This paper investigates the use of constraint logic programming (CLP) in economic modelling
through the design and implementation of two economic models. The first model, the Desai-
Henry model contains only linear equations while the second model, constructed by the author,
contains non-linear elements. In order to implement the second model, a non-linear constraint
solver was constructed. This was necessary because, although CLP is a very powerful
programming paradigm, currently available implementations lack any on-linear constraint
solving mechanisms

Donovan, N

Gilbert, D

English

Brunel University Research Archive

Economic Modelling using Constraint Logic ProgrammingNelson Donovan & David GilbertComputer Science DepartmentCity UniversityNorthhampton SquareLondon EC1V OHBTel: + 44 171 477 8444Fax: + 44 171 477 8587Email: {be140, drg}@soi.city.ac.ukAbstractThis paper investigates the use of constraint logic programming (CLP) in economic modellingthrough the design and implementation of two economic models. The first model, the Desai-Henry model contains only linear equations while the second model, constructed by the author,contains non-linear elements. In order to implement the second model, a non-linear constraintsolver was constructed. This was necessary because, although CLP is a very powerfulprogramming paradigm, currently available implementations lack any on-linear constraintsolving mechanisms.11. IntroductionThe aim of this research was to implement two models of part of the UK economy usingconstraint logic programming (CLP). The first model was to consist of only linear equations;the second model to contain non-linear elements. Implementing the second model had associatedwith it the problem of non-linear constraint solving - no such solving mechanism to the author’sknowledge being currently available in commercial CLP languages. Part of the objective of theproject was therefore to investigate ways of solving non-linear constraints.This paper is organised as follows: Section 2 provides a brief introduction to economics,econometrics and Eclipse - the CLP language used in this research; Section 3 details theimplementation of first model - the Desai-Henry model; Section 4 describes the implementationof the non-linear model and a non-linear constraint solver; and finally section 5 presents theconclusions of the research and suggests ideas for further work.2. Background2.1 Economics and econometricsEconomics is the study of how societies manage, produce and consume resources. Economicknowledge is useful to individuals, businesses and governments alike. Learning about thestockmarket or about interest rates may help individuals manage their finances better;heightened awareness of the determinants of cost and revenue might assist companies in makingbetter business decisions; a knowledge of imperfect competition may help governments controlpricing.Econometrics is the quantitative analysis of economic theory. The end product of econometricsis usually an equation or a series of equations which can be used to build a model describingsome behavioural aspect of an economic system. Models are useful because they force theeconomist to think clearly about the relationships involved in an economic system. They alsoallow him/her to test theories and make forecasts.2.2 EclipseThe constraint programming language used in this project is Eclipse (ECRC Common LogicProgramming System). Eclipse is a Prolog based system which serves as a platform forintegrating various logic programming extensions. The Eclipse kernel is an implementation ofstandard Prolog. It is built around an incremental compiler which compiles the Prolog sourceinto Warren Abstract Machine-like code, and an emulator of this abstract code. The variousconstraint logic programming libraries available with Eclipse are:• library(fd) - finite-domain constraint solver;• library(r) - linear rational arithmetic constraint solver;• library(chr) - constraint handling rules system;• library(propia) - generalised propagation system.The programs written for this project use library(r). The library(r) solver uses a combination ofthe Simplex algorithm and Gaussian elimination to solve a system of arithmetic constraintsconsisting of linear inequalities and inequalities. Instead of the usual syntax for the rationalconstraint relations (i.e. ‘=’, ‘<’, ‘>’, ‘<=’, etc.), the symbols ‘$=’, ‘$<>’, ‘$<’, ‘$<=’, ‘$>’,‘$>=’ are used. The reason for this is that ‘=’ operates over Herbrand terms and the usualarithmetic operators are already used by Prolog.23. Implementation of the Desai-Henry model3.1 Overview of the modelThe model of the UK economy chosen for the first part of this project is a linear  modelproposed by M. Desai and S. G. B. Henry [DH70]. The model was not intended to becomprehensive but rather designed for the purpose of simulating aspects of government policy.The constant parameters were estimated from quarterly data from the period 1955-66 using adevelopment of the Least Squares technique.The Desai-Henry model is defined by the following equations:1. Ct = 2472.43 + 0.2673Ydt-1 + 22.52t2. It = 102.80 + 0.3560(Y - Y*)t-2 + 0.8851It-13. Mt = -230.32 + 0.064Rt-1 + 0.4889Mt-1 + 0.1545Yt-14. Ut = 1994.74 + 17.726t - 0.3999Yt-1 + 0.4499Ut-15. TYt = -256.68 + 0.0836Yt + 0.607TYt-16. Tet = 60.12 + 0.1296Ct + 0.44Tet-17. Yt = Ct + It + Gt + Et - Mt - Tet + St8. Ydt = Yt - TYt9. Rt = Rt-1 + Et - MtwhereC = Consumer expenditureYd = Personal disposable incomeI = InvestmentY = Gross domestic product (GDP) at factor costY* = Moving average of previous four quarters of Y. ThusY*t = (Yt-1 + Yt-2 + Yt-3 + Yt-4) / 4.M = Imports of goods and services.R = Reserves.U = Unemployed (in thousands).t = Time trend (in quarters with 1955Q1 = 1).TY = Tax on the income.Te = Tax on expenditure.G = Government expenditure on goods and services.E = Exports of goods and services.S = Government subsidies.All units are in millions of pounds unless stated.3.2 Implementation requirementsThe two main requirements, regarding the implementation of the model were:1. It should be able to run forward from a starting set of data and produce values for each variable at each time period.2. It should be able to run backwards given certain data at a certain time period and produce results for each variable for each time period.3It was felt that the best way to tackle the design of the Desai-Henry implementation in order toachieve these aims was “bottom-up”. That is, construct a design for each equation in isolationensuring that data produced by each equation was available to other equations. The latter beingnecessary in order to allow the model to run forwards or backwards over time.3.3 Running the model forwardsTo illustrate the design process, a single equation is selected as an example. The design of theother equations following a similar pattern.Recall that the equation for investment is:It = 102.80 + 0.3560(Y - Y*)t-2 + 0.8851It-1This equation can be described in a kind of pseudo logic as follows:investment at time t is Inv ifFind GDP at time t-2,Find GDP_STAR at time t-2,Find Inv at time t-1,Inv = 102.80 + (0.3560 * ((GDP - GDP_STAR) at time t-2)) + (0.8851 * (Inv at time t-1)).where Inv = I  = InvestmentGDP = Y  = Gross domestic productGDP_STAR = Y* =Average of previous four quarters of Y.It will also be necessary to somehow store the value of Inv so it can be used by otherequations (and actually by Inv itself at the next time period). The predicate asserta/1 can beused which adds facts to the program database. The ‘a’ at the end of asserta signifies that thenew fact is to be added in front of the other clauses of the same predicate name. The Eclipsecode is:inv(AnyTime, Inv) :-TimeMinusTwo $= AnyTime - 2,TimeMinusOne $= AnyTime - 1,gdp(TimeMInusTwo, GDP_2),gdp_star(TimeMinusTwo, GDP_STAR),inv(TimeMinusOne, Inv_1),Inv $= 102.80 + (0.3560 * (GDP_2 - GDP_STAR + (0.8851 * Inv_1),asserta(inv(AnyTime, Inv)).Data to start the model can be held as facts. The amount of data needed for each variabledepends on the equation. The equation which requires the most data to be held is GDP_STARwhich needs data on GDP for the previous six quarters. If these quarters are called time periods1-6 then the next time period - the first one to be predicted - will be time period 7. Thoseequations that only require data to be held for the previous quarter will need only single entry attime period 6. For exampleinv(6, 2767).To run the model as a complete unit, a predicate start/2 can be written that calls a predicateeconomy/13 (which calls each model equation in turn) recursively.4start(Time,To):-       Time $> To.start(Time,To):- Time $<= To, nl,     economy(Time, Con, Inv, Imp, Unem, TOI, TOE, GDP, PDI, Res, Exp, GovExp, Subs),        writeln(con(Time, Con)),        writeln(inv(Time, Inv)),writeln(imp(Time, Imp)),........ /*rest of ‘writeln’ statements*/TimePlusOne $= Time + 1,start(TimePlusOne, To).3.4 Running the model backwardsTo run the model backwards in time a different clause is needed. This is due to the fact that ifan equation predicate is called with an instantiated second argument then the equationcontained in the predicate will contain no unknowns. Therefore when the model is runbackwards it will only be able to return the result “yes” or “no” depending on whether theequations balance or not (which they almost certainly will not). However, what is required fromthe model is that it should be possible to assign values to variables at specific times in thefuture and see what effect that has on other variables at the same time period. The Eclipse codefor investment to run the equation backwards is:inv(AnyTime, Inv) :-      AnyTime $>= 9,nonvar(Inv),        Inv $= 102.8 + (0.3560 * (Gdp_2-Gdp_Star_2)) + (0.8851 * Inv_1),        TimeMinusOne $= AnyTime - 1,TimeMinusTwo $= AnyTime - 2,        inv(TimeMinusOne, Inv_1),        gdp_star(TimeMinusTwo, Gdp_Star_2),asserta(gdp(TimeMinusTwo, Gdp_2)),        asserta(inv(AnyTime, Inv)).The line ‘AnyTime $>= 9’ is required to ensure that the equations that run backwards are notcalled at an inappropriate time period. For example, investment depends on the value of GDPtwo quarters previous. As mentioned above, the most recent time period will be called timeperiod 6. Therefore the value of GDP at time 6 will already be stored in the program as a fact.If inv is called earlier than time 9 with a given value then a new value for GDP at time 6 will beproduced.When running the model backwards the order that the predicates for the equations are calledusing economy/13 is important. This is due to the fact that each equation depends on othervariables (and predominantly those from previous time periods). What is required is that thevalue given be used as the starting point and that the value of subsequent variables are derivedfrom it. Therefore, when running the model backwards, the predicate that calculates the givenvariable must be called first.54. Implementation of the non-linear model4.1 Overview of the modelThe following model was constructed by the author. The model was produced using elements ofthe Desai-Henry model combined with new equations based on economic theory. The model isclosed (i.e. there are no variables for which the model generates no values) and was based ondata from the period 1981-1995. The time periods in this model are yearly, rather thanquarterly as in the Desai-Henry model. All the constants for the equations were calculated onthe statistical package Minitab using least squares regression.The model contains six variables. Of these six, only one needs introducing (the other five beingintroduced in the Desai-Henry model), P, which represents yearly inflation. All the variablesare endogenous. All units are in thousands of pounds per capita except for unemployment(millions of people) and inflation (percentage).Below is a list of the equations defining the model:1.  Ct = 477 + 0.903Ydt-1 - 46Ut-12.  Pt = -34.5 + 0.474Pt-1 + 34.4(Yt/Yt-1)3.  Pt = -2.26 + (18.2/Ut)4.  TYt = 268 + 0.0904Yt + 0.349TYt-15.  Yt = 420 + 1.01Yt-14.2 Implementation of equation 3The interesting equation is (3) - the second inflation equation:Pt = -2.26 + (18.2/Ut)Since the model contains another inflation equation (2) the value of P can be determined. Thisleaves U (unemployment) as the only unknown. Therefore this equation can be used to find thevalue of unemployment at time t. The Eclipse code is:unem(AnyTime, Unem) :-inf(AnyTime, Inf),Inf $= -2.26 + (18.2/Unem),asserta(unem(AnyTime, Unem)).Assuming appropriate predicates for the other equations in the model, data for time period 1and a predicate start/2, that behaves the same as the one used for the Desai-Henry model, whathappens when the following call is made:start(2, 3)The result is:con(2, Con)unem(2, Unem)inf(2, 7.73278)gdp(2, 4696.34)pdi(2, 3709.58)toi(2, 986.756)6con(3, Con)unem(3, Unem)inf(3, 6.98578)gdp(3, 5163.3)pdi(3, 4084.16)toi(3, 1079.14)Delayed goals:      $$=(8.6, -2.26 + 18.2 / _424)        $$=(7.73277664, -2.26 + 18.2 / _1575)        $$=(6.98577881, -2.26 + 18.2 / _3858)Linear Store:Con_7320 $= 4523.26855 + -46 * Unem_1_7278Con_5047 $= 4164.99902 + -46 * Unem_1_5005Eclipse could not solve the unemployment constraint because it was non-linear, even though theequation contained only one unknown variable, namely Unem. The above screen dump showsthe unemployment equations at times 1, 2 and 3 in the ‘Delayed goals’ store. The ‘LinearStore’ holds the linear constraints that could not be solved because Unem was unknown.4.3 Solving non-linear constraintsTo overcome this restriction what is needed is:1. A way of solving these constraints.2. A way of detecting what constraints are unsolvable and passing them to the newsolver.As an example, consider the equation y = x2. A well known way of solving this equation is theFormula Iteration Method. The Formula Iteration Method is a particular type of iterativeprocess. The general features of an iterative process for finding the solution to a given equationare:1. A starting value.2. A rule for generating new values (hopefully better approximations to the solution).3. A means of deciding when to stop the calculation.When finding a solution of a given equation by Formula Iteration, the new value is calculatedfrom the old by using some fixed computational rule obtained by rearranging the equation. Forthe equation y = x2, the equation is rearranged in the formx = 1/2 (x + y/x)and the functionx → 1/2 (x + y/x)is implicitly used with starting value x1 to obtain a sequence x1, x2, x3, ... of better and betterapproximations to √y. Also, because the square root of most numbers is a long decimal (ofteninfinite) it is necessary to decide on an appropriate termination point. This can be achieved bycalculating the percentage error. The percentage error in the y = x2 case can be arrived at bytaking x2 away from the current guess, dividing the result by the current estimate and7multiplying the final result by 100. The absolute value of the percentage is then taken.Therefore it is necessary to specify in advance what percentage error will be tolerated. TheEclipse code to solve y = x2 is:sqrt(0, X1, Perr, 0).sqrt(Y, X1, Perr, Answer) :-Y $> 0,X2 $= (X1 + (Y/X1)) / 2,PE $= ((Y - (X2 * X2)) / Y) * 100,absolute(PE, PEabs),check(Perr, PEabs, Y, X2, Answer).where absolute/2 calculates the absolute value of the percentage error and check/5 testswhether the current guess is acceptable (i.e. the current percentage error is less the requiredpercentage error). If it is the result (Answer is returned). Otherwise, sqrt/4 is called again.Now that a predicate to solve a non-linear equation has been written, a way of accessing thedelayed constraints is needed. This is provided for in Eclipse through the use of thedelayed_goals/1 call which binds a given variable to a list of all delayed goals. Each delayedgoal in the list can then be parsed to see if it is of a recognised type (in the above case y = x2)and if it is then the predicate to solve the equation  (i.e. sqrt/4) called. In Eclipse:nlsolver(Answers) :-delayed_goals(ListofGoals), solve(ListofGoals, Answers).solve([], []).solve([D1|Ds], [Result|Results]) :-parse(D1, Result),solve(Ds, Results).Where parse/2 succeeds if the delayed goal is of a recognised type (in this case the form y=x2)and returns the result of the solved equation.A whole library of predicates to solve non-linear equations could be built up. Assuming apredicate to solve non-linear equations of the form y = a + b/x, the nlsolver/1 predicate cannow be used in the non-linear model to solve equation 3. The code for the predicate unem/2now becomes:unem(AnyTime, Unem) :-inf(AnyTime, Inf),Inf $= -2.26 + (18.2/Unem),nlsolver(List),head(List, Unem).5. Conclusions and further workThrough the design and implementation of the Desai-Henry and non-linear models it has beenshown that it is possible to use CLP to build economic models that can run both forwards andbackwards in time. This allows the user to predict values in the future from starting data, andalso to set values at certain future time periods and see what affect this has on the system as awhole. For example, the user may wish to see what the effect on gross domestic product will beif unemployment is 2m in two years time. This flexibility allows the user to test out manydifferent scenarios.8The construction of the non-linear solver for the second model showed that, as long as the set ofdelayed constraints can be accessed, it is possible to write a predicate that can be used toproduce answers to previously unsolved constraints. The non-linear solver can be thought of asa glass-box solver in the sense that the user sees the implementation of the predicate and canadd further non-linear solvers.There are many further areas of work that could be carried out in relation to this research.Some of the more interesting of these are:• the implementation of a meta-interpreter that includes a call to a non-linear solver,such as nlsolver, when the set of goal calls is empty but delayed constraints stillexist;• the extension of the library of non-linear solvers;• the implementation in Eclipse of a larger non-linear model, for example, thetreasury model [HM93] of the UK economy;• the addition of a graphical user interface to the programs;• the placing of the economic programs onto the world wide web.References [CSO] The Central Statistical Office, UK.[CT78] The Course Team. Approximations - The Open University Mathematics Foundation Course, The Open University Press, 1978.[D80] E. T. Dowling. Mathematics for Economists, McGraw-Hill, 1980.[DH70] M. Desai, S. G. B. Henry. Fiscal Policy Simulation for the UK Economy,.1955-66, 1970, contained in [HH70].[ECRC92] ECRC. Eclipse 3.4 User Manual, ECRC, 1992.[ET] Economic Trends, CSO, 1958 - 1994.[FH93] Fruhwirth, A. Herold, V. Kuchenhoff, T. Le Provost, P. Lim, E. Monfroy, M. Wallace. Constraint Logic Programming - An Informal Introduction, technical report ECRC-93-5.[HM93] H.M. Treasury Macroeconomic Model Documentation, UK, 1993.[J82] D. Jackson. Introduction to Economics - Theory and Data, MacMillan, 1982.[K92] P. Kennedy. A guide to Econometrics, Basil Blackwell Ltd, UK, 1992.[MN73] P. Mottershead and J. Naughton. Modelling Economic systems, Module eight of Systems Behaviour Course, The Open University Press, UK, 1973.[S84] J. Stewart. Understanding Econometrics, Unwin Hyman Ltd, UK, 1984.[SN89] P. A. Samuelson, W. D. Nordhaus. Economics, McGraw-Hill, 1989.[SS86] L. Stirling, E. Shapiro. The Art of Prolog, MIT Press, 1986.

A guide to Econometrics,

Constraint Logic Programming - An Informal Introduction, technical report

Fiscal Policy Simulation for the UK Economy,.1955-66,

Introduction to Economics - Theory and Data,

Mathematics for Economists,

Modelling Economic systems, Module eight of Systems Behaviour Course, The Open

The Art of Prolog,

The Central Statistical Office,

The Course Team. Approximations - The Open University Mathematics Foundation Course, The Open

Treasury Macroeconomic Model Documentation,

Understanding Econometrics,

Economic modelling using constraint logic programming

Economic modelling using constraint logic programming

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