,I
THE DEVELOPMENT OF
MATHEMATICAL ECONOMICS
BETWEEN 1711 AND 1838
Hakan Nairn ARDOR*
jJctisatfz/ar sozel afzklama/arzm daha kesin ve a91k bir §ekilde ifade edebilmek i9in matematik biliminin teknik/erinden ve ara9/armdan yaralamrlar. Buna baglz olarak da, iktisadm bir dall o/arak matematiksel iktisat
ortaya flkmt§tzr. Matematiksel iktisadm geli$iminde Cournot'un ve onun
"Recherches" adlz eserinin ozel bir yeri vardzr. $iiphesiz ki, Cournot'dan
once de matematiksel iktisadm geli§imine ~ok onemli katkzlar yapzlml§tlr.
Bu 9ail§mamn amacz da, Cournot'a kadar matematiksel iktisadm geli§imini ve Cournot'un katkzlarzm incelemektir.
" ••• a training in mathematics is helpful by giving command over a
marvelo'usly terse and exact language for expressing clearly some
general relations and some short process of economic reasoning. "
Alfred Marshal/
1. INTRODUCTION
Economics is defined as the study of making the best use of scarce resources, that
is, of maximization subject to constraints. Economists have been trying to find for
example, how households maximize their utilities or how firms maximize their
profits or minimize their losses with respect to some constraints. All constrained
maximization (or minimization) problems have a mathematical structure, which in
turn generates an economic intuition for them. Therefore, mathematics is a
necessary tool for economics.
Mathematical economics is not a separate branch of economics. It is just an
approach to economics analysis, in which the economist uses graphs, mathematical
symbols and technics to supplement verbal explanation. Th~ purpose of this paper is
*
Ar~. Grv., Gazi Uni., i.i.B.F., iktisat Boliimii
140 I HAKAN NAIM ARDOR
to examine the development of mathematical economics between 1711 and 1838.
This examination is divided into two parts. First part examines the development of
mathematical economics between 1711 and 1838 which is the date of publication of
Cournot's "Recherches", and second part examines briefly the method and
contribution of Cournot. One can ask why we add Cournot instead of Bernoulli or
Thompson as a separate section. The answer is because he made the first consistent
and generally successful attempt to apply mathematical analysis to a wide range of
economic problems. He is sometimes considered as the father of mathematical
economics. After Cournot, mathematical economics has developed very rapidly.
There are some attempts to use mathematical analysis in the discussion of
economic problems before Cournot. These attempts begun at the early 1700's. The
mathematical writers are listed in the Irving Fisher bibliography of mathematical
economics which is given at the appendix of this paper for the period of 17111897.(1) It is important to understand that these mathematical :writers did not belong
any school. We can not see any serious attempt to use mathematical analysis at the
mercantilist, physiocrats and classical school. In the mercantilist literature, there is
frequently a consciousness of relationships between varying quantities which is
promising, but mathematical use of the relationships is lacking (Robertson, 1949,
p.523). For example mercantilist writers in Austria estimated that if mining silver
exactly paid for its cost of production, the enterprise was as profitable to the state as
a 100 per cent profit would be to a private person. If the silver sold for one-J:1alf its
cost of production, the profit was 50 per cent, but in either case only the state could
under take the mining (Oser, 1970, p.lO). During the physiocratic era, we also can
not see a serious attempt to use mathematical analysis. (2) As Professor Robertson
(1949) pointed out they could state the Tableau economique mathematically, but
they even did not do this. Also, as Professor Knight pointed out, mathematical
economics appears unnecessary to comment further on for classical economists
(Robertson, 1949, s.523). As a result, we can not say that as a school mercantilist,
the physiocrats and the classicals made attempts to use a mathematical method.
Depending on Fisher's bibliography (with a few exceptions), Robertson (1949),
Theocharis (1983), Henderson (1985) and Gherity (1990) have examined as a some
part of the period of 1711-183 8 which is called pre-Cournot period. (3) According to
(1) This bibliography is taken from Cournot's "Researches in to the Mathematical Principals of the
Theory of Wealth".
(2) Turgot (1727-1781) is exception. He tried to use algebra in his economic analysis.
(3) More specifically, Robertson examined the period of 1738-1838, Henderson examined the period
of 1822-1850, Theocharis's period begun with Aristotle and ended with Coumot. And, Gherity
examined the period of 1771-1826
EKONOMIK YAKLA$/M
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Robertson, mathematical writers made three innovations. These innovations were
the employment of the calculus, the writing of equations without insisting that
function be defined by analytic formulas, and the use of geometry to express two
variable functions (Robertson, 1949, p. 524). Professor Robertson classifies these
writers looking what kind of innovation they made. For example Bernoulli, Frisi and
Buquoy are a class of first user of calculus in economics. Theocharis classify these
writers as nation criteria. For example, the French writers contributions or the
German writers contributions. Instead we follow the order of Fisher's bibliography.
Section 2 examines pre-Cournot period, beginning from Ceva and ending with
Lube~. Section 3 examines Coumot's method and his contributions briefly.
2. PRE-COURNOT PERIOD
Aristotle is the first writer who used mathematics in an economic argument. We
can not see to use of mathematics between Aristotle and Giovanni Ceva.
Giovanni Ceva, an Italian engineer, a writer on money, and the first name of
Jevons-Fischer bibliography, made the first attempt to use mathematics in his
economics analysis after Aristotle. In 1711 Ceva published an essay about
money.(4) He begun his essay describing monetary materials. According to Ceva,
every kind of money can be separated as its intrinsic value and external value. (5)
Then postulating that, ceteris paribus, the external value of a currency is inversely
related to its quantity and external value of money is directly proportional to the
population of a country. He proved his first theorem that the external values of two
currencies are in compound proportion, made up of the direct proportion of their
respective popu1ations and inverse proportion of their quantities (Theocharis, 1983,
p.5). He uses the following illustration which is reproduced by Theocharis.
I
a
b
c
L
a
e
h
K
d
e
f
where the first, second third and fourth columns respectively represent time, population of a country, the quantity of money and the external value of money. At time
(4) His essay's title is "De re numaria, quoad fieri potuit geometrice tractata, ad illustrissimos te
excellentissimos dominos praesidem quaestoresque huius arciducalis Caesarie magistratus
Mantuae".
(5) Intrinsic value is the quantity of the pure metal money contains and external value is the
purchasing power of money.
1421 HAKAN NAIM ARDOR
K the population is d, the quantity of money e and the external value f. He assumes
that at time L the population is equal to the population at time I and the quantity of
money is equal to the quantity of money at time K. First, he compares time I with
time L. When other things are equal (Da=O) external value is inversely related to the
quantity of money:
c:h::e:b
Then he compares the external values at time L and K. He shows that the
external value of money is related population :
h:f::a:d
Then he shows that the rate of two external values is equal to a compound ratio
made up of the inverse ratio of their populations.
c/f=e/bxa/d
Then using similar method he proves· his second theorem which is when the
quantity of gold coins circulating remains unchanged, their value is directly related
to the quantity of silver money.
The important point is for us his method and contributions to development of
mathematical economics instead of his works. According to Schumpeter, Ceva did
not add anything new to the theory of money (Schumpeter, 1961, p.30 1). Probably
Schumpeter is right but his method was different from previous writers. He was
aware of the necessity of mathematics in the economic. According to Ceva,
commerce is so great and complex that the exploration of its nature can not be done
in any other way expect through the use mathematics (Theocharis, 1983, p.4). Also
he used the "ceteris paribus" hypothesis very carefully. These make him an important name at the development of mathematical economics.
The second name in the Jevon's Fisher bibliography of mathematical economics
literature is Esme Mariotte. He published Essai de logigue in 1717. He is accepted
as a precursor of modern utility theory. But it is difficult to see serious attempts to
use mathematics in his economic analysis which consists of some comparisons. But,
as it is mentioned before at the introduction part of this paper, we did not accept
mercantilist writers as a contributor to development of mathematical economics
because they just used basic comparison in their analysis. Due to this reason it is not
easy to understand why Mariotte's name was added to bibliography.
Daniel Bernoulli is the third name in the bibliography. Daniel Bernoulli (17001782), the Swiss mathematician, is generally accepted the first person who really
used the calculus and analytical geometry in economic analysis. He is a precursor of
c
.._
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Gossen, Jevons, Merger, and Walras. He is famous with his utility analysis.(6) The
marginal principle had been applied to income and wealth by Bemoulli in 1738. His
central idea is that, in considering additions in total satisfactions resulting from
incremental additions to one's wealth, it is important to include as criterion of measurement the amount of one's wealth as well as the size of the increment (Robertson,
1936, p.524).
The fourth name is Francois Veron de Forbonnais (1722-1800). He was a businessman and a civil servant. He did a really successful job in analyzing some historic facts. For example, the finance of France from 1595 to 1721 and the finance of
Spain. Forbonnais was the first writer who used mathematical reasoning and symbols in French. The language, he developed, opened the ways to the most important
developments in mathematical economics. His most interesting works are Elemens
du commerce (1754 and 1766) and Principes at observations economiques (1767).
The fifth name is in the bibliography the Italian economist and sociologist Cesare
Beccaria (1738-1794). He tried to show in his works how economic science can be
analytically considered. According to Beccaria, algebra, being a precise and quick
method of reasoning about quantities, can be applied to everything which can
increase or decrease and consequently it can be applied to political sciences but only
up to a point, for political principles depend on a variety of factors which can not be
precisely determined (Theocharis, 1983, p. 21). He published his most important
work "Tentativo analitico sui contrabbandi" in 1764.
Henry Lloyd is the sixth name in the bibliography. A little is known about his
life. His book " An Essay on the Theory of Money" was published in 1771.
According to Lloyd, the price of a commodity is a function of the quantity of money
circulating and the quantity of the good itself. Then, he represent this verbal statement mathematically also (Theocharis, 1983, p.30-31 ).
P=QIM
where P is the price of a good, Q is the quantity of money in circulation and M
is the quantity of the good. He then assumes that y shows the variation of the quantity of the good. If quantity increase from M to yM, price will be P I y assuming the
quantity of money is constant.
PIy
= Q I yM or yP = Q + M/y
(6) His contributions to economic theory are recognized later. Actually, at that time, his contributions
were accepted as a contribution to the theory of probability.
1441 HAKAN NAIM ARDOR
Achylie Nicholas Isnard, French engineer is the seventh name of the bibliography. His
"Traite des richesses" appeared in 1781. Little is known about his life. The first
attempt was made to give mathematical definition and mathematical proof of general equilibrium by Isnard. Until his time, nobody came as close as to general equilibrium like him. In his book, first he interested in the problem of the exchange values of goods. At the beginning of his analysis he points out that one can compare
homogenous things with reference to either their quantities or their magnitudes, but
among heterogeneous things it is necessary to find some homogenous link
(Robertson, 1949 p. 532). According to him, market establish this link (the ratios of
exchange) among goods. He first assumes that money does not exist. Then he
explains how exchange ratios can be expressed. For example, in two goods case (the
first good is M and second is M'), a units of the first good are exchanged for b units
of the second commodity, the exchange value of a unit of the first commodity in
terms of the second is b/a orb: a (Theocharis, 1983, p. 62).
aM=bM'
M/M' =b/ a
The ratio of b I a give the value of goods. With his own words:
The word "value" then express the ratio between two things when they are
compared in exchange. In speaking of economic goods, one rarely uses the
word "value" in absolute sense. The word which properly expresses the
absolute meaning which one would like to give to it is "utility" (Robertson,
1949, p. 532).
Then, he interested in how exchange ratios are determined. With his own words:
If, instead of two goods, we suppose three to be exchanged, or a greater number, it will be the same for the general value of the goods. Each unit of a good
will be equal to the sum of the offers made by the owners of the other goods
divided by the number of units of the good being considered, or what is the
same thing, the values of goods will be directly proportional to the sum of the
offers and inversely proportional to the quantity of the particular good whose
value is being considered. But since the offers are composed of several heterogeneous goods, it is not possible to deduce from the equality, or from the
equation of which we have just spoken, the relation between two particular
goods (Robertson, 1949, p. 533).
Then he continued his analysis for more complicated case adding to model
money also. As a result, it is interesting to see some very similar points with
EKONOMIK YAKLA~IM
1145
Walrasian General Equilibrium model in his model such as his equations system. It
is enough to find relative values of(any) two equations to show equilibrium. We can
call him as progenitor of Walrasian General Equilibrium model.
Marquis de Condercet (1743-1794) is the eight name in the bibliography. He was
one of the best thinkers in social sciences. Besides, he was a mathematician. He used
mathematics in economics only once. His "Vie de Monsieur Turgot" was published
in 1786. In this book, he discussed the ways of changing tax structure from indirect
to direct and the effects of this change. In order to do that, he used some simple
mathematics. His contribution to development of mathematical economics is to use
the sign s as a indication of summation of finite quantities.
Gugliamo Silio is the ninth name. His famous book "Saggio su !'influenza dell'
analisi nelle scienze Politiche ed Economiche" was published in 1792. In this book,
he extended the Becaria's ideas about tariffs and smuggling. He explained the connection between tariffs and smuggling graphically and simple mathematics symbols.
Nicholas Fran9ois Canard who was not a professional economist is the tenth
name we are going to discuss. The French " Institut National des Sciences et Arts"
proposed him a subject which was about an agricultural country, that all taxes fall
ultimately on land (Theocharis, 1983, p. 66). He wrote an essay about it. This essay
was published as a book under the title of "Principes d'economie politique" in 1801.
To write this essay, he had to analyze the sources of wealth and principles of political economy. This essay made Canard one of the most important mathematical economist of his period. He was treated in a way that he did not deserve by his compatriots. In the critiques of his book, compatriots focus on the defects of it, not the merits. Even Cournot criticize him. But later, Coumot admits that Canard gave him a
point of departure (Robertson, 1949, p. 535). Canard made the first attempt of a
comprehensive treatment of the fundamental problem of price determination. Also
he made the first clear attempt at the demand and supply analysis.
The eleventh name in the bibliography is Claus Kroncke. He is the first author
who used mathematics in economics in Germany. He was a road engineer. In 1802,
he wrote "Versuch einer Theorie des Fuhrwerks mit Anwendung auf den
Strassenbau". But, as Theocharis said, it is difficult to find something which are
related to ·economic theory. This is a road engineering book and the only economic
questions dealt with there are of a practical nature like the calculation of costs of
building a new road (Theocharis, 1983, p. 117). His second book "Das SteveiWesen
nach seiner Natur und seinen Wirkungen untersucht" was published in 1804. With
this book, he adds something basic to the theory of money. The velocity of circula-
1461 HAKAN NAIM ARDOR
tion is first used in the equation of exchange formula by Kroncke. Petty, Locke and
especially Cantillon are the first peoples who mentioned about velocity. C7) But,
Kroncke is the first person who explain the role of the velocity of circulation of
money using symbol. According to Kroncke, the quantity of money (r) is determined
by the money value of all goods sold during a certain time (~) and the times that on
the average money is used for buying and selling during the same (m).
Mathematically;
r=~/m
This equation is very familiar for us. If we think quantity of goods produced in
an economy (T) that f is equal to multiplication of total quantity of goods produced
in an economy (T) by their price level (P), we can find that ;
M
= P T I V where r = M, f = P T and m = V
which is Fisher equation (mathematical representation is taken from Theocharis,
1983, p. 103).
Simonde de Sismondi (1773-1842) is the twelfth name in the bibliography. He
was born in Geneva. He was a well known historian, literary critic and economist.
His first book was a study about Tuscon agriculture. After this book, he studied on
history of Italian republics and history of Southern Europe. In 1803, he began to be
known as an author on economics with his book of "De la richesse commerciale ou
principes d'economie politique, appliques a' la legislation du commerce". Actually,
as an economist, he was the follower of Canard's ideas. He is a precursor of modem
international economics. His main attempt to use mathematics in economics analysis is when he tries to show that in both closed and open economy, a country, ceteris
paribus, will be progressing, declining or stationary according to the level of wages
(Theocharis, 1983, p. 79). According to Sismondi, the position of economy depends
on the difference between the national reserve and nation's expenses. The national
product and the necessary salary for the labor which produced the national product
determine the national reserve. Using Theocharis's representation, if D are the
expenses, P is the national product and N is the necessary salary, the nation will be;
progressing if P - N > D
stationary if P - N = D
declining if P - N < D
(7) More information can be found at Schumpeter, 1961, p. 315-16-17.
EKONOMIK YAKLA$/M
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In Sismondi's analysis the salary for necessary work is very important. Because,
it determines the position of economy. According to Sismondi, the salary for necessary work which produced this year's production is equal the last year's salary. This
imply that this year's necessary salary is paid in advance of next year's production.
If this salary is N + x and x is positive, it will mean that next year's production will
be increasing. Sismondi assumes that consumption is equal to production for a
closed economy.
D + (N + x) = P
Therefore
P- N > D ifx > 0
If a country is a permanent debtor, then
D + (N + x) = P + C
where C is the amount of debt and therefore the country will be progressing if
x > C and stationary if x = C.
If a country is permanent creditor, then
D + (N + x)
and therefore
=P - C
p - N > D if X + c > 0
Therefore Sismondi concludes that the difference between this year's necessary
salary and next year's necessary salary determines the position of economy. It is very
interesting to see some basic points of Ohlin's theory of International trade at the
Sismondi's analysis.
The thirteenth name in the bibliography is Joseph Lang. He is known as the first
macro economic mathematical economist. He published two books on economics.
The first one is "Veber den obersten Grundsatz der politichen Oekonomie" which
was published in 1809 and the second is" Grundlinien der politichen Arithmetik",
which was published in 1811. Little is known about his first book. He was affected
by Kroncke and Isnard. He created a macro economic mathematical model where
the problems of distribution, production, money and prices are viewed in their interdependence and the analysis is pursued with consistency and clarity to the ultimate
limits the model will allow (Theocharis, 1983, p. 104). Even his analysis is similar
to Isnard's analysis, he brought the concept of aggregation in the economics.
Georg Von Boquoy (1781-1851) is the fourteenth name in the bibliography. He
was a wealthy, radical and an interes!,il!g. man. When he was very old, he even
attended to the revolution of 1848. He was interested in many fields. The most
important two of them are theoretical mechanics and economics. He wrote a book
named "Die Theorie der Nationalwirthschaft, nach einem neunen Plane und nach
1481 HAKAN NAIM ARDOR
mehreren eigenen Ansichten dargestellt" in 1815. In this book he mentioned about
the managerial side of economics and gave some advises to the farmers about
maximizing their revenue by budding production at a level which the first derivative
disappears and the second becomes negative. According to Buquoy, the revenue and
the cost of cultivating a field are functions of the depth of ploughing (Theocharis,
1983, p. 111). Mathematically;
Y =I (x)
Y = F (x)
where Y is the total revenue, Y is the total cost and x is depth of ploughing. Then,
he defines net revenue subtracting total cost from total revenue;
NR=f (x)- F (x)
Then, he mentions that the first derivative of net revenue function must be zero
and the second derivative must be negative. According to Buquoy, we can fmd
maximum at this point.
f'(x) = F'(x) = 0
We should note that even he did not consider that total revenue and total cost are
a function of output (he thought that these are a function of depth of ploughing), and
use the terms "net revenue" (instead of "marginal revenue" and "marginal cost"),
therefore, his analysis is almost same with the our usual marginal cost-equalsmarginal revenue statement of the maximization. And also, he was aware of the
necessary and sufficient conditions of maximization. These are the contributions of
Buquoy to the development of the mathematical economics.
Luigi Molinari Valeriani (1758-1828) is the fifteenth name in the bibliography.
He published a book under the title of" Del prezzo delle cose tutte mercantili" in
1806. His opinions were about price and supply and demand affected by Verri and
Frisi. Valeriani distinguishes between "value in genere or in the abstract" of a good
and its "specific value" (Theocharis, 1983, p. 37). The " value in genere" is the total
utility of a good. The specific value of a good is determined by " value in genere"
and the quantity of that good. And this specific value determine the exchange value
or price. Even, his opinions were not very original (similar to Verri and Frisi), his
clear mathematical representation gave him a place in the bibliography.
Thomas Perronet Thompson ( 1783-1869), English writer, is the sixteenth name.
He made two contributions to economics: the first one is his mathematical article,
"On the Instrument of Exchange", which was published in 1824. This article
contained the first applications of calculus and geometry to economic analysis by a
EKONOMIK YAKLA$/M
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British author. The importance of this article is to show that total (economic) gains
are maximized where marginal revenue is equal to marginal cost. He proposed that
the government issue on irredeemable paper currency which the public could use to
pay the taxes, thus making it an acceptable "instrument of exchange" (Henderson,
1985, p. 411 ). If the daily issue of such currency was equal to its average daily tax
receipt~, the currency would circulate at par (Henderson, 1985, p. 411 ). Any amount
of currency in excess of this legitimate issue, which Thompson called as a superfluous issue, would depreciate the paper currency. Then, he showed mathematically
t"tat governments gain would be maximized when
(s + p) (d + z) = p (d + z)
where s is the daily superfluous issue, p is the daily legitimate issue, t is time (in
days) and z is the fraction which expresses the depreciation. The left hand side of
this equation is the government's marginal revenue and the right hand side is the
government's marginal cost.
The second is his non-mathematical pamphlet, "The true theory of rent", which
was published in 1826. Even it is not a mathematical, this pamphlet contained some
information about dynamic equilibrium.
L.F.G. De Cazaux, French economist, is the seventeenth name in the bibliography. He published" Elemens d'Economie Privee et Publique" in 1825. This book is
about "value". It is difficult to find a wide information about his life. According to
him, the value is not only the price of something. It is the price as affected by the
"value" of money; but, the value of money is measured, according to Cazaux, by the
rate of interest (Theocharis, 1983, p. 81-82).
The eighteenth name in the bibliography is Francesso Fuoco (1777-1841). He
has two important works. One ofthem is "Saggi economici" which is consist ofsev. eral essays and the other one is "introduzione allo studio della economia industriale"
(1829). "Saggi economici" was published between 1825 and 1827. In this book, his
important essay named" Applicazione dell Algebra all Economia Politica" states his
opinions about mathematical economics. According to Fuoco, economic quantities
like price, value, abundance,scarcity, need, etc. are things which can be expressed
algebraically and reduced in to algorithmic functions once their respective relations
have been determined and the calculus can then be applied and results, like the limits of these functions, and their maximum or minimum values, obtained. Thus, the
application of algebraic language is not only easy and natural but also most useful
as new truths may be discovered (Theocharis, 1983, p. 92). As understand, he is a
strong advocate of using mathematics in economics. His important contribution to
150
I
HAKAN NAIM ARDOR
development of mathematical economics is to determine why economists should use
mathematics in their analysis. Also, he reexamined mathematically some previous
ideas such as Ricardo's theory of rent and Valeriani's theory of value. He also had
very interesting opinions about productivity of labor. In addition to these, he define
economic static and dynamics using the concept of dynamics and static in physics
{Theocharis, 1983, p. 96).
Karl Henrich Rau (1792-1870), a professor, is the nineteenth name in the
bibliography. He published "Lehrbuch der politischen okonomie" which is consist of
three volume in 1826. The first volume of this book is about law, the second volume
is about economic policy and third volume is about public finance (Schumpeter,
1961, p. 503).
Johann Heinrich von Thiinen (1783-850), German agricultural economist, is the
twentieth name. The first volume of his "Der isolirte Staat in Beziehung auf Land
wirthschaft und Natinaol okonomis was published in 1826". He searched for the
empirical verification of some certain ideas at which he had intuitively arrived at an
early age. He constructed an isolated model (he called it ideal state). His isolation
model can be thing as a closed economy. He examined the influence which costs of
transportation 'have on the !~cation of agriculture and on the methods of cultivation
followed with this model (Theocharis, 1983, p. 112).
William Whewell (1794-1866) is twenty first name. He was a scientist and
philosopher at Trinity Colloge, Cambridge. He devoted most of his efforts to
philosophy and the natural sciences. But he made some interesting contribution to
mathematical econotftics. Even Robertson (1949, p.535) called him a "translator",
that his work consisted only of putting Ricardian doctrines into mathematical
language, he was very important name. He gave his name a group of economists
who are called the Whewell Group of Jvfathematical Economists. Whewell published four papers about mathematical economics which were published as a book
later. His first paper " Mathematical Exposition of Some Doctrines of Political
Economy"; was published in 1829.(8) In this paper, he developed a mathematical
measure of price flexibility (Henderson, 1985, p. 408).
w=eu
where w =dp I p, u = dq I q and e is the coefficient of price flexibility.(9) As we
see very easily, his flexibility measurement "e" is demand elasticity. He is known the
first author who formulate a mea~ure for demand elasticity.
(8) I am not going to discuss Whewell's other three papers, because their publication date is out of our
time period.
(9) Later, in this third paper, carrying his analysis further he developed the concept of demand
elasticity.
I
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D. G. Lube', a British author, is the last name of the first part of the
bibliography. He published a book named "An Argument against the Gold Standard
with an Examination of the Principles of the Modem Economists" in 1832. No more
information is available about him in the references I use. Because, he did not made
original contributions to the mathematical economics and the theory of economics.
He was a follower ofValeriani, Kroncke, Rau and Lang. His argument was the gold
standard imposes an arbitrary limit on currency, as the production of gold can not
keep pace with increased demand (Theocharis, 1983, p. 126).
3.COURNOT
Antoine Augustin Coumot was a French mathematician who published books on
mathematics, philosophy, and economics. He was born on 28 August 180 1 in
France. He graduated from mathematics department of Sorbonne in 1823. He took
some courses from Laplace,Lagrance, and Poisson, who are very well known mathematicians even today. He published his first book "Recherches sur les principes
mathematiques de la theorie des richesses" in 1838 when he was the Rector of the
Academy of Grenoble. After this date, he did not publish book on economics until
1877. In 1877, he published his second book on economics named "Revue
Sommaire des doctrines economiques".(lO) Coumot, also published several books
on mathematics and philosophy. Unfortunately, his works on economics were neglected until after his death. But, his works were continued by Jevons, Marshall, and
Fisher.
It is interesting that as a mathematician how Coumot takes great place the development of economics (not just the development of mathematical economics). For
example, his treatment of oligopoly still continue to take place at the new published
microeconomics books. There are two opposite views about this (Theocharis, 1983,
p. 132). The first one believes that the pure theory of mathematics was not popular
at Coumot's days. The fashion was to attempt the extension of the application of the
theory of mathematics to as many practical fields as possible. Due to this reason, he
choose economics. As second view, economics is a side interest for him and he
believes that using mathematics, he could express some economics ideas more clearly. Then he began to study economics and put Smith, Ricardo and Say's arguments
into mathematical form.
(11) This book is the English translation of his "Resherches sur les principes mathematiques de la
theorie des richesses".
1!i2l HAKAN NAIM ARDOR
At the preface of his "The Mathematical Principles of the Theory ofWeaith(ll)
(Cournot, 1838, p. 3) he criticized authors who had written on political economy:
They imagined that the use of symbols and formulas could only lead to
numerical calculations, and as it was clearly perceived that the subject was
not suited to such a numerical determination of values by means of theory
alone, the conclusion was drawn that the mathematical apparatus, if not liable
to lead to erroneous results, was at least idle and pedantic. But, those skilled
in mathematical analysis know that its object is not simply to calculate numbers, but that is also employed to find the relations between magnitudes
which can not be expressed in numbers and between functions whose law is
not capable of algebraic expression.
Also he said that (Coumot, 1838, p. 5);
I have not set out to make a complete and dogmatic treatise on Political
Economy; I have put aside questions, to which mathematical analysis can not
apply, and those which seem to me entirely cleared up already.
We can find almost all clues about his method from these two paragraph. He
showed the relationships between economic quantities by functions. He did not try
to find numerical results between economic quantities. His background as a mathematician affected his economic thinking. Mathematics as a positive sciences based
on a pure theory which is definite and invariable. Using mathematics in economics,
he tried to create the pure theory of economics. In his analysis, he applied the theory of arbitrary functions and the differential calculus to show maximum and minimum values of economic quantities. And he believed that diagrammatic representations could be very usefut(l2)
In his "Recherches sur les principies mathematiques de la theorie des richesses"
"'
he examined several economic
subjects such as price theory, the theory of the rates
of exchange, international trade and theory of social income. Without details and
mathematical representation, I am going to explain these respectively.
Coumot defined and described the downward sloping demand curve and he
proved that equilibrium price is established when the quantity supplied equals the
quantity demanded. He showed that where marginal costs decrease with the expansion of output, a firm's production continues to increase until this affect the market.
1
(11) This book is the English translation of his "Resherches sur les principes mathematiques de la
theorie des richesses".
(12) At the end of his book, there is a appendix which consist often figures.
EKONOMIK YAKLA§IM
1153
In other words, the firm faces a downward sloping demand curve and therefore has
some monopoly power.
It is, moreover, plain under the hypothesis of unlimited competition, and
where, at the same time, the function cpk'[ Dk ] (marginal cost) should be
decreasing one, that nothing would limit the production of the article. Thus
wherever, there is a return on property, or a rent payable for a plant of which
the operation involves expenses of such a kind that the function cpk' [Dk ]
(marginal cost) is a decreasing one, it proves that the effect of monopoly is
not wholly extinct, or that competition is not so great but that the variation of
the amount produced by each individual producer affects the total production
of the article, and its price, to a perceptible extent (Coumot, 1838, p. 91-92).
According to Cournot, larger firms increase their advantage over their smaller
competitors with increasing returns or decreasing cost, therefore competition leads
to monopoly. The goal of monopolist is maximum profit and its decision about price
to maximize profit is determined by his cost and the elasticity of demand. As we
have seen, with some main points, his monopoly analysis is very successful. I think
because it is almost same with today's monopoly analysis. We can say same things
for his oligopoly analysis. In addition of its success, also it began with Cournot.
At the beginning of chapter XI of his book, he said that;
So far we have studied how, for each commodity by itself, the law of demand
in connection with the conditions of production of that commodity, determines the price of it and regulates the incomes of its producer. We considered
as given and invariable the prices of other commodities and the incomes of
other producers ; but in reality the economic system is a whole of which all
the parts are connected and react on each other (Coumot, 1838, p. 127).
Then, he discussed the necessity of consideration which take into account entire
system. But, he admit that this could swpass the power of mathematical ·analysis.
Then, he choose another approach to investigate how changes in the price of consumers' good affect individual incomes and national income. To do this, he defines
social income (or national income):
We will denote by social income the sum, not only of incomes properly so
called, which belong to members of society in their quality of real estate owners or capitalists, but also the wages and annual profits which come to them
in their capacity of workers and industrial agents. We will also include in it
the annual amount of the stipends by means of which individuals or the state
sustain those classes of men which economic writers have characterized as
1541 HAKAN NAIM ARDOR
unproductive, because the product of their 1abor is not anything material or
salable. Usage would doubtless permit the use of these words in a different
acceptation; but we think that the definition which has just given is better
qualified than any other for directing the line of argument toward accurate
deductions and consequences capable of application (Cournot, 1838, p. 128).
As we understand, he defines the social income as the summation of every kind
individual incomes (rents, profits and wages). Then, he shows the effects of changes
in the prices of consumers' good to national income. He could not be so successful
with his this analysis like his the pure theory of price. His arguments was criticized
by Irving Fisher and Vilfredo Pareto.(l3)
Coumot made some contributions to the theory of international trade. For exam~
pie, the determination of foreign exchange, price determination under international
trade, the effect of international trade on national income. These all were very
sue~
cessful attempts for his periods. As his general method, he explains these subjects
very clear and mathematically starting from very simple situation and solution to
general equilibrium solution.
As we mentioned in the introduction of this term paper, he has a great place in
the development of mathematical economics. He is sometimes considered as the
father of mathematical economics. But, we should note that Before Cournot there
are some important attempts to employ mathematical analysis in the discussion of
economic problems as we explained in the section two. And such attempts absolutely affect Cournot's analysis.
References
Blaug, M.,l984, Economic Theozy in Retrospect, Fourth Edition, Cambridge
University Press, Cambridge.
Cournot, A., 1877, The Mathematical Principles at the Theozy of wealth, New York,
The Macmillan Company.
Gherity, J. A., 1990, "The International Flow of Mathematical Economics: From
Lloyd to Thompson", The Manchester School, Vol. LVIII, No. 2, June, p.165
- 172.
Heilbroner, Robert, L., 1992, The Worldly Philosophers, Sixth Edition, Simon &
Schuster Inc., New York.
Henderson, J. P., 1985, "The Whewell Group of Mathematical Economists", The
Manchester School, Vol. 53, no. 4, p. 404 - 431.
(13) The more information can be found in Theocharis, p. 182-185.
EKONOMIK YAKLA§IM
1155
Oser, J., 1970, Ibe Evolution of Economic Ihou!Wt, Second edition, Harcourt,
Brace & World, Inc., New York .
Robertson, R. M., "Mathematical Economics Before Coumot", Journal of
Po1itica1Economy, Vol. 57, no. 6, 1949, p. 523- 536.
Schumpeter, J. A., 1961, Histozy of Economic Analysis, Fourth edition, Oxford
University Press, New York.
Spiegel, H. W., 1983, The Growth Of Economic
University Press, Durham, North Carolina.
Thou~ht,
Second edition, Duke
Theocharis, R. D., 1983, Early Developments in Mathematical Economics, Second
edition, Porcupine Press, Philadelphia .
APPENDIX
BIBLIOGRAPHY
OF
MATHEMATICAL ECONOMICS
FROM CEVA TO COURNOT
(1711-1837)
1711 CEVA, Giovanni. De re numeria, quoad fieri potuit geometrice tractata, ad
iliustrissimos et excellentissimos dominos Praesidem Quastoresque hujus
arciducalis Caesaraie Magistratus Mantuae. Mantova, 4 °, 60pp. [Reviewed
by F. Nicolini, Giom. Degli econ., Oct. 1878. See also Palgrave's Diet.].
1717 MARIOTTE, Esme. Essaie de logique. 2d ed. In collected works. Leide.
[See Principe 97 and 11 me partie, art. Ill.].
1738 BERNOULLI, Daniel. Specimen theoriae novae de mensura sortis.
Commentarri academiae scientiarum imperialis Petropolitanae, vol. V, pp.
175-92. [German transl., 1896, by A Pringsheim: Die Grundlage der modernen Werthlehre. Versuch einer neunen theorie der Weribestimmung von
Glucksfallen (Einleitung von Ludvig Fick). Leipzig (Duncker & Humbl), 60
pp.].
1754 ANON. [F. Forbonnais). Elemens du commerce, 2d. ed., vol. 11, chs. Viii, ix,
Leyde & Paris, 12°.
1764 BECCARIA, Cesare. Tentativoanalitico sui contrabbandi, 1764-5. Estratto
dal foglio preodico intitolato :11 caffe, vol. I, Brescia. [Also in Custodi's
Scrittori classici Italiani di economia politica, parte modema, vol. XII, pp.
235-41, Milano, 1804.
1771 ANON. [probably Maj. Gen. Henry Lloyd]. An essay on the theory of
money. London, 161 pp.
1781 ANON. [A. N. Isnard]. Traite des richesses. London & Lausanne, vols.,
xxiv, 344, 327 pp. [Slightly math.]
I
~
i!
'_!;
15&1 HAKAN NAIM ARDOR
1786 ANON. (Condorcet]. Vie de M. Turgot, pp. 162-9, London. [Eng. Transl.,
1787, pp. 403-9],
1792 SILIO, Gnglielmo. Saggio sull' influenza dell' analisi nelle scienze politiche
ed economiche. Nuova raccolta d' opuscoli di autori Siciliana, vol. V (?),
Palermo.
1801 CANARD, N. F. Principies d'economie politique, ouvrage couronne par
l'Institut. Paris (Buisson), 235 pp. [Reviewed by Francis Homer, Edinburgh
Rev., no. II.]
1802 KRNCKE, Clans. Versuch einer Theorie des Fuhrwerks mitAnwendung auf
den Strassenbau. Giessen.
1803 SIMONDE, J. C. L. De la richesse commerciale, vol. I, pp. 105-9, Geneva.
1804 KRONCKE, Clans. Das Steuerwesen nach seiner Natur und seinen
Wirkungen untersucht. Darmstadt & Giessen (Heyer), xxxii, 440 pp.
1809 LANG, Joseph. Ueber denobersten grundsatz der politischen Oekonomie.
Rig a.
1810 KRONCKE, Clans. Anleitung zur Regulirung der Steuer.
1811 LANG, Joseph. Grundlinien der politischen Arithmetik. Kursk (Langner),
xxii, 216 pp.
1815 BUQUOY, Georg von. Die Theorie der Nationalwirthschaft, nach einem
neuen Plane und nach mehreren eigenen Ansichten dargest~llt. Leipzig, 4 o. [3
Nachtrge: 1816, 1817 (524 pp.), 1818.]
1816 ANON. [L. M. Valeriani]. Apologia della formola p = i I o' trattandosi del
come si determini il prezzo delle cose tutte mercatabilli, contro cio che ne
dice il celebre autore del "Nuova prospetto delle scienze economiche"
Bologna (Marsigli), 62 pp.
1817 ANON. [L. M. Valeriani). Discorso apologetico in cui si sostiene recarsi
invano pelt celebre autore del "Nuovo prospetto delle scienze
economiche control' Apologia della formola p = i I o trattando si del come si
determini il prezzo delle cose tutte mercatabili, cio che il medisimo ha scritto nel tomo 11, il pag. 114-117, 141-146, e nel IV, pag. 214-219,244-263 dell'
opera suddetta. Bologna (Marsigli), 114 pp.
1824 ANON. [T. Perronet Thompson). The instrument of exchange. Westminster
Rev., vol. I, pp. 171-205. [Reprinted separately, 1830, 27 pp.; postscript to
same, Westmin. Rev., 1830, vol. XII, pp. 525-33; reprinted 1842 in:
Exercises, political and others, vol. Ill, pp. 295-343.]
1825 CAZAUX, L. F. G. de. Elemens d'economie privee et publique. Paris
(Huzard) & Toulouse (Douladoure), 251 pp. [Slightly math.]
1825 FUCCO, Francesco. Saggi economici, 1st ser., vol. 2. Pisa, 1825-27.
[Math.?]
EKONOM{K YAf<LA~IM
1157
1826 RAU, K. H. Grundstze der Volkswirthschaftslehre. [Subsequent editions,
1833, '37, '41, '63, '68, Leipzig & Heidelberg. Very slightly math.]
1826 THOMPSON, T. Perronet. On rent.
1826 THUNEN, J. H. von. Der isolirte Staat in Beziehung auf Landwrithschaft
und Nationalkonomie, oder Untersuchungen her den Einfluss, den die
Getreidepreise, der Reichthum des Bodens und die Abgaben auf den
Ackerbau ausben. Ier Th., Hamburg, vii, 290 pp. [2nd ed. Of same, 1842,
Rostock, xv, 391 pp.; 1er fasc. Von 2ter fasc. Von 2ter Th., 1863, Rostock, ix,
444 pp. French transl. By Leverrierre, 1851, Paris (Guillaumin), viii, 340 pp.;
French transl. by Wolkoff (of fasc. Of 1850), 1857, Paris. Slightly math.]
1829 WHEEWELL, William. Mathematical exposition of some doctrines of
political economy. Cambridge Philos. Trans., vol Ill, pp. 191-230, 4°
[Continued, 1831, vol. IV, pp. 155-98; 1850, vol. IX, pp. 128-49, and Part II,
pp. [1-7]; Italian transl. In Biblioteca dell' econ., 1875, 3rd ser., vol. II, pp. 165.]
1832 LUBE'D. G. Argument against the gold standard. London, iv, 192 pp.