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Envelope Theorem
In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem. As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models. The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions \left\ _ that are optimized. Statement Let f(x,\alpha) and g_(x,\alpha), j = 1,2, \ldots, m be real-valued continuously differentiable functions on \mathbb^, where x \in \mathbb^ are choice variables and \alpha \in \mathbb^ are parameters, and consider the problem of choosing x, for a given \alpha, so as to: : \max_ f(x, \alpha) subject to g_(x,\alpha) \geq 0, j = 1,2, \ldots, m an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution Matrix
In bioinformatics and evolutionary biology, a substitution matrix describes the frequency at which a character in a nucleotide sequence or a protein sequence changes to other character states over evolutionary time. The information is often in the form of log odds of finding two specific character states aligned and depends on the assumed number of evolutionary changes or sequence dissimilarity between compared sequences. It is an application of a stochastic matrix. Substitution matrices are usually seen in the context of amino acid or DNA sequence alignments, where they are used to calculate similarity scores between the aligned sequences. Background In the process of evolution, from one generation to the next the amino acid sequences of an organism's proteins are gradually altered through the action of DNA mutations. For example, the sequence ALEIRYLRD could mutate into the sequence ALEINYLRD in one step, and possibly AQEINYQRD over a longer period of evolutionary t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus Of Variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as ''geodesics''. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends up ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Production Economics
Production is the process of combining various inputs, both material (such as metal, wood, glass, or plastics) and immaterial (such as plans, or knowledge) in order to create output. Ideally this output will be a good or service which has value and contributes to the utility of individuals. The area of economics that focuses on production is called production theory, and it is closely related to the consumption (or consumer) theory of economics. The production process and output directly result from productively utilising the original inputs (or factors of production). Known as primary producer goods or services, land, labour, and capital are deemed the three fundamental production factors. These primary inputs are not significantly altered in the output process, nor do they become a whole component in the product. Under classical economics, materials and energy are categorised as secondary factors as they are byproducts of land, labour and capital. Delving further, primary factors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Value Function
The value function of an optimization problem gives the value attained by the objective function at a solution, while only depending on the parameters of the problem. In a controlled dynamical system, the value function represents the optimal payoff of the system over the interval , t1/var> when started at the time-t state variable x(t)=x. If the objective function represents some cost that is to be minimized, the value function can be interpreted as the cost to finish the optimal program, and is thus referred to as "cost-to-go function." In an economic context, where the objective function usually represents utility, the value function is conceptually equivalent to the indirect utility function. In a problem of optimal control, the value function is defined as the supremum of the objective function taken over the set of admissible controls. Given (t_, x_) \in , t_\times \mathbb^, a typical optimal control problem is to : \text \quad J(t_, x_; u) = \int_^ I(t,x(t), u(t)) \, \mathr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roy's Identity
Roy's identity (named after French economist René Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function. Specifically, denoting the indirect utility function as v(p,w), the Marshallian demand function for good i can be calculated as :x_^(p,w)=-\frac where p is the price vector of goods and w is income. Derivation of Roy's identity Roy's identity reformulates Shephard's lemma in order to get a Marshallian demand function for an individual and a good (i) from some indirect utility function. The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income w in the indirect utility function v (p, w), at a utility of u: :v ( p, e(p, u)) = u This says that the indirect utility function evaluated in such a way that minimizes the cost for achieving a certain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Le Chatelier's Principle
Le Chatelier's principle (pronounced or ), also called Chatelier's principle (or the Equilibrium Law), is a principle of chemistry used to predict the effect of a change in conditions on chemical equilibria. The principle is named after French chemist Henry Louis Le Chatelier, and sometimes also credited to Karl Ferdinand Braun, who discovered it independently. It can be stated as: Phenomena in apparent contradiction to Le Chatelier's principle can also arise in systems of simultaneous equilibrium (see response reactions). Le Chatelier's principle is sometimes alluded to in discussions of topics other than thermodynamics. Thermodynamic statement The Le Chatelier–Braun principle analyzes the qualitative behaviour of a thermodynamic system when a designated one of its externally controlled state variables, say L, changes by an amount \Delta L, the 'driving change', causing a change \delta_ M, the 'response of prime interest', in its conjugate state variable M, all other exte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Danskin's Theorem
In convex analysis, Danskin's theorem is a theorem which provides information about the derivatives of a function of the form f(x) = \max_ \phi(x,z). The theorem has applications in optimization, where it sometimes is used to solve minimax problems. The original theorem given by J. M. Danskin in his 1967 monograph provides a formula for the directional derivative of the maximum of a (not necessarily convex) directionally differentiable function. An extension to more general conditions was proven 1971 by Dimitri Bertsekas. Statement The following version is proven in "Nonlinear programming" (1991). Suppose \phi(x,z) is a continuous function of two arguments, \phi : \R^n \times Z \to \R where Z \subset \R^m is a compact set. Further assume that \phi(x,z) is convex in x for every z \in Z. Under these conditions, Danskin's theorem provides conclusions regarding the convexity and differentiability of the function f(x) = \max_ \phi(x,z). To state these results, we define the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Theorem
The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959. The theorem is primarily used in mathematical economics and optimal control. Statement of theorem Maximum Theorem. Let X and \Theta be topological spaces, f:X\times\Theta\to\mathbb be a continuous function on the product X \times \Theta, and C:\Theta\rightrightarrows X be a compact-valued correspondence such that C(\theta) \ne \emptyset for all \theta \in \Theta. Define the ''marginal function'' (or ''value function'') f^* : \Theta \to \mathbb by :f^*(\theta)=\sup\ and the ''set of maximizers'' C^* : \Theta \rightrightarrows X by : C^*(\theta)= \mathrm\max\ = \ . If C is continuous (i.e. both upper and lower hemicontinuous) at \theta, then f^* is continuous and C^*is upper hemicontinuous with nonempty and compact values. As a consequence, the \sup may be replaced by \max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shadow Price
A shadow price is the monetary value assigned to an abstract or intangible commodity which is not traded in the marketplace. This often takes the form of an externality. Shadow prices are also known as the recalculation of known market prices in order to account for the presence of distortionary market instruments (e.g. quotas, tariffs, taxes or subsidies). Shadow Prices are the real economic prices given to goods and services after they have been appropriately adjusted by removing distortionary market instruments and incorporating the societal impact of the respective good or service. A shadow price is often calculated based on a group of assumptions and estimates because it lacks reliable data, so it is subjective and somewhat inaccurate. The need for shadow prices arises as a result of “externalities” and the presence of distortionary market instruments. An externality is defined as a cost or benefit incurred by a third party as a result of production or consumption of a g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. An example of a saddle point is when there is a critical point with a relative minimum along one axial direction (between peaks) and at a relative maximum along the crossing axis. However, a saddle point need not be in this form. For example, the function f(x,y) = x^2 + y^3 has a critical point at (0, 0) that is a saddle point since it is neither a relative maximum nor relative minimum, but it does not have a relative maximum or relative minimum in the y-direction. The name derives from the fact that the prototypical example in two dimensions is a surface that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a riding saddle or a mountain pass between two peaks forming a landform saddle. In te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slutsky Matrix
The Slutsky equation (or Slutsky identity) in economics, named after Eugen Slutsky, relates changes in Marshallian (uncompensated) demand to changes in Hicksian (compensated) demand, which is known as such since it compensates to maintain a fixed level of utility. There are two parts of the Slutsky equation, namely the substitution effect, and income effect. In general, the substitution effect can be negative for consumers as it can limit choices. He designed this formula to explore a consumer's response as the price changes. When the price increases, the budget set moves inward, which also causes the quantity demanded to decrease. In contrast, when the price decreases, the budget set moves outward, which leads to an increase in the quantity demanded. The substitution effect is due to the effect of the relative price change while the income effect is due to the effect of income being freed up. The equation demonstrates that the change in the demand for a good, caused by a pric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
