Amsterdam: North Holland Publishing Company. Mathematical Economics. Twenty Papers of Gerard Debreu. Cambridge: Cambridge University Press. Milgate and P. Newman eds. The New Palgrave. New York and London: W. Dierker, E. Topological Methods in Walrasian Economics. Berlin-New York- Heidelber Springer. Hahn, F. Hara, C. Twenty Papers by Gerard Debreu. Kamiya, K. Existence and uniqueness of equilibria with increasing returns. Mantel, R.
Debreu, Gerard [WorldCat Identities]
Mas-Colell, A. D Whinston, J. Green, McKenzie, L. Momi, T. Nash, J. Negishi, T. Nikaido, H. Introduction to Sets and Mappings in Modern Economics. Amsterdam: North-Holland Publishing Company. Amsterdam: North Holland Publishing Company. Mathematical Economics. Twenty Papers of Gerard Debreu. Cambridge: Cambridge University Press.
Milgate and P.
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Kamiya, K. Existence and uniqueness of equilibria with increasing returns. Mantel, R. Mas-Colell, A. D Whinston, J. Green, McKenzie, L.
Momi, T. Nash, J. Negishi, T. Nikaido, H. Introduction to Sets and Mappings in Modern Economics. Amsterdam: North-Holland Publishing Company. The development of paper money is one of the dramatic developments by human societies. In the past, the value of money was "secured" by precious metal usually gold but recently many currencies don't guarantee that you can "turn in" and receive gold or silver in exchange. In fact, there are now cryptocurriences that are exchanged on markets that have no government backing at all.
Mathematics had been used to understand the way that changes in the " money supply " affect what goes on in a country's or the world's economic system. In modern times many economists talk about being influenced by the economist John Maynard Keynes Modern acrimonious discussions of economics often center on "Keynesian" approaches to government activity in the economy. Some of this plays out in Keynes's views of Say's Law. While many know Keynes's name only as an economist, he was also an extremely talented mathematician.
In particular, he was the 12th highest scorer for 12th wrangler on the famous examination known as the Mathematical Tripos at Cambridge University. The person with the highest score was known as the senior wrangler. The Mathematical Tripos was known to be a very difficult examination to perform well on, but students who did often got high recognition for their accomplishment. This encouraged some students to devote great amounts of time to doing well on this examination, even if it meant they neglected going to their "classes.
The distinguished number theorist G. Hardy was very unhappy with the way that the Mathematical Tripos distorted the mathematical experience for students at Cambridge and worked hard to have it downgraded in importance. He felt performing well on the Tripos had become an end in itself rather than a means to an end. Hardy himself had been a 4th wrangler in Many today see some of the international comparison exams related to mathematics as distorting what goes on in individual countries because good performance on these examinations has become an end in itself for some countries rather than having the mathematics education system of the country serve the needs of the students in the country.
Keynes is particularly known for his work in probability theory. He authored the book A Treatise on Probability in , while he was still at Cambridge.
His views on probability were complex and not fully in accord with what came before. His work set off a reaction by Frank Ramsey , and many other philosophers and mathematicians were inspired to take a closer look at how probability theory could be used in the "real world. Historically, probabilities were often interpreted as "stabilized" relative frequencies.
This approach may make sense for situations where a biased or fair coin is being tossed or a fair die is being rolled. But for the kinds of issues that often drew Keynes's attention and that of others concerned with the making of decisions in an economic context, this approach could not be defended. Many decisions were made in environments that had never been seen before and would not be seen again. This led to probability being thought of in terms of degrees of belief, and gave birth to what is often described as subjective or personal probability.
Leonard Savage , whose doctorate degree in mathematics at Michigan was entitled "The Application of Vectorial Methods to the Study of Distance Spaces," wrote about this approach in his important book of about probability theory. In particular, the idea of utilitarianism discussed below sometimes involves decision making based on the notion of "expected utility.
Gérard Debreu, 1921-2004
Utilities and expected utilities also play a big part in mathematical game theory which is an important part of economics and mathematical economics. Many people who contributed to economics did so in ways that are also viewed as contributions to philosophy. One point of view involves what has come to be called utilitarianism. This term refers to the ethical notion that individuals and governments should maximize "utility," which in this context refers to well being. The pioneers though there are other "ancestors" for the ideas involved of this way of thinking were Jeremy Bentham and John Stuart Mill.
However, what does "utility" mean and how can we measure it? From a more recent perspective, utility is sometimes described as the pleasure, satisfaction, or value of some action or having possession of some item. One gets "utility" from owning a house, from getting a college education, from proving a new theorem, etc.
Thus, for example, he argued against slavery and for women's rights at a time when slavery was widespread, including legal in the United States. He also wrote about women's rights at a time when women could not vote in the US or Great Britain, and the rights of women within their marriages not to be "subservient" to their husbands.
On the subject of happiness, certainly a notion that means different things to different people, he wrote: "the ultimate end, for the sake of which all other things are desirable whether we are considering our own good or that of other people is an existence as free as possible from pain and as rich as possible in enjoyments.
Later I will look at some of the issues about how to measure "good" or "happiness" and whether one can sum the individual happiness of people in a group to get the "happiness" of the group. However, it is noteworthy that simple examples show it is possible to give great "happiness" to a very small group of people in a big group, making it look like the happiness of the group is great when in fact the average mean happiness for the group is not truly large.
One billionaire living in a small town may make it look as if the mean wealth of the people in the community is high when in fact the wealth of a single person is distorting the impression of the wealth of the group.
Something similar can happen for "utility," and one thing that might go into measuring the "utility" of a group is the wealth of its members. It is surprisingly recent that the American philosopher John Bordley Rawls looked at issues related to calling into question whether the greatest good for the largest number of people was the "proper" optimization measure for policy. Vastly oversimplifying Rawls, instead of arguing for the greatest good for the greatest number of people he argued for a maximin principle--acting in a way that made the outcome for the worst well-off person to be as favorable best as possible.