r 0 . . .00 (3)

p E.- Y° O . . . c . . . 00 (4)

Ye = Yd - Y' -00 . . . 0 . . . 00 (5)

Yd = f(-p) 0 . . .o0 (6)

r = p - c -00-0 . . . oo (7)

Proceeding from that state, price will reach land-mark value c . No other move is possible in thissimple model . With price equal to costs, supplywill remain constant . Price will move on de-creasing, forcing decreasing return and increas-ing demand . Excess demand will increase withincreasing demand and constant supply . I willskip the further process of qualitative simulationin this paper, because the method is well known .The dynamic course is shown in figure 4 . The

diagram a.) in figure 4 shows the course of Yeand r . This diagram is not appropriate, obvi-ously . Excess demand is still moving, even ifreturn equals zero . Diagram b) shows the inter-action of the dynamic variables Y' and p. Theisoclines are figured for a better understanding .

o--i

r

Figure 4: Qualitative simulation

r

P

The figures just mark one point : This qual-itative simulation is not able to show whetherthe dynamic process will reach a stable equilib-rium, will explode or keep a constant orbit . Thecyclic characteristic of the dynamic behavior onthe other hand is obvious . With the assumednegative effect of price on demand the adjust-ment process should result in a stable equilib-rium, see the mathematical analysis above . Thecycles only remain on a constant orbit, if thegi = 0 isocline lays parallel to the p-axis, if de-mand is independent of price . The qualitativevalues are too coarse to express damped cycles .

Qualitative simulationand orders of magnitudeTo achive more specific results qualitative val-ues must be more expressive to analyse a processaproaching cyclic an equilibrium . The assump-tions in the stated example are not sufficient tocompare the crucial values, especially the valuesof demand and supply . Both remained in themeaningless range (0, oo) during simulation . So,if demand is not equal to supply, price moves, in-dependent of the amount of excess demand . Inthe egg market for example, price would rise inthe same way in reaction to an excess demand often eggs or of ten million eggs . It would be muchmore plausible to assume, price will change veryslowly, if at all, when excess demand is near zero .Then demand and supply are of the same orderof magnitude .

Following RAIMAN a scale Close is introduced,including scale Id, see (16) for the details . Un-til now, a market has been called cleared if thesingle element sets Id(variable) overlaped :

Id(Yd ) - Id(Y')

q

yd = Y'

To analyse an economic model a market now isassumed to be cleared, if the sets Close(variable)overlap :

Close(Yd) ;Z~ Close(Y')

q

{y1(1- e )Yd < y < (1+e)Yd,s > 0,-small} fl{YJ( 1- e)Y' < y < (1+e)Y',e > 0,esmall}

540

To compare these coarse values the ;z:~ relationwas introduced for equal orders of magnitude .

As usual = is true for equal values . The relation< denotes smaller values in order of magnitude .Applied on qualitative simulation this yields

in different adjustment of dynamic variables ifthe determining value on the right hand side ofthe respective equation is Close(0) . The valueof Y` is in Close(0) if Close(Yd) ~Zz Close(Y')holds .To continue the example above the qualitative

description of the model remains unchanged tothe set of equations (3)-(7) . In addition thosepairs of variables are .grouped together which hasto be compared in orders of magnitude duringthe qualitative simulation process :

Y d=

{(0, 00) ; T}

Yd ;?~ Ys

Close(Yd ) < Close(Ys)

If c is not assumed to be constant but a functionof production quantity, p

c would have to beadded to 8 . However, it would not change the

following simulation if p

c is added despite cconstant .The start situation is the same as above, added

the order of magnitude relation between yd andY'

{(0, 00) ; T}

r

=

f(-oc), 0);1}

p =

{(C, 00) ; l}

Y' = f( - oo, 0) ; ?}

The first simulation step yields price equal tocosts . Supply keeps constant, while demand in-creases . The negative excess demand moves to-wards zero :

"

=

{(0, OG) ; -}

r = {0 ;1}

p

=

{c;1 }

Y°

=

{( - C)0, 0) ; T }yd = {(0, 00) ; T}

Close(Yd ) <

Close(Y')

Decreasing p leaves landmark value c . The dis-tance between demand and supply is reduced be-cause of increasing demand . Prices adjustmentslows down.

Y" = {(0, 00);1}

r = 1(-OO, 0);1}

p =

{(0, C);1}

Y°

=

f(-OO, 0) ; T}Yd = {(0,00);T} Close(Yd) ;Z:~ Close(Ys)

Excess demand is equal to landmark value zero .Price keeps constant below costs, demand keeps

konstant, too . Supply decreases because of neg-ative return :

= {(0, Co) ;1}

r = f( - oo, 0) ; -}" = {0 ; T}

yd=

f(0,00);_} Id(Yd) ~ Id(Ys)p =

{( 0 , C) ; -}

Excess demand leaves landmark zero, demand isstill near supply. Price starts to increase . Supplyand demand decreases :

Y" _ {(0, 00);1}

r = 1(-OC, 0) ; T}

p = {(0, c) ; T}

Y" _

{(0, oo) ; ?}Yd = {(0,00);1} Close(Yd )

Close(YO)

In this crucial situation the damping effect ofprice elastic demand does its work . With de-creasing supply and demand, excess demandkeeps positive but close to zero . Price adjustsslowly increasing until it is equal to costs .

" = {(0 , 00) ; -}

r =

{0 ; T}p = {c;T}

Y` = {(0,00) ;1}Yd = {(0,00) ; 1}

Close(Yd) .:

Close(Y')

Simulation does not stop in this situation .Still some variables will move. But the furthersteps, which I have skipped here, result in a cy-cle close to the equilibrium . The process will notleave the range close to the equilibrium again .Qualitatively, the adjustment process stops inequilibrium .

Figure 5 shows the dynamic course . The thicklines are the respective isoclines . The range be-tween the thin lines to the right and left of thethick lines is close to the isoclines . The intersec-tion of both ranges is close to the equilibrium .For a better understanding the range close tothe Y'=0 isocline is added, despite the fact thatprice and costs have not been compared in ordersof magnitude during simulation .

Simulation starts in the point right to bothisoclines . Unlike above it is obvious that the cy-cles are damped . After leaving the p=0 isoclineadjustment does not leave the range close to thatisocline again . That is because of the stabelizingeffect of price elastic demand .

I YS

Conclusion

P

Figure 5: Qualitative simulation and orders ofmagnitude

The application of qualitative reasoning on eco-nomic models appears downright forcing in com-parison to the application on physical processes .In economics the basis of proved physical laws ismissing . In contrast to natural science, in eco-nomics it is not possible to investigate or provebasic laws under laboratory conditions . Missingor incomplete data is not the main characteris-tic of economic analysis . Real data are no objectin the context of equilibrium analysis . Relation-ships are assumed, and according to these as-sumptions the economist analyses the behaviorof the corresponding system .From this point of view the critique of CEL-

LIER in (1) on the qualitative physics approachhas to be weakened, but remains a motivationfor future work . Cellier compares the qualita-tive simulation of a differential equations systemaccording to KUIPERS (11) with an analysis ofthe same system by qualitative mathematics . Itis not remarkable that the results of qualitativesimulation are less specific . Especially qualita-tive simulation is not able to state parameter re-lations depicting the relations yielding in stableor unstable behavior of the model.

In his example Cellier describes a system ofthree linear differential equations with parame-ters a, b, and c . The mathematical analysis ofthe model yields a stable adjustment if c < ab

holds. In comparison with that the system istransformed into qualitative notation . KuipersQSIM algorithm grants to find every type of be-havior that is physically possible . In this ex-ample it will be the result that stable, unsta-ble, or neutral behavior is possible . The qual-itative description of the model does not allowstatements beyond these . Cellier says, "Unfor-tunately, this does not tell us very much." Thefact that this chosen model formulation allowsevery noted kind of adjustment, is already a re-sult . On the other hand this result is not suf-ficient . The next step is to find the conditionswhich determine the kind of behavior . In thispaper it is done via adding orders of magnitudeto qualitative simulation .

Future Work

In contrast to the stated example above, in morecomplex models more than only one move willbe possible in a certain qualitative situation .The simulation of a qualitative model is directlyanalogeous to the application of a productionsystem . This can be used for following differentpaths . Production systems implement knowl-edge based systems . The major components arei) a collection of rules, each of which is composedof a condition and an action, ii) a working mem-ory which contains the developing informationthat. defines the current state of the system, andiii) a control loop which cycles continually. Inevery cycle one rule is applied . The conditionsof the rules are matched against the informationin the working memory . A rule whose condi-tions evaluate as true can fire, see (10) or (15)for example . In our case the rules are the quali-tatively noted coherences between the economicvariables in the working memory . In every situa-tion defined by given values of the economic vari-ables, the presuppositions for reaction of morethan one variable hold . The resulting behav-ior of the system depends on the decision, as towhich variable will react first. . Because of thecombinatorical explosion of the number of pathsthat would have to be considered, it is impossibleto inspect every possible development. However,the inspection of a path can be stopped when aspecial behavior is discovered. It is to assume

that such a behavior will be recognizable after afew production cycles .The chosen conflict resolution method decides

which variable will react faster than the others,if several variables can react in a situation. In-specting and modifiying the corresponding pa-rameters of the- conflict resolution mechanismcan help to identify the conditions of model be-havior . A statement like "if variable x reactsfaster than variable y this will result in explo-sive behavior" can be expected. It has to bestudied whether more complex coherences canbe achieved, like e.g . the parameter based state-ment c < ab .

(2) Lev E. El'sgol'c . Qualitative Methods inMathematical Analysis. American Mathe-matical Society, 1964 .

(4) Peter Flaschel . Macrodynamics: incomedistribution, effective demand and cyclicalgrowth . Verlag Peter Lang, Frankfurt/M .,1993 .

(6) Ciancarlo Gandolfo . Qualitative Analy-sis and Econometric Estimation of Con-tinuous Time Dynamic Models . North-Holland, Amsterdam, 1981 .

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