|
Indirect Utility
__NOTOC__ In economics, a consumer's indirect utility function v(p, w) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of income w. It reflects both the consumer's preferences and market conditions. This function is called indirect because consumers usually think about their preferences in terms of what they consume rather than prices. A consumer's indirect utility v(p, w) can be computed from his or her utility function u(x), defined over vectors x of quantities of consumable goods, by first computing the most preferred affordable bundle, represented by the vector x(p, w) by solving the utility maximization problem, and second, computing the utility u(x(p, w)) the consumer derives from that bundle. The resulting indirect utility function is :v(p,w)=u(x(p,w)). The indirect utility function is: *Continuous on R''n''+ × R+ where ''n'' is the number of goods; *Decreasing in prices; *Strictly increasing in income; *Homogeno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes what's viewed as basic elements in the economy, including individual agents and 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 these elements. Other broad distinctions within economics include those between positive economics, describing "what is", and normative economics, advocating "what ought to be"; between economic theory and applied economics; between rati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a utility function that represents a single consumer's preference ordering over a choice set but is not comparable across consumers. This concept of utility is personal and based on choice rather than on pleasure received, and so is specified more rigorously than the original concept but makes it less useful (and controversial) for ethical decisions. Utility function Consider a set of alternatives among which a person can make a preference ordering. The utility obtained from these alternatives is an unknown function of the utilities obtained from each alternative, not the su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Income
Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For example, a person's income in an economic sense may be different from their income as defined by law. An extremely important definition of income is Haig–Simons income, which defines income as ''Consumption + Change in net worth'' and is widely used in economics. For households and individuals in the United States, income is defined by tax law as a sum that includes any wage, salary, profit, interest payment, rent, or other form of earnings received in a calendar year.Case, K. & Fair, R. (2007). ''Principles of Economics''. Upper Saddle River, NJ: Pearson Education. p. 54. Discretionary income is often defined as gross income minus taxes and other deductions (e.g., mandatory pension contributions), and is widely used as a basis to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Utility Maximization Problem
Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. 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 (income), the prices of the goods and their preferences. Utility maximization is an important concept in consumer theory as it shows how consumers decide to allocate their income. Because consumers are rational, they seek to extract the most benefit for themselves. However, due to bounded rationality and other biases, consumers sometimes pick bundles that do not necessarily maximize their utility. The utility maximization bundle of the consumer is also not set and can change over time depending on their individual preferences of goods, price changes and incr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Function
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''degree''; that is, if is an integer, a function of variables is homogeneous of degree if :f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n) for every x_1, \ldots, x_n, and s\ne 0. For example, a homogeneous polynomial of degree defines a homogeneous function of degree . The above definition extends to functions whose domain and codomain are vector spaces over a field : a function f : V \to W between two -vector spaces is ''homogeneous'' of degree k if for all nonzero s \in F and v \in V. This definition is often further generalized to functions whose domain is not , but a cone in , that is, a subset of such that \mathbf\in C implies s\mathbf\in C for every nonzero scalar . In the case of functions of several real variables and rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconvex 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 ... [...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 u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expenditure Function
In microeconomics, the expenditure function gives the minimum amount of money an individual needs to spend to achieve some level of utility, given a utility function and the prices of the available goods. Formally, if there is a utility function u that describes preferences over ''n '' commodities, the expenditure function :e(p, u^*) : \textbf R^n_+ \times \textbf R \rightarrow \textbf R says what amount of money is needed to achieve a utility u^* if the ''n'' prices are given by the price vector p. This function is defined by :e(p, u^*) = \min_ p \cdot x where :\geq(u^*) = \ is the set of all bundles that give utility at least as good as u^*. Expressed equivalently, the individual minimizes expenditure x_1p_1+\dots +x_n p_n subject to the minimal utility constraint that u(x_1, \dots , x_n) \ge u^*, giving optimal quantities to consume of the various goods as x_1^*, \dots x_n^* as function of u^* and the prices; then the expenditure function is :e(p_1, \dots , p_n ; u^*)=p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gorman Polar Form
Gorman polar form is a functional form for indirect utility functions in economics. Motivation Standard consumer theory is developed for a single consumer. The consumer has a utility function, from which his demand curves can be calculated. Then, it is possible to predict the behavior of the consumer in certain conditions, price or income changes. But in reality, there are many different consumers, each with his own utility function and demand curve. How can we use consumer theory to predict the behavior of an entire society? One option is to represent an entire society as a single "mega consumer", which has an aggregate utility function and aggregate demand curve. But in what cases is it indeed possible to represent an entire society as a single consumer? Formally: consider an economy with n consumers, each of whom has a demand function that depends on his income m^i and the price system: :x^i(p,m^i) The aggregate demand of society is, in general, a function of the price syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hicksian Demand Function
In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essentially, a Hicksian demand function shows how an economic agent would react to the change in the price of a good, if the agent's income was compensated to guarantee the agent the same utility previous to the change in the price of the good—the agent will remain on the same indifference curve before and after the change in the price of the good. The function is named after John Hicks. Mathematically, :h(p, \bar) = \arg \min_x \sum_i p_i x_i : \ \ u(x) \geq \bar . where ''h''(''p'',''u'') is the Hicksian demand function, or commodity bundle demanded, at price vector ''p'' and utility level \bar. Here ''p'' is a vector of prices, and ''x'' is a vector of quantities demanded, so the sum of all ''p''''i''''x''''i'' is total expenditure o ... [...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] |
