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Convex Preferences
In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly corresponds to the concept of diminishing marginal utility without requiring utility functions. Notation Comparable to the greater-than-or-equal-to ordering relation \geq for real numbers, the notation \succeq below can be translated as: 'is at least as good as' (in preference satisfaction). Similarly, \succ can be translated as 'is strictly better than' (in preference satisfaction), and Similarly, \sim can be translated as 'is equivalent to' (in preference satisfaction). Definition Use ''x'', ''y'', and ''z'' to denote three consumption bundles (combinations of various quantities of various goods). Formally, a preference relation \succeq on the consumption set ''X'' is called convex if whenever :x, y, z \in X where y \succeq x a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is the social 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 analyzes what's viewed as basic elements in the economy, 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 analyzes the economy as a system where production, consumption, saving, and investment interact, and factors affecting it: employment of the resources of labour, capital, and land, currency inflation, economic growth, and public policies that have impact on glossary of economics, these elements. Other broad distinctions within economics include those between positive economics, desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Utility
In economics and consumer theory, a linear utility function is a function of the form: ::u(x_1,x_2,\dots,x_m) = w_1 x_1 + w_2 x_2 + \dots w_m x_m or, in vector form: ::u(\overrightarrow) = \overrightarrow \cdot \overrightarrow where: * m is the number of different goods in the economy. * \overrightarrow is a vector of size m that represents a bundle. The element x_i represents the amount of good i in the bundle. * \overrightarrow is a vector of size m that represents the subjective preferences of the consumer. The element w_i represents the relative value that the consumer assigns to good i. If w_i=0, this means that the consumer thinks that product i is totally worthless. The higher w_i is, the more valuable a unit of this product is for the consumer. A consumer with a linear utility function has the following properties: * The preferences are strictly monotone: having a larger quantity of even a single good strictly increases the utility. * The preferences are weakly convex, bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shapley–Folkman Lemma
The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ross M. Starr. The lemma may be intuitively understood as saying that, if the number of summed sets exceeds the dimension of the vector space, then their Minkowski sum is approximately convex. Related results provide more refined statements about ''how close'' the approximation is. For example, the Shapley–Folkman theorem provides an upper bound on the distance between any point in the Minkowski sum and its convex hull. This upper bound is sharpened by the Shapley–Folkman–Starr theorem (alternatively, Starr's corollary). The Shapley–Folkman lemma has applications in economics, optimization and probability theory. In economics, it can be used to extend results proved for convex preferences to non-convex preferences. In op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-continuous Function
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, roughly speaking, the function values for arguments near x_0 are not much higher (respectively, lower) than f\left(x_0\right). A function is continuous if and only if it is both upper and lower semicontinuous. If we take a continuous function and increase its value at a certain point x_0 to f\left(x_0\right) + c for some c>0, then the result is upper semicontinuous; if we decrease its value to f\left(x_0\right) - c then the result is lower semicontinuous. The notion of upper and lower semicontinuous function was first introduced and studied by René Baire in his thesis in 1899. Definitions Assume throughout that X is a topological space and f:X\to\overline is a function with values in the extended real numbers \overline=\R \cup \ = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level Set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is called a level curve, also known as ''contour line'' or ''isoline''; so a level curve is the set of all real-valued solutions of an equation in two variables and . When , a level set is called a level surface (or ''isosurface''); so a level surface is the set of all real-valued roots of an equation in three variables , and . For higher values of , the level set is a level hypersurface, the set of all real-valued roots of an equation in variables. A level set is a special case of a fiber. Alternative names Level sets show up in many applications, often under different names. For example, an implicit curve is a level curve, which is considered independently of its neighbor curves, emphasizing that such a curve is defined by an implicit e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-convex Function
In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. ''Univariate'' unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex. Definition and properties A function f:S \to \mathbb defined on a convex subset S of a real vector space is quasiconvex if for all x, y \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indifference Curve
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the consumer with equal levels of utility, and the consumer has no preference for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. In other words, an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent preferences rather than something from which preferences come. The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles. There are infinitely many indifference curves: one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Function
In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of a function, graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (mathematics), epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include the quadratic function x^2 and the exponential function e^x. In simple terms, a convex function refers to a function whose graph is shaped like a cup \cup, while a concave function's graph is shaped like a cap \cap. Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leontief Utility Function
In economics, especially in consumer theory, a Leontief utility function is a function of the form: u(x_1,\ldots,x_m)=\min\left\ . where: * m is the number of different goods in the economy. * x_i (for i\in 1,\dots,m) is the amount of good i in the bundle. * w_i (for i\in 1,\dots,m) is the weight of good i for the consumer. This form of utility function was first conceptualized by Wassily Leontief. Examples Leontief utility functions represent complementary goods. For example: * Suppose x_1 is the number of left shoes and x_2 the number of right shoes. A consumer can only use pairs of shoes. Hence, his utility is \min(x_1,x_2). * In a cloud computing environment, there is a large server that runs many different tasks. Suppose a certain type of a task requires 2 CPUs, 3 gigabytes of memory and 4 gigabytes of disk-space to complete. The utility of the user is equal to the number of completed tasks. Hence, it can be represented by: \min(, , ). Properties A consumer with a Leonti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Utility
In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a Goods (economics), good or Service (economics), service describes how much pleasure or satisfaction is gained by consumers as a result of the increase or decrease in Consumption (economics), consumption by one unit. There are three types of marginal utility. They are positive, negative, or zero marginal utility. For instance, you like eating pizza, the second piece of pizza brings you more satisfaction than only eating one piece of pizza. It means your marginal utility from purchasing pizza is positive. However, after eating the second piece you feel full, and you would not feel any better from eating the third piece. This means your marginal utility from eating pizza is zero. Moreover, you might feel sick if you eat more than three pieces of pizza. At this time, your marginal utility is negative. In other words, a negative marginal utility indicates that every unit of good ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jerry Green (economist)
Jerry Richard Green (born December 15, 1946) is the John Leverett Professor in the University and the David A. Wells Professor of Political Economy at Harvard University. He is known for his research in economic theory, as well as writing the most commonly used microeconomic theory textbook for graduate school with Andreu Mas-Colell and Michael Whinston, ''Microeconomic Theory''. Biography Green received his bachelor's degree from the University of Rochester in 1967 and his Ph.D. in economics in 1970. He then joined Harvard's economics faculty. He was Harvard's Provost from 1992-1994 and chaired the economics department from 1984-1987. He is a recipient of the J. Kenneth Galbraith Prize for excellence in teaching. He is a Senior Fellow at the Harvard Society of Fellows and a fellow of the Econometric Society. He was elected fellow of the American Academy of Arts and Sciences in 1994 and fellow of the Society for the Advancement of Economic Theory The Society for the Advancement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
