Influence of Math on Economics
Michelle Balmer, Marcie Holland, Beverly Segars, Israel Figueroa, and Porshia Cross MTH 110
Rigoberto Martinez
March 17, 2012

The Influence of Math on Economics

The history of mathematics is an investigative study of the discoveries of mathematics methods and notations from the past. The study of mathematics began in the 6th century BC with the Pythagoreans who coined the ancient Greek term mathematics with the meaning subject of instruction. Before the spread of knowledge, mathematics was written expressions of the development of Babylonian, Egyptian, and Pythagorean Theorem, which demonstrate the basics of arithmetic and geometry. Arithmetic is one of the oldest forms of mathematics used by Antoine-Augustin Cournot and Joseph Louis Francois Bertrand for tasks of simple day-to-day counting to science and business activities. Arithmetic involves the study of quantity as it relates to addition, subtraction, multiplication, and division. Mathematicians refer to the more advanced term of number theory.
Antoine-Augustin Cournot and Joseph Louis Francois Bertrand’s influence on mathematics lay the foundation of the economic effect of the application method on economic theory and analysis of how it will affect others. Each allows individuals to form a meaningful understanding of the complex expressions in various languages. Cournot and Bertrand have used the importance of mathematics to optimize the maximum values of economic functions. The development of Hindu and Arabic numerals devise place value concept as it relates to business operations. In recent years, new research trends have developed that indicate the meaningful functions of other branches of mathematics that use modern techniques of set theory. This assignment will express the impact Antoine-Augustin Cournot and Joseph Louis François Bertrand’s theories had on the economies need for supply and demand. Their contributions to science and mathematics had a positive effect on their society as well as modern society as it is seen today.
Antoine-Augustine Cournot born August 28, 1801 in France and Joseph Louis Francois
Bertrand born March 11, 1822 in Paris France were two known mathematicians who have influence the history of mathematics. At the age of nineteen Cournot enrolled in preparatory school for mathematical studies in Beasancon, France. Soon after in 1821 he was admitted to Eole Normale Superieure in Paris. In 1823 he became the literary advisor to Marshal Gouvoin Saint Cyr and began to tutor his son in mathematics. Cournot wanted to be the best so in 1829 he earned his doctorate in Science concentrating on astronomy and mechanics. Cournot was the first economist who was competent in both subjects. Cournot endeavored to apply mathematics to the treatment of economics. Some of his great works include his book Researches on the Mathematical Principals of the Theory of Wealth.
In 1838, Cournot was hired as rector and professor of mathematics at the Academy of Grenoble. Three years later he assumed the role of inspector general of education and published one of his best-known works Recherches sur les principes mathématiques de la théorie des richesses (Researches on the Mathematic Principles of the Theory of Wealth). Here he introduced the idea of mathematical function and probability into economic analysis (“New World Encyclopedia,” n.d.). In particular he presented his ideas about monopoly, duopoly (or oligopoly) and perfect competition. Furthermore, he showed the affect taxes might have on production costs. Many economists believe that this book may have caused a paradigm shift toward modern economic analysis and provided a break from previously held theories (“New World Encyclopedia,” n.d.).
Cournot’s theories on monopolies and "duopolies" are still famous today. His definition of a market is still the basis for that presently used in economics (“Today in Science," n.d.):
Economists understand by the term Market, not any particular market place in which things are bought and sold, but the whole of any region in which buyers and sellers are in such free intercourse with one another that the prices of the same goods tend to equality easily and quickly.
“Researches” makes Cournot a pioneer of mathematical economics, 25 years before Stanley Jevons (O’Conner & Robertson, 1996). Jevons’ book “The Theory of Political Economy” is credited with beginning the mathematical method in economics (“Standford Encyclopedia of Philosophy,” January 2007). While Cournot's economic masterpiece is credited with presenting the theories we still use today, he received hardly any response (or when there was a response, it was highly critical) when it came out in 1838. According to New World Encyclopedia (n.d.), “the members of the French Liberal School, who dominated the economics profession in France at the time, took no notice of it, leaving Cournot crushed and bitter” (Biography).
Furthermore, there are other mathematicians who dared to challenge Cournot work such as Joseph Louis Francois Bertrand. To make matters worse, not only was Cournot’s book ignored by the economic ‘experts’ of the time, but to have a mathematician come from behind and disprove your work might be considered a rude insult at best! Nevertheless, Joseph Louise Bertrand theorized, after reviewing Cournot’s work on oligopolies (duopolies) that Cournot had reached a misleading conclusion.
Joseph Louis Francois Bertrand 1822-1900 was intelligent as a child. At the age of nine years old algebra and elementary geometry were no competition for him. Just like Cournot Bertrand earned his degree at an early age in math. He was seventeen years old when he earned a doctorate for his thesis on thermodynamics. Bertrand was very busy in 1839 due to his first paper being published, the mathematical theory on electricity. Bertrand published works include the differential geometry and the probability theory. There is one thing they both had in common and that is the study of duopoly and the model competition used in economics. Bertrand, at the time employed as a professor at École Polytechnique and Collège de France, reworked Cournot's duopoly model using prices rather than quantities as the strategic variables, showing that the equilibrium price was simply the competitive price (Wikipedia, n.d.)
In Bertrand’s Competition, he put forth that two firms in an oligopoly will compete based on price, rather than quantities. It should be noted, however, that Bertrand’s theory has been criticized for its unrealistic approach and has not been shown to be a better model than Cournot's model. In fact, the models cater to different market structures, they have almost the same assumption but the implications are not. Bertrand suggests that duopoly is enough to drop the prices to marginal cost level. Perfect competition will arise from duopoly and some will state neither model is better than the other but it depends on the industry market. As stated in Bertrand Competition Versus Cournot “If capacity and output can be easily changed, Bertrand is generally a better model of duopoly competition”. If output and capacity are difficult to adjust, then Cournot is generally a better model. The Cournot model can be recast as a two stage model, where in the first stage firm choose capacities, and in the second they compete in Bertrand fashion.” However, even though this model cannot be applied to real life models, it still holds an important place in economics, as they explain how firms can compete (Buzzle, n.d.).

The impact of Antoine-Augustin Cournot and Joseph Louis Francois Bertrand’s work is that it laid the foundations of economics, microeconomics and eventually macroeconomics.
Antoine-Augustin Cournot and Joseph Louis Francois Bertrand’s work on microeconomics helped lay the foundations for understanding perfect competition. Antoine-Augustin Cournot “…considered conditions for equilibrium with monopoly, duopoly and perfect competition” (O’Connor and Robertson, 1996, para 7). Antoine-Augustin Cournot discovered that in the perfect competitive situation,” price is equal to marginal cost” (Fonseca, n.d., para 12). By understanding the point where price equals marginal cost in a perfect competition, John Nash was able to identify the Nash equilibrium. Nash equilibrium is “a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from his or her chosen strategy after considering an opponent's choice” (Nash Equilibrium, 2012, para 1 ). The Nash equilibrium is based on the idea of a perfect competition idea from Antoine-Augustin Cournot and Joseph Louis Francois Bertrand. The idea that a firm does not have any incentive to move from their chosen strategy because they view it as the equilibrium based on opponent’s choice. The idea that a perfect equilibrium exists helped John Nash determine that in a microeconomic viewpoint deviation from the perfect equilibrium would not occur. Companies would not try to gain a larger market share because their current share has the largest profitability.
Antoine-Augustin Cournot and Joseph Louis Francois Bertrand foundations of economics lead to the study of macroeconomics. An example of how their work was used in macroeconomics is the general equilibrium theory. Antoine-Augustin Cournot’s duopoly theory and Joseph Louis Francois Bertrand’s revision of the theory lead to the development of Leon Walras’ general equilibrium theory. “General equilibrium theory studies supply and demand fundamentals in an economy with multiple markets, with the objective of proving that all prices are at equilibrium” (General Equilibrium Theory, 2012, para 1) With the basis of Antoine-Augustin Cournot and Joseph Louis Francois Bertrand work, Leon Walras was able to change the view of economics by applying it to the economy as a whole. “Before Walras, economists had made little attempt to show how a whole economy with many goods fits together and reaches an equilibrium” (Leon Walras, 2008, para 1).
The works of Antoine-Augustin Cournot and Joseph Louis Francois Bertrand on microeconomics inspired Leon Walras and other economists to view the market as a whole. Economists would later determine ways to change the market as a whole based on the findings of economists before them in order to create the ideal market. With a basic understanding of supply and demand with two firms from Antoine-Augustin Cournot and Joseph Louis Francois Bertrand, economists were able to determine different triggers that impact the whole market’s demand and supply. This allows for economists to make predictions on future events based on current conditions and also allows economists to make suggestions on how to correct the problems in the economy.

References
Buzzle. (n.d.). Retrieved from http://www.buzzle.com/articles/bertrand-competition.html
Fonseca, G.L. (n.d). Antoine Augustin Cournot, 1801-1877. The History of Economic Thought Website. Economics New School. Retrieved on March 17, 2012 from: http://homepage.newschool.edu/~het/profiles/cournot.htm
General Equilibrium Theory. (2012). Investopedia, Inc. Retrieved on March 17, 2012 from: http://www.investopedia.com/terms/g/general-equilibrium-theory.asp#axzz1pODx9XbU Leon Walras. (2008). Library of Economics and Liberty. Liberty Fund, Inc. Retrieved on March 17, 2012 from: http://www.econlib.org/library/Enc/bios/Walras.html Nash Equilibrium. (2012). Investopedia, Inc. Retrieved on March 17, 2012 from: http://www.investopedia.com/terms/n/nash-equilibrium.asp#axzz1pODx9XbU New World Encyclopedia. (n.d.). Retrieved from http://www.newworldencyclopedia.org/entry/Antoine_Augustin_Cournot#Monopoly

O'Connor, J.J. & Robertson, E.F. (1996). Antoine Augustin Cournot. School of Mathematics and Statistics: University of St Andrews, Scotland. Retrieved on March 17, 2012 from: http://www-history.mcs.st-andrews.ac.uk/Biographies/Cournot.html

O'Connor, J.J. and Robertson, E.F. (December 1996). School of Mathematics and Statistics,

University of St. Andrews . Retrieved from http://www-history.mcs.st-andrews.ac.uk/Biographies/Cournot.html

Stanford Encyclopedia of Philosophy. (January 2007). Retrieved from http://plato.stanford.edu/entries/william-jevons/
Today in science. (n.d.). Retrieved from http://www.todayinsci.com/8/8_28.htm#birth
Wikipedia. (n.d.). Retrieved from http://en.wikipedia.org/wiki/Joseph_Louis_Fran%C3%A7ois_Bertrand www.groups.dcs.st-and.sc.uk/history