|
Debreu's Representation Theorems
In economics, the Debreu's theorems are preference representation theorems—statements about the representation of a preference ordering by a real-valued utility function. The theorems were proved by Gerard Debreu during the 1950s. Background Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). All the responses are recorded and form the person's ''preference relation.'' Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single ''utility function'' - a function that maps a real number to each option, such that the utility of option A is larger than that of option B if and only if the agent prefers A to B. Debreu's theorems address the following question: what conditions on the preference relation guarantee the existence of a representing utility function? Existence of ordinal utility function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interactions of Agent (economics), economic agents and how economy, economies work. Microeconomics analyses what is viewed as basic elements within economy, economies, including individual agents and market (economics), markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyses economies as systems where production, distribution, consumption, savings, and Expenditure, investment expenditure interact; and the factors of production affecting them, such as: Labour (human activity), labour, Capital (economics), capital, Land (economics), land, and Entrepreneurship, enterprise, inflation, economic growth, and public policies that impact gloss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Topology
In mathematics, the real coordinate space or real coordinate ''n''-space, of dimension , denoted or , is the set of all ordered -tuples of real numbers, that is the set of all sequences of real numbers, also known as ''coordinate vectors''. Special cases are called the ''real line'' , the ''real coordinate plane'' , and the ''real coordinate three-dimensional space'' . With component-wise addition and scalar multiplication, it is a real vector space. The coordinates over any basis of the elements of a real vector space form a ''real coordinate space'' of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension , ( Euclidean line, ; Euclidean plane, ; Euclidean three-dimensional space, ) form a ''real coordinate space'' of dimension . These one to one correspondences between vectors, points and coordinate vectors explain the names of ''coordinate space'' and ''coordinate vector''. It allows using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann–Morgenstern Utility Theorem
In decision theory, the von Neumann–Morgenstern (VNM) utility theorem demonstrates that rational choice under uncertainty involves making decisions that take the form of maximizing the expected value of some cardinal utility function. The theorem forms the foundation of expected utility theory. In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function, where such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility. Neumann, John von and Morgenstern, Oskar, '' Theory of Games and Economic Behavior''. Princeton, NJ. Princeton University Press, 1953. That is, they proved that an agent is (VNM-)rational ''if and only if'' there exists a real-valued function ''u'' defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of ''u'', which can then be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Preference Representation Theorem
In economics, a utility representation theorem shows that, under certain conditions, a preference ordering can be represented by a real-valued utility function, such that option A is preferred to option B if and only if the utility of A is larger than that of B. The most famous example of a utility representation theorem is the Von Neumann–Morgenstern utility theorem, which shows that any rational agent has a utility function that measures their preferences over lotteries. Background Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). If the agent prefers A to B, we write A\succ B. The set of all such preference-pairs forms the person's '' preference relation.'' Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single ''utility function'' - a function ''u'' that assigns a real number to each opt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinal Utility
In economics, a cardinal utility expresses not only which of two outcomes is preferred, but also the intensity of preferences, i.e. ''how much'' better or worse one outcome is compared to another. In consumer choice theory, economists originally attempted to replace cardinal utility with the apparently weaker concept of ordinal utility. Cardinal utility appears to impose the assumption that levels of absolute satisfaction exist, so magnitudes of increments to satisfaction can be compared across different situations. However, economists in the 1940s proved that under mild conditions, ordinal utilities imply cardinal utilities. This result is now known as the von Neumann–Morgenstern utility theorem; many similar utility representation theorems exist in other contexts. History In 1738, Daniel Bernoulli was the first to theorize about the marginal value of money. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separable Space
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence ( x_n )_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence. Like the other axioms of countability, separability is a "limitation on size", not necessarily in terms of cardinality (though, in the presence of the Hausdorff axiom, this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every continuous function on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset. Contrast separability with the related notion of second countability, which is in general stronger but equivalent on the class of metrizable spaces. First examples Any topological space that is itself finite or countably infinite is separable, for the whole space is a countable dense subset of itself. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann–Morgenstern Utility Theorem
In decision theory, the von Neumann–Morgenstern (VNM) utility theorem demonstrates that rational choice under uncertainty involves making decisions that take the form of maximizing the expected value of some cardinal utility function. The theorem forms the foundation of expected utility theory. In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function, where such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility. Neumann, John von and Morgenstern, Oskar, '' Theory of Games and Economic Behavior''. Princeton, NJ. Princeton University Press, 1953. That is, they proved that an agent is (VNM-)rational ''if and only if'' there exists a real-valued function ''u'' defined by possible outcomes such that every preference of the agent is characterized by maximizing the expected value of ''u'', which can then be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinal Utility
In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask ''how much'' better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility. For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by a function ''u'' such that: :u(A)=9, u(B)=8, u(C)=1 But critics of cardinal utility claim the only meaningful message of this function is the order u(A)>u(B)>u(C); the actual numbers are meaningless. Hence, George's preferences can also be represented by the following function ''v'': :v(A)=9, v(B)=2, v(C)=1 The functions ''u'' and ''v'' are ordinally equivalent – they represent George's preferences equally well. Ordinal utility contrast ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-infinity
In mathematics, \ell^\infty, the (real or complex) vector space of bounded sequences with the supremum norm, and L^\infty = L^\infty(X,\Sigma,\mu), the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces \ell_1 of absolutely summable sequences, and L^1 = L^1(X,\Sigma, \mu) of absolutely integrable measurable functions (if the measure space fulfills the conditions of being localizable and therefore semifinite). Pointwise multiplication gives them the structure of a Banach algebra, and in fact they are the standard examples of abelian Von Neumann algebras. Sequence space The vector space \ell^\infty is a sequence space whose elements are the bounded sequences. The vector space operations, addition and scalar multiplication, are applied coordinate by coordinate. With respect to the nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Tangent
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry. They are used to express Lorentz boosts as hyperbolic rotations in special relativity. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, and fluid dynamics. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Diction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Cuts
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two sets ''A'' and ''B'', such that each element of ''A'' is less than every element of ''B'', and ''A'' contains no greatest element. The set ''B'' may or may not have a smallest element among the rationals. If ''B'' has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between ''A'' and ''B''. In other words, ''A'' contains every rational number less than the cut, and ''B'' contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographic Preferences
In economics, lexicographic preferences or lexicographic orderings describe comparative preferences where an agent prefers any amount of one good (X) to any amount of another (Y). Specifically, if offered several bundles of goods, the agent will choose the bundle that offers the most X, no matter how much Y there is. Only when there is a tie between bundles with regard to the number of units of X will the agent start comparing the number of units of Y across bundles. Lexicographic preferences extend utility theory analogously to the way that nonstandard infinitesimals extend the real numbers. With lexicographic preferences, the utility of certain goods is infinitesimal in comparison to others. Etymology ''Lexicography'' refers to the compilation of dictionaries, and is meant to invoke the fact that a dictionary is organized alphabetically: with infinite attention to the first letter of each word, and only in the event of ties with attention to the second letter of each word, etc. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
