A budget set or opportunity set includes all possible consumption bundles that someone can afford given the prices of goods and the person's income level. The budget set is bounded above by the budget line. In set notation, for consumption goods with associated prices , the budget set is
A price is the quantity of payment or compensation given by one party to another in return for one unit of goods or services.. A price is influenced by both production costs and demand for the product. A price may be determined by a monopolist or may be imposed on the firm by market conditions.
Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms.
Sets are fundamental objects in mathematics. Intuitively, a set is merely a collection of elements or members. There are various conventions for textually denoting sets. In any particular situation, an author typically chooses from among these conventions depending on which properties of the set are most relevant to the immediate context or on which perspective is most useful.
where is income, and the consumption set is assumed to be the nonnegative orthant in .
In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions.
Graphically speaking, all the consumption bundles that lie inside the budget constraint and on the budget constraint form the budget set or opportunity set.
In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Consumer theory uses the concepts of a budget constraint and a preference map to analyze consumer choices. Both concepts have a ready graphical representation in the two-good case.
By most definitions, budget sets must be compact and convex.
In geometry, a convex set or a convex region is a subset of a Euclidean space, or more generally an affine space over the reals, that intersects every line into a line segment. Equivalently, this is a subset that is closed under convex combinations. 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.
In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication.
In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is indifferent/curve that shows all combinations of two products that will provide the consumer with equal levels of utility. That is, 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.
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. Once stationary points have been identified from the first-order necessary conditions, the definiteness of the bordered Hessian matrix determines whether those points are maxima, minima, or saddle points.
The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their preferences subject to limitations on their expenditures, by maximizing utility subject to a consumer budget constraint.
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
In economics and particularly in consumer choice theory, the income-consumption curve is a curve in a graph in which the quantities of two goods are plotted on the two axes; the curve is the locus of points showing the consumption bundles chosen at each of various levels of income.
In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration. This is done assuming that this can be done with a single chart. The generalized velocities are the time derivatives of the generalized coordinates of the system.
In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function.
In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem. It consists of choosing how much of each available good or service to consume, taking into account a constraint on total spending as well as the prices of the goods.
There are three fundamental theorems of welfare economics. The first theorem states that a market will tend toward a competitive equilibrium that is weakly Pareto optimal when the market maintains the following two attributes:
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.
Revealed preference theory, pioneered by economist Paul Samuelson, is a method of analyzing choices made by individuals, mostly used for comparing the influence of policies on consumer behavior. Revealed preference models assume that the preferences of consumers can be revealed by their purchasing habits.
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. The system of equations and inequalities corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations and inequalities.
The theory of quantum error correction plays a prominent role in the practical realization and engineering of quantum computing and quantum communication devices. The first quantum error-correcting codes are strikingly similar to classical block codes in their operation and performance. Quantum error-correcting codes restore a noisy, decohered quantum state to a pure quantum state. A stabilizer quantum error-correcting code appends ancilla qubits to qubits that we want to protect. A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space. This highly entangled, encoded state corrects for local noisy errors. A quantum error-correcting code makes quantum computation and quantum communication practical by providing a way for a sender and receiver to simulate a noiseless qubit channel given a noisy qubit channel whose noise conforms to a particular error model.
Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations:
In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function where is strictly concave. A nice property of the quasilinear utility function is that, the Marshallian/Walrasian demand for does not depend on wealth and therefore is not subject to a wealth effect. The absence of a wealth effect simplifies analysis and makes quasilinear utility functions a common choice for modelling. Furthermore, when utility is quasilinear, compensating variation (CV), equivalent variation (EV), and consumer surplus are algebraically equivalent. In mechanism design, quasilinear utility ensures that agents can compensate each other with side payments.
A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe or checkers (draughts). In other words, it extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite.
The Brownian motion models for financial markets are based on the work of Robert C. Merton and Paul A. Samuelson, as extensions to the one-period market models of Harold Markowitz and William F. Sharpe, and are concerned with defining the concepts of financial assets and markets, portfolios, gains and wealth in terms of continuous-time stochastic processes.
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.
Andreu Mas-Colell is a Spanish economist, an expert in microeconomics and one of the world's leading mathematical economists. He is the founder of the Barcelona Graduate School of Economics and a professor in the department of economics at Pompeu Fabra University in Barcelona, Catalonia, Spain. He has also served several times in the cabinet of the Catalan government. Summarizing his and others' research in general equilibrium theory, his monograph gave a thorough exposition of research using differential topology. His textbook on microeconomics, co-authored with Michael Whinston and Jerry Green, is the most used graduate microeconomics textbook in the world.
Michael Whinston is an American economist and currently the Sloan Fellows Professor at Massachusetts Institute of Technology. Previously he was the Robert E. and Emily H. King Professor at Northwestern University and is also a Fellow to the American Academy of Arts and Sciences and Econometric Society.
The International Standard Book Number (ISBN) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.
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