University of London
School of O riental and African Studies D epartm ent of Economics
M athem atical Economics and Control Theory: Studies in Policy O ptim isation
M a so u d D e r a k h sh a n -N o u
Thesis S u b m itted for th e Degree of D octor of Philosophy
November 1996
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A bstract
C h ap ter 1 deals w ith th e origin and lim itations of m a th e m a tic a l econom ics and its im plications for econom ic applications of o ptim al control theory. Using an historical approach, we have proposed a hypothesis on th e origin and lim itatio ns of classical and m odern m ath em a tic al economics. Sim ilar hypotheses proposed by C ournot, W alras, von N eum ann-M orgenstern and D ebreu are shown no t to be convincing. C onditions are established under which applications of m a th e m a ti
cal m etho ds, in general, and o ptim al control theory, in p articu la r, m ay produce econom ic results of value.
C h ap ter 2 concerns th e form ation and developm ent of o p tim al control ap
plications to econom ic policy optim isation. It is shown th a t th e application of m athem atical control theory (as com pared w ith engineering control) m ay signifi
cantly co n trib u te to m athem atical economics (as com pared to econom etrics). T he developm ent of o p tim al grow th th eory has been exam ined as an exam ple. W ith in th e context of econom ic policy optim isation, a critical exam ination of th e recent developm ents in m acroeconom ic m odelling, th e relationship betw een th eo ry and observation, ratio n al expectations, th e Lucas critique and th e problem of tim e- inconsistency is presented.
C h ap ter 3 provides th e first illu stratio n of th e m ain th em e of th e earlier chap
ters. Using th e generalised H am iltonian in P o n try ag in ’s m axim um principle, as well as using B ellm an’s dynam ic program m ing, we have o b tained a n um b er of new results on th e m a th em a tic al properties of op tim al consum ption un der liquid
ity constraints. For exam ple, we have dem o nstrated how th e response of op tim al consum ption to liquidity constraints is conditioned by th e consum er’s in te rte m poral elasticity of su b stitu tio n . Considered as a m ath em atical stru c tu re , this is shown to cap tu re th e effects of th e following variables on th e o p tim al consum ption p ath : p ure preference p aram eters, th e in terest rates variations and th e stru c tu ra l p aram eters prevailing in th e credit m arkets.
In chapter 4, th e dynam ic Leontief m odel, which according to th e conditions established in ch ap ter 1, is one of th e m ost successful applications of m a th em a tic al m ethods to econom ic policy analysis, is first considered as a control problem . We have th e n ob tained th e o p tim al consum ption p a th for determ inistic and stochastic dynam ic Leontief m odels w ith su b stitu te activities which are in tu rn fo rm u lated in determ inistic and stochastic environm ents. O ur solution uses P o n try ag in ’s m ax
im um principle, B ellm an’s m eth o d and A stro m ’s Lem m a on sto ch astic dynam ic program m ing.
Acknowledgements
I wish to express m y sincere g ra titu d e to m y supervisors Professor Laurence H arris an d D r. M assoud K arshenas. T heir sup po rt and encouragem ent th rou gh out th e progression of this work are highly appreciated. I have also benefited greatly from th e W eekly Sem inars by Econom ic Research S tudents. T h e advice and critical rem arks of Professor Ben Fine, who supervised these sem inars, to geth er w ith th e com m ents of th e p articip an ts, have been a continuous source of in spiratio n. M ost of all, I am grateful to m y wife P arv in and to our chil
dren, N azanin an d Jam sh id , for th eir patience, u nd erstan din g and help during th e w riting of th is dissertation.
To Professor Laurence Harris
my supervisor
to whom I shall remain indebted
Contents
Chapter 1
The Origin and Limitations of Mathematical Economics and its Implications for Economic Applications of Optimal Control:
An Historical Approach
1.1 Introduction 9
1.2 M athem atisation o f economics: hypotheses on the origin and significance of m athem atical economics 12 1.2.1 D ebreu’s incidentality hypothesis o f early developments in
mathematical economics 14
1.2.2 C ournot’s hypothesis: erroneous presentations and the poor mathematical knowledge 17
1.2.3 W alras’s hypothesis: the narrowness o f ideas 20
1.2.4 von Neumann and M orgenstem ’s hypothesis: the unfavourable circumstances 22
1.2.5 The hypothesis o f one-dimensionalisation o f economic analysis:
the reduction o f economic life to mechanical economic science 25 1. Mathematical economics and the formation o f mechanical
economic science 27
i) The nature o f classical mathematical economics 28 ii) Mathematical economics: a remedy to multi-dimensional
political economy 30
2. Mathematical economics and Marxian economics 35 1.3 The origin and formation o f modern mathematical economics 39
1.3.1 Refutation o f D ebreu’s hypothesis on von Neumann and M orgenstem ’s epoch-making contribution 41
1.3.2 The hypothesis o f co-ordinated research programmes 44
1.4 Classical vs modern m athem atical economics: attitudes and limitations 1.4.1 Attitudes 49
1.4.2 Limitations 53
-Can mathematical methods discover economic truths? 56 1.5 The rocky lane to successful co-ordination: the developm ent o f the
relationship between m athem atical economics and econometrics 60
1.5.1 The turning point in the rocky lane to co-ordination: the emergence o f alternative strategies 64
1.6 Economic applications of optimal control theory as an illustration 66 1.6.1 Limitations o f economic applications o f optimal control theory 71 1.7 The logic of abstraction: the origin of limitations in mathem atisation
of economics and its implications for optimal control applications 73 1.8 Summary and concluding remarks 78
Chapter 2
Control Theory and Economic Policy Optimisation:
Developments, Challenges and Prospects
2.1 Introduction 88
2.2 The beginning: classical control theory and economic stabilisation 93
2.3 Early applications o f modern control theory to optimal economic policies 97
2.3.1 Optimality conditions in models o f economic grow th 98 2.3.2 Engineering control theory and econometrics 103
1. Contributions o f control engineers and control theorists 104 2. Contributions o f control engineering institutions 109
2.4 Stochastic and adaptive control applications to optimal economic policy design
2.4.1 Stochastic control applications 110 2.4.2 Adaptive control applications 113
2.5 The relationship between th eoiy and observation: a critical analysis of the recent developments in macro-econometric modelling and the role of dynamic optimisation 116
2.5.1 Responses to the Cowles Commission’s traditional strategy 117 2.5.2 Data-instigated vs theory-based econometric models: the origin o f
the gap, the role o f dynamic optimising models and a critical analysis o f the attempts to bridge it 124
-A critical examination o f Smith and Pesaran on the interplay o f theory and observation 128
2.5.3 Speculations on the future course o f developments 132 2.6 Rational expectations, the Lucas critique and the policy
ineffectiveness debate 137
2.7 Tim e-inconsistency and the optimal control of macro-econometric models with rational expectations 144
2.7.1 Time-inconsistency, reputation and the stochastic environment 151 2.7.2 Rational expectations and econometric modelling in practice 154
2.8 Summary and concluding remarks 157
Chapter 3
Consumption Behaviour Under Liquidity Constraints:
An Application of Optimal Control Theory
3.1 Introduction 164
3.2 The importance and implications of liquidity constraints in consumption models 167
3.3 Optimal consumption properties using the maximum principle 173 3.3.1 Consumer’s rate o f time preference, interest rates and the
optimal path o f consumption 174 3.3.2 The Bernoulli case 177
3.3.3 Pontryagin’s maximum principle and Hall’s random walk hypothesis 180
3.4 Optimal consumption properties using the dynamic program m ing 181 3.4.1 Optimal consumption paths by direct search 184
3.4.2 Properties o f the optimal consumption path 190
3.4.3 The Bernoulli case and optimal consumption functions 194
3.5 Properties o f the optimal consumption path with liquidity constraints 198
3.6 The generalised Ham iltonian, liquidity constraints and the rejection of Hall’s random walk hypothesis 200
3.7 Time-varying interest rates and the properties of optimal consumption path under liquidity constraints 204
3.7.1 Time-varying interest rates and liquidity constraints 206 3.7.2 Liquidity constraints and the interaction between time-varying
interest rate and the utility discount rate 208
3.7.3 Interest rates, intertemporal elasticity o f substitution and liquidity constraints 211
3.8 Optimal consumption in a stochastic environment 216
3.8.1 Uncertain lifetimes and the optimal consumption behaviour 217 7
3.8.2 Income uncertainty and the optimal consumption behaviour 219 3.8.3 Implications o f income uncertainty on the applications o f
optimal control theory to dynamic consumption decisions 225 3.9 Summary and concluding remarks 229
Chapter 4
Optimal Control of Dynamic Leontief Models
4.1 Introduction 234
4.2 The L eontief model and mathematical economic modelling:
background, importance and the optimal control approach 236 4.3 The dynam ic L eontief model as a control problem 240
4.4 Optimal consumption policies for dynamic Leontief models using Pontryagin’s maximum principle 243
4.5 Optimal consumption policies for the dynamic L eontief model using Bellm an’s dynam ic programming 246
4.6 Optimal control o f the dynam ic Leontief model with substitution:
problem formulation 253
4.7 A dynam ic programming solution of the Leontief substitution system 257
4.8 Optimal stochastic control of dynamic Leontief models with future uncertainties and a stochastic substitution 263 4.8.1 A dynamic programming formulation o f the problem 264 4.9 Summary and concluding remarks 270
References 275
Chapter One
The Origin and Lim itations of M athem atical Economics and its
Implications for Economic
Applications of Optim al Control: An Historical Approach
1.1 In tr o d u c tio n
O ur sta rtin g po in t is th e recognition of th e fact th a t o ptim al control is no m ore th a n an advanced m a th em a tic al m eth o d in th e field of dynam ic o ptim izatio n theory. Its applications to economic analysis are therefore constrain ed by th e lim itations in m a th em a tic al tre a tm e n t of economics. It follows th a t m any ques
tions on econom ic applications of op tim al control cannot successfully be exam ined w ithout a direct reference to th e origin and lim itations of th e m a th e m a tiz a tio n of economics. As an exam ple, consider th e following questions: W h a t is th e logical ju stificatio n for o ptim al control applications to economic analysis? W h a t are th e salient features of op tim al control th eory which have m ade it so a ttra c tiv e to th e com m unity of m a th em a tic al econom ists and econom etricians? W h a t factors have co n trib u ted to its successful applications and w hat have been th e underlying con
ditions responsible for its p artial failure in satisfying th e earlier g reat optim ism ? T he key to all these, questions lies in th e m ath em atical n a tu re of o p tim al control theory and on its capabilities and lim itations in handling specific problem s in
econom ic analysis.
We s ta rt th e analysis of th e origin of m ath em atical econom ics by presenting th e following questions in section 1.2: W hy is C ournot (1838) unanim ously agreed as th e b irth of m a th em a tic al economics while 38 research work on this su bject had been published before th a t? W hy was th e m ath em atical econom ics of C ournot to tally ignored by classical econom ists for m ore th a n 30 years u n til Jevons (1871) revived it? W hy is th e im p ortance and significance of applications of m a th e m a t
ics to econom ics still an u n settled issue whereas physical sciences can h ardly do w itho ut m ath em atics? To answer these questions, we have first exam ined th e hy
potheses p u t forw ard by D ebreu, C ournot, W alras and von N eum ann-M orgenstern in sections 1.2.1 to 1,2.4, respectively. We have found th a t none of these hy pothe
ses is satisfactory.
O ur hypothesis of one-dim ensionalisation of economic analysis is discussed in section 1.2.5. H istorically, advances in classical economics to g eth er w ith theo
retical developm ents in M arxian economics had produced a n um b er of different econom ic doctrines which were considered by th e advocates of m a th em a tic al eco
nomics as a chaotic state, T he “scientific” or m ath em a tic al approach and th e socio-political approach as two possible responses to such an environm ent of m u lti
dim ensionality in econom ic studies are exam ined. M ath em atical economics and th e fo rm ation of econom ic science is discussed in section 1.2.5 w here “th e n atu re of classical m ath em a tic al econom ics” and “m ath em atical econom ics as a rem edy to m ulti-dim ensional political econom y” are analysed. T he hypothesis th a t m a th em atical econom ics has been developed as a response to M arxian economics is another topic which is discussed in section 1.2.5.
T h e origin and form ation of m odern m a th em atical econom ics is th e su b ject of section 1.3. T he origin an d n a tu re of forty years recession in theoretical develop-
m ents in m a th em a tic al economics from M arshall(1890) to th e em ergence of a new era in m a th e m a tic a l economics are explained in this section. D eb reu’s h y po th e
sis w hich regards th e p ublication of von N eum ann and M o rg en stern’s Theory o f G ames and Econom ic Behaviour (1944) as th e startin g poin t of m odern m a th e m atical economics is critically exam ined and rejected in section 1.3.1. In section 1.3.2, we have proposed our hypothesis th a t th e creation of th e Econom etric So
ciety in 1930, th e Cowles Com m ission in 1932 and th e concom itant advances in coordinated research program m es in m ath em a tic al economics can be considered as epoch-m aking events which have m arked th e beginning of m o dern m a th em a tic al economics.
To identify th e salient features of m odern m a th em atical econom ics, we have com pared th e a ttitu d e s of classical m ath em a tic al econom ists and th e ir percep
tions on th e lim itation s of econom ic applications of m a th em a tic al m eth od s w ith those of m odern m a th em a tic al econom ists in sections 1.4.1 and 1.4.2, respectively.
T h e theo retically im p o rta n t question of w hether economic tru th s are discoverable th rou gh th e in stru m en tality of m ath em atics is discussed in section 1.4.2.
T h e objectives of th e Cowles Com m ission and th e E conom etric Society in co
ordinatin g th e m u tu a l p en etratio n of econom ic theory, m a th em a tic al m eth od s and statistics, discussed in section 1.3.2, have been seriously challenged by th e recent developm ents of th e relationship betw een m a th em a tic al econom ics and econom et
rics. This problem is discussed in section 1.5 w here th e n a tu re of th e disparity betw een m a th em a tic al economics and econom etrics in building up m odels for em pirical analysis is discussed. This provides a background for section 2.5 in C h ap ter two w here a discussion of th e critique of m acro-econom etric m odels is presented in th e light of th e ongoing debates on th e relationship betw een th eo ry and obser
vation.
H aving discussed th e origin of m a th em a tic al economics, th e salient features of m od ern m a th em a tic al economics and its lim itations and th e problem s asso
ciated w ith th e relationship betw een m ath em atical economics and econom etrics, th e econom ic applications of optim al control theory is discussed in section 1.6 as an illu stratio n . In section 1.7 a tte m p ts are m ade to identify th e sources of lim ita
tions in th e m a th e m a tiz a tio n of economics. T he im plications of such lim itatio n s are also stu d ied for econom ic applications of optim al control. It is hypothesised th a t th e logic of ab stractio n em ployed in obtaining basic econom ic concepts of narrow com ponents which facilitate q u an titativ e form ulations plays th e key role in generating such lim itations. Conditions u nder which econom ic applications of o p tim al control can produce m ore reasonable results are also discussed in this section. F inally a su m m ary and concluding rem arks are presen ted in section 1.8.
1.2 M a th e m a tiz a tio n o f E con om ics: H y p o th e s e s on th e O rigin and S ign ifican ce o f M a th e m a tic a l E co n o m ics
We present our argum ent in an historical context. This will, hopefully, pro
vide a basis for fu tu re speculations. It is now agreed th a t Civa (1642-1734), an Italian m a th em atician , is th e first au th o r to apply m ath em a tic al m etho ds to eco
nom ic problem s. His work on money, w ritten in 1711, is th e first tru e exam ple of m a th em a tic al economics in which th e ideas of definitions, p o stu lates, rem arks, propositions, theorem s and corollaries are used in analysing money. T his work, however, was com pletely ignored u n til 1871 when it appeared in Jevo ns’s L ist o f M athem atico-Econom ic Books, M em oirs, and Other Published W ritings.
127 years after Civa’s work, C ournot, professor of m ath em a tic s at Lyon and th e R ector of th e A cadem y of G renoble, published his epoch-m aking co n tribu tion to econom ics u nd er th e title Recherches sur les Principes M athem atiques de la
Theorie des Rechesses in 1838. Econom ists today unanim ously agree th a t th e sym bolic b irth of m ath em a tic al economics is th e year in which C ournot published his book. T h e first key question is th a t why C ournot (1838) and n ot Civa (1711) or any o th e r work am ong th e 38 research work published before C ournot on m a th em atical econom ics,1 is not considered to be th e pioneering work in th is field?
W h at has m ad e C o u rn o t’s work to be recognised as an epoch-m aking contribu
tion? Has C o u rn o t’s m ath em a tic al excellence been responsible for th is success or has it been realized th a t this work can be regarded as sta rtin g point for a new curren t of th ou gh ts in political economy? We will come back to these questions in section 1.2.5.
However, C o u rn o t’s book received little or no a tten tio n a t th e tim e: “For several years not a single copy of th e book was sold. In 1863 th e au th o r tried to overcom e th e indifference of th e public by recasting th e work and om ittin g th e algebraic form ulae. T his tim e th e book was called Principes de la Theorie des Richesses. In 1876 he published it again in a still m ore elem en tary form
/
and u n der th e title of Revue Som m aire des Doctrines Econom iques b u t w ith th e sam e re su lt” .2 J. B. C herrim an, a C anadian m athem atician, published a te n page review on C ournot in 1857. This was th e only published recognition of C o u rn o t’s book. C o u rn o t’s significant con trib ution to m a th em atical econom ics was finally revived by Jevons (1871). On page 26 in th e preface, Jevons s ta te d th a t “This work m u st occupy a rem arkable position in th e history of th e su bject. It is strange th a t it should have rem ained for m e am ong Englishm en to discuss its value” .3
A ccording to F isher (1891, p .109) “T he intro du ctio n of m a th e m a tic a l m eth od 1For a list of 38 work before Cournot, i.e. during the period 1711-1838 and 62 work from Cournot to Jevons, i.e. 1838-1871, published on mathematical economics, see Jevons’s L ist o f Mathematico-Economic Books, M emoirs and Other Published Writings, pp. 322-339, in his
Theory of Political Economy, 1871.
2See Charles Gide and Charles Rist (1909, 1948), p. 499.
3All references to Jevons (1871) made in this chapter are from its 4th edition, London:
Macmillan, 1911, 339 pages.
m arks a stage of grow th -perhaps it is not extravagant to say, th e entrance of political econom y on a scientific era ... Before Jevons all th e m an y a tte m p ts at m a th e m a tic a l tre a tm e n t fell flat. Every w riter suffered com plete oblivion u n til Jevons u n earth ed th e ir volum e in his bibliography” . This will lead us to th e second key question: W hy C o urno t’s significant contribution to g eth er w ith previous work on m a th e m a tic a l economics were com pletely ignored, or were not tak en seriously, by classical econom ists?
A m ore fu nd am en tal point which is related to th e above m entioned two key questions is th e following: W hy has th e application of m a th em a tic al m ethods in econom ics not been very successful? In o ther words, if physical sciences can hardly develop w itho ut m ath em atics why is th e im p ortance and significance of m ath em a tic s in econom ic analysis not yet a settled question? We exam ine these points in th e context of th e following hypotheses in sections 1.2.1-1.2.5.
1.2,1 D e b r e u ’s I n c id e n ta lity H y p o th e s is o f E a r ly D e v e lo p m e n ts in M a th e m a tic a l E c o n o m ic s
T h e hypothesis th a t m ath em atical economics has em erged from now here and has grown w ith no aim s while being ind ependent of any curren t of econom ic thoughts has received su pp orts from a n um ber of econom ic h istorians and even from m a th em atical econom ists. G herity (1990) held th a t “For m any years h istorians of econom ics saw those who pioneered th e application of m ath em a tic s to economics as individuals who had appeared out of nowhere, spoken th e ir piece and fallen back into oblivion w ithout im pact or influence on th e ir contem poraries or on those who cam e afte r” . In a sim ilar b u t m ore elab orated line of arg um ent, De- breu (1986, p. 1259) regarded th e em ergence of m a th em a tic al econom ics sim ply as an historical coincidence: “[The early progress of m a th em a tic al economics] is m arked by several m ajo r scientific accidents. One of th e m occurred in 1838 ...
was responsible for two others of those accidents, W hen Leon W alras delivered his first professional lecture th ere on 16 D ecem ber 1870, he h ad held no previous academ ic app ointm ents; he h ad published a novel and short sto ry 4 b u t nothing on econom ic th eo ry and he was exactly 36 ... For Vilfredo P areto , who succeeded W alras in his chair in 1893, it was also a first academ ic app o in tm en t; like his predecessor he h ad n ot published anything on economic th eo ry before; and he was 45” .5
T h e hypothesis of incidentality of developm ents in m a th e m a tic a l economics has also been rep o rted in R obertson (1949, p. 535). He speaks of m a th em a tic al econom ists as “m ore or less isolated figures who cannot be said to have co n trib u ted to a current of th o u g h t because th ere is no discernible flo w ". T heocharis (1983, p.
1) has tried to p artially im prove this hypothesis by saying th a t “in m any instances 4In here, Debreu has referred to Francis Sauveur, published by Walras in 1858, Paris: E.
Dentu.
5See, also, Debreu (1987, p. 399). It should be noted that there are a number of errors in De
breu’s (1986) comments on Walras and Pareto. The following facts reveal such shortcomings: 1) Walras, unhappy with his engineering studies at Ecole des Mines and dissatisfied with literature and journalism as his second academic challenge, was persuaded by his father, an economist, to study economics at the age of 24 to continue his father’s research on m athem atical economics.
It was after 12 years of hard work that this self-taught economist was offered the new chair of political economy at the University of Lausanne. 2) The reason that Walras had held no previous academic position was his lack of any officially recognised educational credentials in economics. 3) Walras presented a paper on Taxation in 1860 in an international conference on taxation in Lausanne which remarkably impressed the audience. [For 1, 2 and 3 see Jaffe (1954), the translator’s forward to Walras (1874), pp. 5-6]. 5) During 1859 to 1862, when Walras was working as a journalist for the Journal des Economistes and La P resse, he published VEconomie Politique et le Justice, Paris: Guillaunin, 1860, in which he strongly attacked the normative eco
nomic doctrines of P.-J. Proudhon, [See Donald A. Walker (1987), p. 852]. 6) Vilfredo Pareto, graduated in m athematical and physical sciences in 1867 and engineering in 1870, started to write and publish articles, as early as 1872, on commerce, the state of Italian industry, railways, advantages and disadvantages of public and private use of the railway system and support of free trades to prevent any form of state intervention in economic activity. Pareto was one of the founders of the Adam Smith Society, which spread and upheld the doctrine of economic liber
alism. In October 1891, Pareto published his controversial article “L’ltalie economique” which was followed by another critical work in April 1892 on Italian Government economic policies.
In 1890, Maffeo Pantaleoni, the famous Italian economist, advised him to study the work of Walras on mathematical economics, and Pareto met Walras himself on September 1890, before accepting the chair of Walras in political economy in 1893. [See G. Busino (1987), p. 800]. The above facts clearly reject Debreu’s hypothesis on the incidentality of mathem atical economics in Lausanne school.
these w riters were fam iliar w ith th e work of th eir predecessors and did, in fact, build u pon th e m ” .
We can classify th e pre-C o u rn o t’s early m ath em atical econom ists, startin g from Civa (1711), as purely academ ic and intellectual exercises in which econom ic concepts were being tran slated into m ath em atical symbols an d operations. These work all lacked any sense of direction. On th e contrary, pioneers in m a th em a tic al economics in th e 19th century, i.e. Cournot (1838), Jevons (1871), W alras (1874), M arshall (1890), Fisher (1891) and P areto (1896), were all com pletely aware of th e ir backgrounds, th e ir cu rrent positions and, m ost im p o rta n t, th e ir aim s. In this section, we establish th e validity of our hypothesis for C ournot, which is m ore controversial due to his historical isolation of being 30 years before Jevons. In th e course of our fu tu re analysis, th e validity of this hypothesis will be established for Jevons, W alras and P areto.
To reject D ebreu ’s hypothesis of incidentality of C o u rn o t’s con trib ution we re
fer to th e first p arag raph of th e preface in Cournot (1838). He has clearly adm ired and ap preciated th e one hundred years of developm ents in political econom y be
fore him , b u t a t th e sam e tim e has urged th e necessity of developing a positive economics due to th e fact th a t th e public has becom e so tired of theories of differ
ent econom ic system s and doctrines: “T h e science known as P o litical Economy, which for a century has so m uch in terested thinkers, is to-day m ore generally dif
fused th a n ever before. It shares w ith politics proper th e a tte n tio n of th e great journals, which are to-day th e m ost im p o rtan t m eans of spreading inform ation;
bu t th e public is so tired of theories and system s th a t now th e d em an d is for so-called “positive” m a tters, i.e. in political economy, custom -house ab stracts, statistic al docum ents and governm ent reports, such as will throw th e light of ex
perience on th e im p o rtan t questions which are being ag itated before th e country
and which, so greatly in terest all classes of society” .6
D espite th e fact th a t C o u rn o t’s prim e objective was to su p p o rt an econom etric ty p e analysis, th e shortages of organised d a ta and th e lack of ap p ro p ria te statis
tical m eth od s of estim ation , forced him to concentrate on pu re theorization of econom ic concepts tow ards building up a positive economics. T h e second p ara
graph in his preface explains this point: “I will only observe th a t Theory ought not to b e confounded w ith system s ... and th a t, to a m an of m y position in p articu lar, m ore th a n to any other, it should be perm issible to consider from an exclusively theoretical stan d p o in t, a su bject of general in terest which has so m any different sides” .
T h e few facts presented above reject th e hypothesis th a t m a th em a tic al eco
nom ics has come from now here and has developed w ith 110 clear aim s. B u t th e question rem ains why th e early m ath em a tic al econom ists failed to achieve th eir objectives? Or equivalently, why em inent classical econom ists did not em ploy m a th em a tic al m ethods in th e ir econom ic analysis? In this regard, we exam ine th e following hypothesis.
1 .2 .2 C o u r n o t’s H y p o th e s is: E rro n eo u s P r e s e n ta t io n s an d P o o r M a th e m a tic a l K n o w le d g e
A ccording to this hypothesis, th e in accu rate early w ritings in m a th em a tic al eco
nom ics and th e ir weak econom ic contents together w ith th e fact th a t th e com
m u n ity of classical econom ists were no t well equipped w ith basic m a th em a tic al knowledge, were th e significant factors which hindered th e pace of developm ents in m a th em a tic al economics. This hypothesis which was first proposed by C ournot (1838) has received su pp ort from a n um ber of econom ists including F isher (1891).
6All references to Cournot (1838), made in this Chapter, are from its English translation by Nathaniel T. Bacon: Researches into the Mathematical Principles of the Theory o f Wealth, New York: Macmillan, 1897, reprinted 1927.
A ccording to C ournot “T he a tte m p ts which have been m ade in this direction have rem ain ed very little known and I have been able to learn only th e titles of th em , except one, Les Pricipes de VEconomie Politique by C anard, a sm all work published in 1801 and crowned by th e Institut. These preten ded principles are so radically at fau lt and th e application of th em is so erroneous, th a t th e approval of a distinguished body of m en was unable to preserve th e work from oblivion. It is easy to see why essays of this n a tu re should not incline such econom ists as Say and R icardo to algebra” [Cournot (1838), preface].
If th e above hypothesis was tru e, th e n C ou rn ot’s work, a concise, original and well presented work on m ath em a tic al tre a tm e n t of econom ics, should have a ttra c te d th e a tte n tio n of econom ists of his tim e; b u t we know th a t his work was absolutely ignored by econom ists for m ore th a n 30 years u n til Jevons (1871) revived it. T he subsequent developm ents in m a th em a tic al econom ics can also provide useful evidence to reject C o u rn o t’s hypothesis. For exam ple, th e concise
/
and m a th em atically elabo rated co ntrib utio n of W alras, i.e. E lem ents d ’Econom ie Politique Pure was hardly noticed in France during the tw enty-five years after its publication in 1874. Even Alfred M arshall, a m ath em atician and an econom ist, has only m entioned W alras in th e briefest of com m ents in his Principles o f Eco
nom ics (1890) and did n o t take W alras’s general equilibrium seriously at all.
Eighty years after W alras, m ath em a tic al econom ists of th e 20th cen tu ry such as A brah am W ald, John von N eum ann, Jo h n Hicks, Frank H ahn, O scar Lange, P au l Sam uelson, Lionel M cKenzie, G erad D ebreu, K enneth Arrow and M ichio Mor- ishim a acknowledged W alras’s co ntrib ution and paid atte n tio n tow ards fu rth er developm ents in W alras’s general equilibrium analysis.7
Let us now exam ine th e hypothesis of poor m a th em a tic al knowledge. Ac
cording to this hypothesis th e com plete oblivion of early developm ents in m athe-
m atical econom ics was m ainly due to th e inadequacy of m a th em a tic al knowledge am ong econom ists. This argum ent is not satisfactory either. D espite th e fact th a t econom ists, in a rare unanim ous agreem ent, would select Jevons (1871), a m a th e m atician, a logician and ed ucated in chem istry b u t self-taught in economics, as th e first econom ist who m ade known to th e world th e rem arkable position of C ournot in th e h isto ry of economics, Jevons him self confessed th a t he could not m a th e
m atically u n d erstan d all p arts of C ou rno t’s book. On page xxx in th e preface to th e 2nd edition of his book (1879), Jevons m aintained th a t “Even now I have by no m eans m astered all p a rts of it, m y m ath em atical power being insufficient to enable m e to follow C ournot in all p arts of his analysis” .
T he above q u otatio n raises th e question th a t if Jevons, like m any o th er econom ists before him , did n ot com pletely u n d erstan d C ournot, w hat m ade him pronounce th e forgotten C ournot as th e m ost influential m ath em atical econom ist of th e early 19th century? A detailed analysis of this question is given in section 1.3 below.
However, to com plete th e present argum ent, let us refer to th e fact, as discussed in section 1.2.5, th a t Jevons and W alras were trying to design a scientific eco
nom ics which was characterized m ainly by its m a th em atical natu re. T his is ex
actly w hat C ournot had in m ind. It is not surprising, therefore, th a t Jevons appraised C ournot w ithout fully un derstanding him . It was th e com patibility of C o u rn o t’s m ethodology an d his a ttitu d e w ith those of Jevons and W alras which, after all, brought him recognition after 30 years. In sum m ary, th e higher levels of m a th em a tic al knowledge am ong econom ists in th e 1870’s as th e m ain factor in un derstan d in g C ournot and thus reviving his work, do not count since Jevons him self did not possess such a m ath em atical knowledge.
1 .2 .3 W a lr a s’s H y p o th e s is: T h e N a rr o w n ess o f Id ea s
A ccording to W alras,8 th e dichotom y betw een deduction and ind uction or be
tw een p ure reasoning and experience which had separated sciences from a rts was th e m ain reason th a t classical econom ists disregarded th e use of m ath em atics in th e ir work. “If n in eteenth century ... has com pletely ignored [m athem atical economics], th e fau lt lies in th e idea, so bourgeois in its narrow ness, of divid
ing education in to two sep arate com partm ents: one tu rn in g out calculators w ith no knowledge w hatsoever of sociology, philosophy, history, or economics; and th e other cu ltivating m en of letters devoid of any notion of m a th em a tic s” [Elements o f Pure M athem atics (1900), 4 th edition, p. 48], W alras claim ed th a t by em ploying b o th deductive and inductive reasoning, m ath em atical economics can be ranked w ith sciences such as astronom y and mechanics: “T he tw en tieth century, which is not far off, will feel th e need, even ... of entru sting th e social sciences to m en of general cultu re who are accustom ed to thinking b o th inductively and deduc
tively and who are fam iliar w ith reason as well as experience. T he m a th em a tic al econom ics will ran k w ith th e m a th em a tic al sciences of astronom y and mechanics;
and on th a t day ju stice will be done to our work” (ibid, p. 48).
D evelopm ents of m a th em atical economics in th e 20th centu ry have strongly rejected W alras’s hypothesis. T he question rem ains however th a t why W alras did not sim ply add th e experim ental dim ension (i.e. q u a n tita tiv e analysis and m easurem ents) to classical economics? In other words, if according to W alras, th e fa m ilia rity o f economists with reason as well as experience would have ranked economics w ith th e acknowledged m ath em atical sciences, why, in stead of com
pletely ignoring th e w ell-established classical economics, did he not m ake an effort to represent classical economics m athem atically for th e purpose of q u a n tita tiv e
8All references to Walras (1874), made in this chapter, are from its English translation by W illiam Jaffe: Elements of Pure Economics, or the theory of social wealth, London: George Allen and Unwin, 1954, 620 pages.
analysis and em pirical m easurem ents?9
On th e contrary, an exam ination of th e Elem ents o f Pure Econom ics indicates th a t W alras did no t m ake any significant contribution eith er tow ards inductive thin king in econom ics or in m easuring econom ic relations and advancing exper
im ental aspects of econom ic theorization. His work, in stead of tak in g classical economics one step tow ards experim entations, com pletely erased th e em pirical contents of classical political economy. In sum m ary, W alras’s actu al contribu
tions to economics did not follow his injunctions on the objectives of econom ic studies.
It is in terestin g to note th a t th e academ ic life of V ilferedo P areto confirms th e con tradiction existing betw een W alras’s prim e objective in econom ic theoriza
tion and his actu al contributions. Recall th a t P areto, accepted W alras’s chair at Lausanne in 1892. A fter m aking a num ber of contributions m ainly to W alras’s theory of general equilibrium ,10 he realized th e weakness of p ure econom ics and its possible applied versions. In Cours d } Econom ie Politique (1896-7) he stated th a t “... p ure economics shows us th e general form of th e phenom enon; applied economics provides a second approxim ations; b u t neith er will even be able to show us how to m anage th e economic life of every individual” [Busino (1987), p.
801]. In Cours he clearly sta te d the im portance of interrelations of economics and social phenom ena. In 1905, P areto published his Manuale d ’E conom ia Politico,}
his words at th e end of this book are clearly a departure from W alras’s principles:
“W hoever w ants to m ake a scientific stu d y of the social facts has to ta k e account of reality and not of ab stract principles and th e like P areto th e n gave up
9For example, William Whewell, the Cambridge mathematician, represented m athematically some doctrines of political economy in general and Ricardo’s system in particular. See Whewell (1829, 1831, 1850). His work were completely ignored by Walras.
10It should be noted that after accepting Walras’s chair in 1892, “Pareto spent the whole of the next year writing a refutation of Marx’s theory of value which was published in Paris in 1893 as the introduction to an anthology of passages by Paul Lafargue taken from Marx’s Das K apitaF . See Busino (1987), p. 801.
econom ics and co ncentrated exclusively on sociology, (ibid, p. 802).
In sum m ary, w ith regard to th e above facts, it is very difficult to accept W al
ra s ’s hypothesis th a t th e n in eteen th century econom ists ignored m a th em a tic al econom ics sim ply due to th e prevailing narrow ness of ideas in discrediting exper
im en tatio n s in econom ic analysis. O n th e contrary, evidence is m ore in favour of th e hypothesis th a t W alras’s own contributions have fu rth e r advanced and stren g th en ed such narrow ness of ideas.
1 .2 .4 v o n N e u m a n n an d M o r g e n s te r n ’s H y p o th e s is : T h e U n fa v o u r a b le C ir c u m sta n c e s
von N eum an n and M orgenstern (1944) have exam ined th e problem of m a th e m a tizatio n of econom ics w ithin a w ider context. If economics is a science why, in co n trary to o th er sciences w here m a th em atics has been applied w ith g reat suc
cess, has its use not been highly successful? M ost sciences could h ard ly m ake any progress w ith o u t m a th em atics and yet th e real co ntrib utio n of m a th em a tic s to economics has rem ained an u n settled question. A ccording to von N eum ann and M orgenstern th e com bination of th e following unfavourable circum stances are th e m ain factors at work.
1. T h e V a g u e n e s s o f B a s ic E c o n o m ic C o n c e p ts , von N eu m ann and M or
genstern (1944, p. 4) have pointed out th a t “Econom ic problem s were n o t form u
la ted clearly and are often sta te d in such vague term s as to m ake m a th em a tic al tre a tm e n t a priori appear hopeless because it is quite u n certain w hat th e prob
lem s really are. T h ere is no point in using exact m ethods w here th e re is no clarity in th e concepts and issues to which they are to be applied” . T his is in sharp co n trast to th e general view held am ong m ath em atical econom ists th a t th e m a th em atizatio n of economics facilitates a m ore concise exposition of problem s and avoid th e digressions of vague argum entation.
von N eum ann and M orgenstern’s claim im plies th e following contradiction:
F u rth er developm ents in m ath em a tic al economics as a science depends entirely on prior developm ents in “non-scientific” descriptive economics. To provide fu rth er evidence to our claim , we refer to page 4 (ibid): “Consequently, th e in itial ta sk is to clarify th e knowledge of th e m a tte r by fu rth er careful descriptive w ork” . T his is in d isp u te w ith th e established view in th e profession th a t if econom ics is to be a science it m u st be m ath em atical. M oreover, von N eum ann and M orgenstern have not specified th e conditions u nder which careful and concise advances in descriptive econom ics can be attain e d -i.e. w ith or w ithout m ath em atics. If th e la tte r holds, th e uniqueness of m a th em a tic al economics as an exact science will collapse.
2. I n a d e q u a t e E m p ir ic a l E c o n o m ic F a c ts . A ccording to von N eum ann and M orgenstern, m a th em a tic al economics has not achieved very m uch because th e em pirical background of econom ic science has been definitely inad eq uate. T hey held, however, th a t th eir com m ent should not be construed as a disparagem ent of statistical-econom ic research program m e which was very prom ising a t th e tim e, (p. 5, ibid).
T h e recognition of th e fact th a t statistical-econom ic research work, or w hat is now known as econom etrics, could have m ade progress while m a th em a tic al eco
nom ics was stag nating refers to th e point th a t von N eum ann and M orgenstern have a d m itte d th a t economics as a science can m ake advances outside th e re
stric ted fram ew ork of Jevons’s calculus of pleasure and pain or W alrasian u tility and profit m axim ization.
3. L i m i t a t i o n s in M a t h e m a t i c a l T r e a t m e n t o f H u m a n B e h a v i o u r . It ap
pears th a t von N eum ann and M orgenstern are th e first m a th em a tic al econom ists of rep u ta tio n in th e tw entieth century who have acknowledged th e fu n d am en tal
o b jection th a t econom ic th eo ry cannot be m odelled in th e sam e fo rm at as physical sciences for it is a science of social and hum an phenom ena which has to take into account a nu m ber of non-econom ic elem ents such as psychological, historical and cu ltu ral factors. This im plies lim itatio ns in m ath em atical form ulations of hum an behaviour: “We should a tte m p t to utilise only some com m onplace experience con
cerning h um an behaviour w hich lends itself to m ath em a tic al tre a tm e n t” (ibid. p.
5).
von N eum ann and M orgenstern have m ade a very im p o rta n t p o in t th a t th ere are u n certain ties ab ou t th e exact m ath em a tic al m ethods which should be used in econom ic analysis in general and even in m a th em atizatio n of th a t class of hum an behaviour which lends itself to m a th em a tic al trea tm e n t. T h e existing tools in m a th em a tic al econom ics such as calculus of variations or differential equations m ight not be th e right in stru m en ts for economic analysis since th e y are m ainly developed for physical sciences: “I t is therefore to be expected -or feared- th a t m a th em a tic al discoveries of a sta tu re com parable to th a t of calculus 11 will be needed in order to produce decisive success in [economics]. ... A fo rtio ri it is unlikely th a t a m ere rep etitio n of th e tricks which served us so well in physics will do for th e social phenom ena to o” (ibid. p. 6).
In sum m ary, von N eum ann and M orgenstern’s final recom m end atio n to m a th em atical econom ists is to w ait for new discoveries in m a th em a tic al m eth od s which are m ore ap p ro p ria te for th e analysis of social sciences. B ut have th e y proved th e existence of such m a th em a tic al m ethods? T h e answer is no. M oreover, before th e discovery of such m ath em a tic al m ethods, how can th e real econom ic prob
lems “scientifically” be exam ined for policy recom m endations? A ccording to von N eum ann and M orgenstern, econom ists cannot exam ine such real econom ic prob-
11In here, von Neumann and Morgenstern have referred to the role played by infinitesimal calculus in the creation of the discipline of mechanics.
lems sim ply because they are not yet qualified: How to stabilise em ploym ent, how to increase th e natio nal income, or how to d istrib u te it adequately? N obody can really answ er these questions and we need not concern ourselves w ith th e pretension th a t th ere can be scientific answers at p resen t” (ibid. p. 6).
1 .2 .5 T h e H y p o th e s is o f O n e -d im e n sio n a lisa tio n o f E c o n o m ic A n a ly sis: T h e R e d u c tio n o f E c o n o m ic L ife t o M e c h a n ic a l E c o n o m ic S c ie n c e
We now propose a hypothesis to exam ine th e th ree questions m entioned ear
lier. T h e growing desire for one-dim ensionalisation of econom ic analysis in th e second h alf of th e nin eteen th century strongly m o tivated th e m a th em a tiz atio n of economics. Let us s ta rt by a brief reference to th e underlying factors which encouraged this approach in economic analysis.
Advances in classical economics ensued from contributions of S m ith (1776), R icardo (1817) and Mill (1848) together w ith fu rth er developm ents in theoretical socialism in general and M arxism in p articu la r [Marx (1848, 1856-57 and 1867)], produced a sta te of m ulti-dim ensionality in economic analysis in w hich econom ic issues were stu died in relation to o th er dim ensions such as historical, sociological and political aspects. T h ere could be th e following two in ter-related responses to such an environm ent of m ulti-dim ensionality in economic studies: “scientific”
approach and “non-socio-political” i.e. “m a th em a tic al” approach.
1. Scientific and m ath em atical developm ents of th e 18th and 19th centuries had a rem arkable im pact on m any authors in shaping th e ir a ttitu d e s tow ards searching for an economic science sim ilar to physical or exact sciences. Close connections in all aspects of social life m ake a special stu dy of any one of th e m not sufficiently productive. However, these authors favoured th e idea th a t econom ists should assum e th e ir distinctive role and not devote them selves to stu d y th e laws
of a unified social science. T he rap id progress in physical sciences in th e 19th century was m ainly regarded to be th e result of breaking up th e broad problem s into th e ir com ponent p arts. Such an outlook m o tivated som e auth ors to accept th e view th a t political economy should no longer be seen as if it were a single undivided and indivisible science.
T h e nex t step tow ards m aking a science of economics, in th e sam e fashion as physical sciences, was to discover th e general laws of econom ics which rem ain th e sam e th ro u g h o u t all different ages and conditions: “J u st as th e re is a general science of m echanics, so we m ust have a general science or th eo ry of economy.
... T h e th eory of econom y proves to be, in fact, th e m echanics of u tility and self-interest” [Jevons (1876), pp. 198-199]. However, to som e au tho rs, economic science was identified w ith m ath em a tic al or pure economics w hich was confused on occasion w ith w hat is known today as econom etrics. Since advances in m echanics had th e greatest im p act on th e form ation of m ath em atical econom ics, th e role of calculus in m a th em a tiz atio n of economics has always been profound.
2. W ith in an environm ent of conflicting theoretical developm ents betw een th e m ain stream classical economics and th e newly established M arxian econom ics in th e second-half of th e 19th century, any significant th eo retical contributions in classical economics usually had political and social im plications, escalating th e prevailing tension in political and social dimensions. S eparation of econom ic is
sues from o ther related social subsystem s could have been th e rem edy to this inter-related, m ulti-dim ensional, “chaotic” sta te of econom ic studies. T h e con
cept of pure or m athem atical economics, which was identical to econom ic science, was invented to represent a positive or politically n eu tra l system of econom ic knowledge.
The Failure to Recognize Cournot?
Before sta rtin g to exam ine th e role of calculus in one-dim ensionalisation of eco
nom ics, we are now in a position to provide an answer to our earlier question of how can th e 30 years tim e lag in th e recognition of C ournot (1838) be ex
plained? Recall th a t according to C ournot (1838, p. 1), th e wide varieties of different theories and doctrines had m otiv ated th e public desire for “positive”
economics. However, despite C o u rn o t’s view, th e social, political and econom ic conditions were n ot chaotic enough in C o u rn o t’s tim e to generate a strong driving force for one-dim ensionalisation of economics. D uring th e 30 years from C ournot to Jevons, th e diffusion of socialism an d M arxism had produced a different en
vironm ent. W hen Jevons (1871) and W alras (1874) sta rte d th e ir cam paigns to popularise m a th em a tic al economics, th e work of C ournot to g eth er w ith 38 work on m a th e m a tic a l economics before C ournot and 62 such work from C ournot to Jevons12, were all revived to provide an arm y of supportive lite ra tu re for th e suc
cess of this new school of thou gh t in economics. Even th e an tiq u ate d work of Civa in 1711 was needed to give m ore stren g th to this arm y w hich was ab ou t to launch an a tta c k upon th e trad itio n of great classical econom ists as well as on K arl M arx ’s new unified and all-em bracing approach to econom ic studies. We will come back to this point in section 2 below.
1. M athem atical Economics and the Formation of Mechanical Eco
nomic Science
M ath em atical economics, in th e sense discussed above, was developed in con
ju n c tio n w ith th e concept of pure economics. We will show in section 1.5 th a t m a th em a tic al economics cannot be defined properly w ith ou t using th e concept of p ure economics and vice versa. T his will play an im p o rta n t role in under-
i2See Jevons (1871), pp. 322-339, for a complete list of these contributions.
stan ding th e underlying factors at work in lim itations of m odern m a th em a tic al econom ics in general and economic applications of optim al control theory in p ar
ticular. In this section, we first exam ine th e n a tu re of classical m a th em a tic al econom ics before presenting an analysis of m ath em atical economics as a rem edy to m ulti-dim ensional political economy.
i) T h e N a t u r e o f C la s s ic a l M a t h e m a t i c a l E c o n o m ic s
A ccording to Jevons (1871, p. v ii in preface), since economics “deals thro ug ho ut w ith q u an tities, it m ust be a m ath em atical science in m a tte r if not in language.
... T h e T h eory of Econom y thu s tre a te d presents a close analogy to th e science of S tatistic al M echanics and Laws of Exchange are found to resem ble th e Laws of E qu ilibriu m of a lever as determ ined by th e principle of v irtu a l velocities.
T he n a tu re of W ealth and Value is explained by th e consideration of indefinitely sm all am ounts of pleasure and pain, ju s t as th e theory of S tatics is m ad e to rest upon th e eq uality of indefinitely sm all am ounts of energy” . F urth erm ore, on page 3 (ibid) he w rites: “It is clear th a t economics, if it is to be a science at all, m ust be a m a th em a tic al science ... My theory of Econom ics, however, is purely m a th em a tic al in ch aracter” .
By m a th em a tic s, Jevons basically m eant differential calculus: “T h e theory consists in applying th e differential calculus to th e fam iliar notions of w ealth, utility, value, dem and, supply, capital, interest, labour and all th e o ther quan
tita tiv e notions belonging to th e daily operations of in d u stry ” (p. 3). However, he held th a t th e m inim ization of costs in fulfilling the u tility of an individual is th e u ltim a te objective of economic science which he defined as th e the Calculus o f Pleasure and P ain (1871, p. m, preface). He w rote on page 27 th a t “th e calculus of u tility aim s a t studying th e ordinary wants of m an a t th e least cost of lab o u r” . A lthough Jevons has been usually praised for introducing into econom ic analysis
th e id ea of m axim ization (or m inim ization), th e origin of these ideas traces back to C ournot in 1838 w hen he w rote on page 44 in chapter 4 of his book th a t “we shall invoke b u t a single axiom , or, if you prefer, m ake b u t a single hypothesis, i.e. th a t each one seeks to derive th e greatest possible value from his good or his la b o u r” .
From this point of view, th e m odern definition of econom ics as th e allocation of scarce resources for op tim um satisfaction of altern ativ e objectives is in fact th e generalization of Jev on s’s calculus of pleasure and pain; and m o dern m a th em a tic al op tim izatio n techniques in general and op tim al control th eo ry in p articu la r, which are th e m ost efficient m ath em a tic al tools for achieving such optim u m satisfaction, are noth in g b u t th e advanced versions of elem entary calculus em ployed by Jevons.
T he n a tu re of general m ath em a tic al m ethods in econom ic analysis has been rem ained basically unchanged since Jevons; although, as will be discussed in sec
tion 1.4, its role has been rem arkably im proved. According to Jevons (1879), th e m eth o d consists in assum ing certain sim ple conditions of th e functions as com form able to experience and then disclosing by sym bolic inference th e im plicit results of these conditions” (p. xxxi preface to th e 2nd edition). As for the role of m ath em a tic s, we refer to Fisher (1891, p. 119) where he s ta te d th a t “T he efforts of th e econom ist is to see, to p icture th e interplay of econom ic elem ents.
... M athem atics is th e la n tern by which w hat before was dim ly visible now looms up in firm , bold outlines. We see b e tte r. We also see fu rth e r” .
It is interesting to note th a t this idea can also be traced back to C ournot (1838). A ccording to him th e objective of using m a th em a tic al sym bols is “to facilitate th e exposition of problem , to render it m ore concise, to open th e way to m ore extended developm ents and to avoid th e digressions of vague argum en
ta tio n ” (p. 3). A sim ple com parison of C o u rn o t’s definition of th e objective of
m a th em a tic al economics w ith sim ilar m odern definition expressed by th e editor of th e Journal o f M athem atical Econom ics reveals th e fact th a t th e re has not been any significant change during th e p ast 157 years. B. Allen (1995), th e editor, expresses his view in th e Jo u rn a l’s statem en t of aims as follows: th e form al m a th e m a tic a l expression of econom ic ideas is of vital im p o rtan ce to economics.
Such an expression can determ ine w hether a loose economic in tu itio n has a coher
ent logical m eaning. Also a full form al developm ent of econom ic ideas can itself suggest new econom ic concepts and intuitions. T h e p rim ary ob jective is ... to express econom ic ideas using form al m ath em atical reasoning” .
ii ) M athematical Economics: a Remedy to Multi-dimensional Political Economy
From th e early 19th century, it was felt th a t th e prevailing philosophical and his
torical approach to econom ic issues had produced a s ta te of m ulti-dim ensional political economy. Some econom ists, confused by such m ulti-dim ensionalities and also im pressed by rapid advances in pure and applied sciences, regarded this situ ation as th e source of th e prevailing alleged chaotic sta te of econom ic knowledge.
Let us quote again from C ournot (1838) on this point: “T h e public is so tired of theories and system s th a t now th e dem and is for so-called ‘p o sitiv e’ m a tte rs ” (p. 1). A sim ilar idea is expressed 40 years later by Jevons (1879): “T he present chaotic sta te of Econom ics arises from th e confusing together of several branches of know ledge” (p. xvi preface to th e 2nd edition). W alras (1874) has also expressed sim ilar view s.13
13 “There are today heaven knows how many schools of political economy: the deductive school and the historical school, the school of laisser-faire and the school of State intervention or Socialism, of the Chair, the Socialist school properly so-called, the Catholic school, the Protestant school, etc. For my part, I recognize only two: the school of those who do not demonstrate and the school, which I hope to see founded, of those who do demonstrate their conclusions. By demonstrating rigorously first the elementary theorems of geometry and algebra and then the resulting theorems of the calculus and mechanics, in order to apply them to experimental data, we have achieved the marvels of modern industry. Let us follow the same procedure in economics and, without doubt, we shall eventually succeed in having the same control over the nature of
Subdivisionism was generally regarded to be a rem edy to this alleged chaotic s ta te of economics. For C ournot this subdivisionism m ean t con stru ctin g a pure or positive econom ic theorization which should be developed indepen dent of th e prevailing political system s: “I will only observe th a t theory ought no t to be confounded w ith system s ... and th a t ... it should be perm issible to consider from an exclusively th eo retical stan d p o in t, a subject of general in terest w hich has so m an y different sides” (1838, pages 1 and 2 in th e preface). Jevons (1871, p.
20) clearly signifies th e necessity for subdividing economic knowledge: “Political econom y is in a chaotic s ta te at present, because th ere is need for subdividing a too extensive sphere of knowledge” .14 M oreover, “we m ust distinguish th e em pirical elem ents from th e a b strac t theory, from th e applied theory, and from th e m ore detailed a rt of finance and adm in istration . Thus will arise various sciences, such as com m ercial statistics, th e m a th em a tic al theory of econom ics, sy stem atic and descriptive economics, econom ic sociology and fiscal sciences ... T h en will be division according to th e m ann er of trea tin g th e branches of sub ject. T he m anner m ay be theoretical, em pirical, historical, or practical; th e su b ject m ay be cap ital and labour, currency, banking, ta x atio n , land tenure, etc. -an d not to speak of th e m ore fu n d am en tal division of th e science as it trea ts of consum ption, p roduction, exchange and d istrib u tio n of w ealth” (1879, p. xvii, 2nd edition).
D ifferentiating betw een scientific and literary tem p er in econom ic analysis can best be seen in W alras (1874) who nam ed his book E lem ents d } Econom ie P oli
tique Pure. T he fact th a t he was indebted to Cournot (1838) for using calculus in econom ic analysis im plies th a t by pure economics he basically m ean t m a th e m a t
things in the economic and social order as we already have in the physical and industrial order”
[Walras (1874), English translation (1954), p. 471].
14 A similar idea is expressed by Jevons in a lecture on the Future of Political Economy delivered at the University College, London in 1878: “... One hundred years after the first publication of the Wealth of Nation, we find the state of the science to be almost chaotic. There is certainly less agreement now about what political economy is than there was thirty or fifty years ago”
[see Jevons (1876), p. 191].
