Use of Quasiconcave Utility Functions in Economics

Quasiconcave is a mathematical concept that has several applications in economics. To understand the significance of the  terms applications in economics, it is useful to begin with a brief consideration of the origins and  meaning of the term in mathematics. Origins of the Term The term quasiconcave was introduced in the early part of the 20th century in the work of John von Neumann, Werner Fenchel and Bruno de Finetti, all prominent mathematicians with interests in both theoretical and applied mathematics, Their research in  fields such as probability theory, game theory and topology eventually laid the groundwork for an independent research field known as generalized convexity.  While the term quasiconcave: has applications in many areas, including economics, it originates in the field of generalized convexity as a topological concept. Definition of Topology Wayne State Mathematics Professor Robert Bruners brief and readable explanation of topology  begins with the understanding that topology is a special form of geometry. What distinguishes topology from other geometrical studies is that topology treats geometric figures as being essentially (topologically) equivalent if by bending, twisting and otherwise distorting them you can turn one into the other. This sounds a little strange, but consider that if you take a circle and begin squashing from four directions, with careful squashing you can produce a square. Thus, a square and a circle are topologically equivalent. Similarly, if you bend one side of a triangle until youve created  another corner somewhere along that side, with more bending, pushing and pulling, you can turn a triangle into a square. Again, a triangle and a square are topologically equivalent.   Quasiconcave as a Topological Property Quasiconcave is a topological property that includes concavity. If you graph a mathematical function and the graph looks more or less like a badly made bowl with a few bumps in it but still has a depression in the center and two ends that tilt upward, that is a quasiconcave function. It turns out that a concave function is just a specific instance of a quasiconcave function—one without the bumps. From a laypersons perspective (a mathematician has a more rigorous way of expressing it), a quasiconcave function includes all concave functions and also all functions that overall are concave but that may have sections that are actually convex. Again, picture a badly made bowl with a few bumps and protrusions in it.   Applications in Economics One way of mathematically representing consumer preferences (as well as  many other behaviors) is with a utility function. If, for example, consumers prefer good A to good B, the utility function U expresses that preference as:   Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  Ã‚  U(A)U(B) If you graph out this function for a real-world set of consumers and goods, you may find that the graph looks a bit like a bowl—rather than a straight line, theres a sag in the middle. This sag generally represents consumers aversion to risk. Again, in the real world, this aversion isnt consistent: the graph of consumer preferences looks a bit like an imperfect bowl, one with a number of bumps in it. Instead of being concave, then, its generally concave but not perfectly so at every point in the graph, which may have minor sections of convexity. In other words, our example graph of consumer preferences (much like many real-world examples) is quasiconcave.  They tell anyone wanting to know more about consumer behavior—economists and corporations selling consumer goods, for instance—where and how customers  respond to changes in good amounts or cost.

Notes on Art History - 2917 Words

Note: This statue of an Ancient Egyptian woman clearly shows the way of dressing in Ancient Egypt. This statue belongs to the New Kingdome period. Body: The woman is wearing a kalasiris, a typical ancient Egyptian dress. The length of the dress tells us that this woman probably belonged to the upper class, as working woman mostly wore a kalasiris in a shorter variant. The dress is made of linen, which was perfect for Egypt’s hot climate. The dress is undecorated, but does have a lot of pleats going in various directions, which must have been very labour intensive. Head: The woman is wearing a wig, made of human hair or date palm fibres. The hair is curled and was kept together by wax. The wigs were only worn by wealthy people.†¦show more content†¦In that era, all men wore the same type of cloth, but the fabric and way it was worn, marked important social differences. Agustus, being an emperor is therefore a clear example of how people of higher social status would wear their clothes. Body: Augustus is wearing a tunica, an essential garment in the warderobe of the ancient Roman man. It consists of two pieces of rectangular fabric, one set on top of the other. It was sewn together at the sides and the top wit holes left for the head and arms. This tunica has long and wide sleeves. On top of the tunica, Augustus wears a toga. A long cloak which is draped around the body. Head: Augustus has short cut hair and his toga is draped around his head to form a hood. Court of Emperor Justinian with Macimian and court officialsand Praetorian Guards, Basilica of San Vitale in Ravenna, Italy, 547 AC, Mosaik Note: A clear example of clothing from the Byzantium era. Body: Justinian, in the centre, is wearing an imperial chlamys with a large lozenge shaped goldon panel (tablion) and an opulent brooch. Underneath that he wears a white silk tunic with tight sleeves (Divistion), which is decorated with god stripes and circular roundles and gold embroidered rd shoes. The Dignitaries wear a white chlamys with a purple tablion an an embroidered tunic. The archbishop Maximianus wears a goldon chasuble and a white dalmatic with very wideShow MoreRelatedThe Realism Movement Of France1606 Words   |  7 Pagesthe Realism movement, defined Realism as a human conclusion which awakened the very forces of man against paganism, Greco-Roman art, the Renaissance, Catholicism, and the gods and demigods, in short against the conventional ideal† (New World Encyclopedia). 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The Long Telegrams Impact on the Truman Doctrine Free Essays

pThe long Telegram and It’s impact on the Truman Doctrine George Kennan’s long telegram had many impacts on the American foreign policy. Its influence can be seen directly through the Truman Doctrine in 1947. The Long Telegram influenced Truman’s doctrine of containment, and it opened the eyes of many Washington officials as to how dangerous the Soviets and more specifically communism was. We will write a custom essay sample on The Long Telegrams Impact on the Truman Doctrine or any similar topic only for you Order Now George Kennan’s Long Telegram was a reply as to why the Soviets weren’t supporting the World Bank and the International Money Fund. At the time, the  American view of the  Soviet was friendly, yet cautious because the  Soviets had been  allied with the US in  WWII. The long telegram, which wasreleased to the public in 1947, basically created the foundation on which the Cold War stood from the US’ point of view. By claiming that the Soviets were completely against capitalism, would side with Marxists, and separated from reality, Kennan set up a standard system of beliefs  that the US would follow throughout the Cold War, and even gave ideas as to how to combat the Soviets if need be. The Long Telegram gave the US clear reason to consider war with Russia inevitable. It fostered further fear in America of the Soviets by talking  about how  Russia couldn’t be trusted and great lengths should be taken to ensure the absence of Soviets and their influence  in America. The Truman Doctrine was a contribution made to foreign policy by Harry Truman in 1947, after the Long Telegram had come into being. Truman declared, kind of like Wilson’s Moral Diplomacy, that it was the US’ duty to spread democracy and battle communism. Truman asked Congress to send aid to Greece and Turkey. After being ravaged by Germany in WWII,  Greece had been about to lose aid from Great Britain; Turkey had also depended greatly on the help of the US and Britain. The US would  have been  in danger of losing power over the Soviets and in Europe and Asia had Russia gained the two countries. This proves that what George Kennan wrote helped Truman realize that he needed to help other nations and create the foreign policy. Kennans telegram greatly impacted many other aspects of the Truman doctrine, for example the truman Doctrine all but promised that the United States would resist any Soviet expansion. This refects Kennan’s advise that the United States must carefully choose its points of resistance. Kennan’s notion of containment which was expressed in the Long Telegram was also illustrartes in the Truman Doctrine. George Kennans writings inspired America and thanks to the Long Telegram and the Truman Doctrine, America opened their eyes and did what was necessary to not allow themselves to be over ruled by the soviets. How to cite The Long Telegrams Impact on the Truman Doctrine, Essay examples